Speaker: Hong Wang, Courant Institute, New York University

Title: Restriction estimates using decoupling theorems and incidence inequalities for tubes

Abstract: Suppose f is a function with Fourier transform supported on the unit sphere in R^d, Stein conjectured that the L^p norm of f is bounded by the L^p norm of its Fourier transform, for p> 2d/(d-1). We propose to study this conjecture using Bourgain-Demeter decoupling theorems and incidences estimates for tubes. In this talk, we will describe a geometric conjecture on the number of incidences for tubes that would imply Stein's restriction conjecture. We prove this geometric conjecture in R^2 and use it to prove a restriction estimate in R^3 for p> 3+ 1/7, which implies Wolff's hairbrush Kakeya estimate. This is joint work with Shukun Wu.

Speaker: Georg S. Weiss, University of Duisburg-Essen

Title: Classification of global solutions to the obstacle problem in the plane (joint work with Anthony Salib)

Abstract: Global solutions to the obstacle problem were first completely classified in two dimensions by Sakai using complex analysis techniques. Although the complex analysis approach produced a very succinct proof in two dimensions, it left the higher dimensional cases, and even closely related problems in two dimensions, unresolved. A complete classification in dimensions $n\geq 3$ was recently given by Eberle, Figalli and Weiss, forty years after Sakai published his proof. In this paper we give a proof of Sakai's classification result for unbounded coincidence sets in the spirit of the recent proof by Eberle, Figalli and Weiss. Our approach, in particular, avoids the need for complex analysis techniques and offers new perspectives on two-dimensional problems that complex analysis cannot address.

Speaker: Gabor Szekelyhidi, Northwestern University

Title: Singular Kahler-Einstein metrics and RCD spaces

Abstract: An important problem is to understand the metric geometry of singular Kahler-Einstein metrics. The RCD property is a notion of Ricci curvature lower bound for metric measure spaces, that has been extensively studied recently. I will show that singular Kahler-Einstein spaces that can be approximated in a suitable sense by constant scalar curvature metrics satisfy this condition, and I will discuss some applications.

Speaker: Changfeng Gui, University of Macau

Title: On a classification of steady solutions to two-dimensional Euler equations

Abstract: In this talk, I shall provide a classification of steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the whole plane must be the whole circle unless the flow is a parallel shear flow. In an infinitely long horizontal strip or the upper half-plane supplemented with slip boundary conditions, besides the two types of flows appeared in the whole space case, there exists an additional class of steady flows for which the set of flow angles is either the upper or lower closed semicircles. This type of flows is proved to be the class of non-shear flows that have the least total curvature. As consequences, we obtain Liouville-type theorems for two-dimensional semilinear elliptic equations with only bounded and measurable nonlinearity, and the structural stability of shear flows whose all stagnation points are not inflection points, including Poiseuille flow as a special case. Our proof relies on the analysis of some quantities related to the curvature of the streamlines. This talk is based on a joint work with Huan Xu and Chunjing Xie.

Speaker: Jianxiong Wang, Rutgers University

Title: The Method of Moving Spheres on Hyperbolic Space and Higher Order Conformal Equations

Abstract: The classification of solutions of semilinear partial differential equations, as well as the classification of critical points of the corresponding functionals, have wide applications in the study of partial differential equations and differential geometry. The classical moving plane and the moving sphere method on $\mathbb{R}^n provide an effective approach to capturing the symmetry of solutions. In this talk, we focus on the equation $P_ku = f(u)$ on hyperbolic spaces $\mathbb{H}^n, where $P_k$ denotes the GJMS operators on $\mathbb{H}^n$ and $f : R → R$ satisfies certain conditions. We develop a moving sphere approach for integral equations on $\mathbb{H}^n, to obtain the symmetry property positive solutions. Our methods also rely on Helgason-Fourier analysis and Hardy-Littlewood-Sobolev inequalities on hyperbolic spaces together with a Kelvin transform we introduce on $\mathbb{H}^n in this paper. If time permits, we will also talk about several applications regrading the higher order prescribing Q-curvature problem.

Speaker: Xavier Cabre, ICREA (Institucio Catalana de Recerca i Estudis Avancats) & Universitat Politecnica de Catalunya

Title: Stable solutions to semilinear elliptic equations are smooth up to dimension 9

Abstract: The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. It was initiated by a work of Crandall and Rabinowitz, motivated by the Gelfand problem in combustion theory. The theory experienced a revival in the mid-nineties after new progress made by Brezis and collaborators. I will present these developments, as well as a recent work, in collaboration with Figalli, Ros-Oton, and Serra, which finally establishes the regularity of stable solutions up to the optimal dimension 9. I will also describe a more recent paper of mine which provides full quantitative proofs of the regularity results.

Speaker: Hung Tran, University of Wisconsin at Madison

Title: Periodic homogenization of Hamilton-Jacobi equations: some recent progress

Abstract: I first give a quick introduction to front propagations, Hamilton-Jacobi equations, level-set forced mean curvature flows, and homogenization theory. I will then show the optimal rates of convergence for homogenization of both first-order and second-order Hamilton-Jacobi equations. Based on joint works with J. Qian, T. Sprekeler, and Y. Yu.

Speaker: Yan Guo, Brown University

Title: Stability of Contact Lines

Abstract: Contact lines (e.g. coffee meets with the coffee cup, boundary of a droplet on table) occurs naturally when a free fluid surface, a capillary surface, meets with a fixed solid boundary. Despite its importance in fluid theory and applications, the dynamical law for contact line has been only studied in relative recent history. We review recent work to establish well-posedness of classical fluid models for describing contact line dynamics.

Speaker: Yannick Sire, Johns Hopkins University

Title: Harmonic maps between singular spaces

Abstract: After reviewing briefly the classical theory of harmonic maps between smooth manifolds, I will describe some recent results related to harmonic maps with free boundary, emphasizing on two different approaches based on recent developments by Da Lio and Riviere. This latter approach allows in particular to give another formulation which is well-suited for such maps between singular spaces. After the works of Gromov, Korevaar and Schoen, harmonic maps between singular spaces have been instrumental to investigate super-rigidity in geometry. I will report on recent results where we introduce a new energy between singular spaces and prove a version of Takahashi's theorem (related to minimal immersions by eigenfunctions) on RCD spaces.

Speaker: Federico Franceschini, Institute for Advanced Study

Title: The dimension and behaviour of singularities of stable solutions to semilinear elliptic equations

Abstract: Let $f(t)$ be a convex, positive, increasing nonlinearity. It is known that stable solutions of $-\Delta u =f(u)$ can be singular (i.e., unbounded) if the dimension $n \ge 10$. Brezis conjectured that if $x=0$ is such a singular point, then $f'(u(x))$ blows-up like $|x|^{2-n}$. Villegas showed that such a strong statement fails for general nonlinearities. In this talk, we prove - for all nonlinearities - a version of Brezis conjecture, which is essentially the best one can obtain in view of the counterexamples of Villegas. Building on this result we then show that the singular set has dimension $n-10$, at least for a large class of nonlinearities that includes the most relevant cases. This is a joint work with Alessio Figalli.

Speaker: Mikaela lacobelli, Institute for Advanced Study & ETH

Title: Stability and singular limits in plasma physics

Abstract: In this talk, we will present two kinetic models that are used to describe the evolution of charged particles in plasmas: the Vlasov-Poisson system and the Vlasov-Poisson system with massless electrons. These systems model respectively the evolution of electrons, and ions in a plasma. We will discuss the well-posedness of these systems, the stability of solutions, and their behavior under singular limits. Finally, we will introduce a new class of Wasserstein-type distances specifically designed to tackle stability questions for kinetic equations.

Speaker: Chongchun Zeng, Georgia Institute of Technology

Title: Water waves linearized at monotonic shear flows

Abstract: We consider the 2-dim water wave problem -- the free boundary problem of the Euler equation with gravity and possibly surface tension -- of finite depth linearized at a uniformly monotonic shear flow $U(x_2)$. Our main focuses are eigenvalue distribution and inviscid damping. We first prove that in contrast to finite channel flow and gravity waves, the linearized capillary gravity wave has two unbounded branches of eigenvalues for high wave numbers. They may bifurcate into unstable eigenvalues through a rather degenerate bifurcation. Under certain conditions, we provide a complete picture of the eigenvalue distribution. Assuming there are no singular modes (i.e. embedded eigenvalues), we obtain the linear inviscid damping. We also identify the leading asymptotic terms of velocity and obtain stronger decay for the remainders. The linearized gravity waves will also be discussed briefly if time permits. This is a joint work with Xiao Liu.

Nonlinear Analysis Seminar ---- Spring 2024 (Organizers: Yanyan Li, Dennis Kriventsov)

Speaker: Marta Lewicka, University of Pittsburg

Title: The Monge-Ampere system and the isometric immersion system: convex integration in arbitrary dimension and codimension

Abstract: The Monge-Ampere equation det abla^2 v =f posed on a d=2 dimensional domain omega and in which we are seeking a scalar (i.e. dimension k=1) field v, has a natural weak formulation that appears as the constraint condition in the Gamma-limit of the dimensionally reduced non-Euclidean elastic energies. This formulation reads: frac{1}{2} abla votimes abla v) + sym abla w= - (curl curl)^{-1}f and it allows, via the Nash-Kuiper scheme of convex integration, for constructing multiple solutions that are dense in C^0(omega), at the regularity C^{1,alpha} for any alpha<1/3, no matter the sign of the right hand side function f. Does a similar result hold in higher dimensions d>2 and codimensions k>1? Indeed it does, but one has to replace the Monge-Ampere equation by the Monge-Ampere system, by altering curl curl to the corresponding operator that arises from the prescribed Riemann curvature problem, similarly to how the prescribed Gaussian curvature problem leads to the Monge-Ampere equation in 2d. Our main result is a proof of flexibility of the Monge-Ampere system at C^{1,alpha} for alpha<1/(1+d(d+1)/k). This finding extends our previous result where d=2, k=1, and stays in agreement with the known flexibility thresholds for the isometric immersion problem: the Conti-Delellis-Szekelyhidi result alpha<1/(1+d(d+1)) when k=1, as well as the Kallen result where alphato 1 as ktoinfty. For d=2, the flexibility exponent may be even improved to alpha<1/(1+2/k), using the conformal invariance of 2d metrics to the flat metric. We will also discuss other possible improvements and parallel results and techniques valid for the isometric immersion system.

Speaker: Hans Ringstrom, KTH

Title: A quiescent regime for big bang formation

Abstract: Recently, many results concerning stable big bang formation have appeared. Most of the results concern stability of spatially homogeneous and isotropic solutions. However, a recent result of Fournodavlos, Rodnianski and Speck (FRS) covers the full regime in which stability is to be expected. On the other hand, it is restricted to the stability of spatially homogeneous and spatially flat solutions. In this talk, I will present a new result (joint work with Hans Oude Groeniger and Oliver Petersen) in which we identify a general condition on initial data ensuring big bang formation. The solutions need, in this case, not be close to symmetric background solutions. Moreover, the result reproduces previous results in the Einstein-scalar field and Einstein-vacuum settings. Finally, the result is in the Einstein-non-linear scalar field setting, and therefore yields future and past global non-linear stability of large classes of spatially locally homogeneous solutions.

Speaker: Lili He, Princeton University

Title: The linear stability of weakly charged and slowly rotating Kerr-Newman family of charged black holes

Abstract: I will discuss the linear stability of weakly charged and slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman solution plus a pure gauge term. The proof uses tools from microlocal analysis and a detailed description of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator at low frequencies.

Speaker: Luc Nguyen, University of Oxford

Title: On the minimality of the degree-one vortex solution in the Ginzburg-Landau theory

Abstract: It has been conjectured for some time that, in the Ginzburg-Landau theory of superconductivity, the degree-one vortex solution is energy minimizing. In this talk, we review recent progresses on this problem.

Speaker: Alexis Vasseur, University of Texas at Austin

Title: Viscous perturbations to discontinuous solutions of the compressible Euler equation

Abstract: The compressible Euler equation can lead to the emergence of shocks-discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent viscosities.The mathematical study of this problem is however very difficult because of the destabilization effect of the viscosities. Bianchini and Bressan proved the inviscid limit to small BV solutions using the so-called artificial viscosities in 2004. However, until very recently, achieving this limit with physical viscosities remained an open question. In this presentation, we will present the basic ideas of classical mathematical theories to compressible fluid mechanics and introduce the recent a-contraction with shifts method. This method is employed to describe the physical inviscid limit in the context of the barotropic Euler equation.

Speaker: Hui Yu, National University of Singapore

Title: Rigidity of global solutions to the thin obstacle problem

Abstract: The thin obstacle problem is a free boundary problem concerning the shape of an elastic membrane resting on a lower-dimensional obstacle. In this talk, we discuss some rigidity properties of solutions in the entire space. We see very rigid behaviors when the solution grows quadratically at infinity. When the solution has higher rate of growth, we see no rigidity. This talk is based on joint works with Simon Eberle (BCAM) and Xavier Fernandez-Real (EPFL).

Nonlinear Analysis Seminar ---- Fall 2023 (Organizers: Yanyan Li, Dennis Kriventsov)

Speaker: Javier Gomez Serrano, Brown University

Title: Generic global existence for the modified SQG equation

Abstract: In this talk we will present a construction of global existence of small solutions of the modified SQG equations, close to the disk. The proof uses KAM theory and a Nash-Moser argument, and does not involve any external parameters. We moreover prove that this phenomenon is generic: most solutions satisfy it. Joint work with Alex Ionescu and Jaemin Park.

Speaker: Chao Li, Courant Institute, New York University

Title: Gluing theorems in positive scalar curvature

Abstract: Given two Riemannian manifolds with suitable curvature conditions and isometric boundary, can we glue them along their isometric boundary, then desingularize the result while preserving the same curvature conditions? A positive answer to such questions enables one to study deep rigidity and flexibility phenomena for the curvature conditions. In this talk, I will focus on related desingularization results concerning such problems for positive scalar curvature and mean convex (or minimal) boundary conditions. This is based on joint work with Alessandro Carlotto.

Speaker: Ravi Shankar, Princeton University

Title: A doubling approach for sigma-k PDEs

Abstract: The interior regularity for viscosity solutions of the sigma-2 equation is the remaining case of the Monge-Ampere sigma-k family to be understood. The dimension two case was done in the 1950’s by Heinz, and the dimension three case was done in 2008 by Warren and Yuan. In joint works with Yu Yuan, we show regularity in dimension four for sigma-2. Our argument uses a doubling inequality for the Hessian. A similar argument gives a proof of interior regularity for strictly convex solutions of the Monge-Ampere (sigma-n) equation in all dimensions, using a geometric approach distinct from Pogorelov’s maximum principle of the 70’s.

Speaker: Inigo Urtiaga Erneta, Rutgers University

Title: Regularity of stable solutions to semilinear elliptic problems

Abstract: A classical problem in PDEs concerns the regularity of stable solutions to nonlinear elliptic equations. This turns out to be a delicate question, even for apparently simple semilinear problems. In this setting, the smoothness of stable solutions depends on the dimension. Namely, for the Laplacian, Cabré, Figalli, Ros-Oton, and Serra (2020) have shown that stable solutions are smooth up to dimension 9, while singularities may appear in higher dimensions. In this talk, I will discuss analogous results for more general operators while giving an overview of the regularity theory for stable solutions.

Speaker: Ayush Khaitan, Rutgers University

Title: The ambient metric and the singular Ricci flow

Abstract: We establish a weighted version of the Fefferman-Graham ambient metric to be the natural geometric framework for studying singular Ricci flow. We also construct infinite families of fully non-linear analogues of Perelman's F and W functionals, and study their monotonicity under several natural conditions. Many of these constructions are completed using a "Ricci flow vector field" in the ambient space, which also allows us to provide short proofs of the monotonicity of Perelman's F and W functionals.

Speaker: Paul Minter, Princeton University

Title: Area minimising currents: singular set rectifiability and tangent cone uniqueness

Abstract: One of the great achievements of geometric measure theory is Almgren’s work (1983) on area minimising n-currents of codimension larger than 1, showing that such objects are smoothly embedded away from a (generally unavoidable) singular set of dimension at most n-2. In this talk, I will discuss recent work with Camillo De Lellis (IAS) and Anna Skorobogatova (Princeton) showing that the singular set is countably (n-2)-rectifiable, with unique tangent cones at almost every point.

Speaker: Hao Jia, University of Minnesota

Title: Dynamics of linearized 2d Navier Stokes equations in high Reynolds number regime

Abstract: Over the last decade there has been tremendous progresses in the study of fine dynamics of the linearized two dimensional Euler equations around shear flows or vortices, leading to a more or less complete understanding of important physical phenomena such as inviscid damping and vorticity depletion. For the corresponding Navier Stokes equations with a small viscosity, similar results are much harder to prove. On the technical level, this is a singular perturbation problem, and instead of Rayleigh equation (2nd order) one now needs to study, very precisely, the Orr-Sommerfeld equations (4th order) when the spectral parameter is close to the cluster of eigenvalues. In this talk, we will review some recent advances in this direction and outline a unified approach to study uniform-in-viscosity inviscid damping, vorticity depletion and enhanced dissipation. Joint work with Rajendra Beekie (Duke) and Shan Chen (UMN).

Speaker: Ling Xiao, University of Connecticut

Title: Generalized Minkowski inequality via degenerate Hessian equations on exterior domains

Abstract: In this talk, we will talk about the proof of a sharp generalized Minkowski inequality that holds for any smooth, strictly $(k-1)$-convex, star-shaped domain $\Omega$. Our proof relies on the solvability of the degenerate k-Hessian equation on the exterior domain $R^n\setminus\Omega.$

Nonlinear Analysis Seminar ---- Spring 2023 (Organizers: Yanyan Li, Dennis Kriventsov)

Speaker: Georg Weiss, University of Duisburg-Essen

Title: Complete classification of global solutions to the obstacle problem

Abstract: The characterization of global solutions to the obstacle problem in R^N, or equivalently of null quadrature domains, has been studied over more than 90 years. In this paper we give a conclusive answer to this problem by proving the following long-standing conjecture: The coincidence set of a global solution to the obstacle problem is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base.

Speaker: Mimi Dai, Princeton University and University of Illinois Chicago

Title: A path of understanding fluid equations: from Leray to recent breakthroughs

Abstract: The mathematical theory of incompressible fluids, a classical topic, still poses challenges for us today. The pioneering work of Leray in 1930s built the foundation for the Navier-Stokes equation (NSE), the governing equation of fluid motion. It also raised some important questions. One renowned question is regarding the appearance of singularity of solutions to the 3D NSE in finite time; another one concerns the well-posedness of the Leray-Hopf weak solution. We will talk about some major breakthroughs toward resolving these problems, sparked by empirical laws in physics and techniques from other disciplines of mathematics.

Speaker: Lei Zhang, University of Florida, Gainesville

Title: Non-simple Blowup solutions and vanishing estimates for singular Liouville equations

Abstract: The singular Liouville equation is a class of second order elliptic partial differential equations defined in two dimensional spaces: $$\Delta u+ H(x)e^{u}=4\pi \gamma \delta_0 $$ where $H$ is a positive smooth function, $\gamma>-1$ is a constant and $\delta_0$ stands for a singular source placed at the origin. It is well known that this equation is related to Nirenberg problem and is a reduction of Toda system which comes from a rich background. When a sequence of solutions tends to infinity near the origin, they are called blowup solutions. If $\gamma$ is a multiple of $4\pi$, the blowup solutions may violate spherical Harnack inequality around the origin, in which case they are called <200b>``non-simple blowup solutions". There were quite a few major challenges related to non-simple blowup solutions. In this talk I will report my recent joint works with Juncheng Wei and Teresa D'Aprile. Our results lead to new understanding not only on the structure of solutions of the singular Liouville equation, but also on Toda systems connected with Lie Algebra and Algebraic Geometry.

Speaker: Yifu Zhou, Johns Hopkins University

Title: Singularity formation for the Landau-Lifshitz-Gilbert equation in dimension two

Abstract: Landau-Lifshitz-Gilbert equation (LLG), which models the evolution of spin fields in continuum ferromagnetism, can be viewed as a coupling between the harmonic map heat flow and the Schrödinger map flow. In this talk, we shall report some recent construction of finite-time singularities for LLG in dimension two. The construction is done via the inner-outer gluing method, and to overcome the difficulties caused by the presence of the dispersion, technical ingredients such as distorted Fourier transform and sub-Gaussian estimates are employed. This is based on a joint work with J. Wei and Q. Zhang.

Speaker: Dennis Kriventsov, Rutgers University

Title: Rectifiability of interfaces with positive Alt-Caffarelli-Friedman limit via quantitative stability

Abstract: The Alt-Caffarelli-Friedman (ACF) monotonicity formula captures fine behavior of pairs of nonnegative harmonic functions which vanish on the mutual boundary of complementary domains. I will describe the following theorem: the set of points with positive limit for the ACF formula, corresponding roughly to where both functions have linear growth away from the interface, is countably n−1 rectifiable. The proof leverages a new quantitative stability property for the ACF formula, which in turn is based on a new quantitative stability result for the Faber-Krahn inequality. Indeed, we show that if λ(Ω) is the first Dirichlet eigenvalue of a domain Ω of volume one and uΩ is the first eigenfunction, then |u_Ω − u_B| 2 ≤ C[λ(Ω) − λ(B)] for some ball B with |B| = |Ω| = 1. Our proof of this relies on regularity theory for some "critical" modifications of Bernoulli-type free boundary problems. This is based on joint work with Mark Allen and Robin Neumayer.

Speaker: Jiajie Chen, Courant Institute, New York University

Title: Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data

Abstract: Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is an outstanding open problem. In this talk, we will first review recent progress in singularity formation in incompressible fluids. Then, we will present a result inspired by the Hou-Luo scenario for a potential 3D Euler singularity, in which we prove finite time blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. To establish the blowup results, we develop a constructive proof strategy with computer assistance and prove the nonlinear stability of an approximate self-similar blowup profile. In the stability analysis, we decompose the linearized operator into the leading order operator and the remainder. We develop sharp functional inequalities using optimal transport and the symmetry properties of the velocity kernels to estimate the nonlocal terms from the velocity and use weighted energy estimates to establish the stability analysis of the leading order operator. The key role of computer assistance is to construct an approximate blowup profile and approximate space-time solutions with rigorous error control, which provides critical small parameters in the energy estimates for the stability analysis and allows us to control the remainder perturbatively. This is joint work with Tom Hou.

Speaker: Dallas Albritton, Princeton University

Title: Kinetic shock profiles for the Landau equation

Abstract: Compressible Euler solutions develop jump discontinuities known as shocks. However, physical shocks are not, strictly speaking, discontinuous. Rather, they exhibit an internal structure which, in certain regimes, can be represented by a smooth function, the shock profile. We demonstrate the existence of weak shock profiles to the kinetic Landau equation. Joint work with Matthew Novack (Purdue University) and Jacob Bedrossian (UCLA).

Speaker: Han Lu, University of Notre Dame

Title: A non-compactness result for $\sigma_2$-Yamabe problem

Abstract: Let (M,g) be a compact Riemannian manifold of dimension $n\geq 3$. The $\sigma_k$-Yamabe problem is concerned with finding metric of constant $\sigma_k$-curvature in the conformal class of g. The compactness of the $\sigma_k$-Yamabe problem is still widely open when $n/2> k\geq 2$. We will present a non-compactness result for the $\sigma_2$-Yamabe problem when n is large. This is a joint work with Bin Deng and Juncheng Wei.

Speaker: Daniela De Silva, Barnard College

Title: The Alt-Phillips functional for negative powers

Abstract: We discuss the one-phase free boundary problem associated to the Alt-Phillips functional $$\int_\Omega (|\nabla u|^2 + u^\gamma \chi_{\{u>0\}}) dx$$ for $\gamma \in (-2,0).$ We establish partial regularity results for the free boundary and discuss the rigidity of global minimizers when $\gamma$ approaches the extreme of the admissible values.

(Joint with Complex Analysis and Geometry Seminar)

Speaker: Yuxin Ge, Institut de Mathematique de Toulouse, University Paul Sabatier

Title: Compactness of asymptotically hyperbolic Einstein manifolds in dimension 4 and applications

Abstract: Given a closed riemannian manfiold of dimension 3 (M^3,[h]), when will we fill in an asymptotically hyperbolic Einstein manifold of dimension 4 (X4,g+) such that h = r2g+ on the boundary M^3 = \del X^4 for some defining function r on X4? This problem is motivated by the correspondance AdS/CFT in quantum gravity proposed by Maldacena in 1998 et comes also from the study of the structure of asymptotically hyperbolic Einstein manifolds. In this talk, I discuss the compactness issue of asymptotically hyperbolic Einstein mani- folds in dimension 4, that is, how the compactness on conformal infinity leads to the compactness of the compactification of such manifolds under the suitable conditions on the topology and on some conformal invariants. As application, I discuss the uniqueness problem and non-existence result. It is based on the joint works with Alice Chang. *********************************************************************

Nonlinear Analysis Seminar ---- Fall 2022 (Organizers: Yanyan Li, Dennis Kriventsov)

Speaker: Zihui Zhao, University of Chicago

Title: Quantitative boundary unique continuation I

Abstract: Unique continuation property is a fundamental property of harmonic functions, as well as solutions to a large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes at a point to infinite order, it must vanish everywhere (in the connected set containing that point). In the same spirit, we are interested in quantitative unique continuation results, which are to use the local information about the growth rate of a harmonic function, to deduce its global properties. In this talk, I will focus on recent progress about quantitative unique continuation at the boundary. In particular, I will talk about X. Tolsa's work on the Bers problem and my joint work with C. Kenig estimating the size of the singular set of a harmonic function at the boundary.

Speaker: Alexandru Ionescu, Princeton University

Title: Linear and nonlinear stability of shear flows and vortices

Abstract: I will talk about some recent work on the stability problem of shear flows and vortices as solutions of the Euler equations in 2D. Our results include nonlinear stability for monotonic shear flows and point vortices. This is joint work with Hao Jia.

Speaker: Yannick Sire, Johns Hopkins University

Title: Harmonic maps with free boundary and beyond

Abstract: I will introduce a new heat flow for harmonic maps with free boundary. After giving some motivations to study such maps in relation with extremal metrics in spectral geometry, I will construct weak solutions for the flow and derive their partial regularity. The introduction of this new flow is motivated by the so-called half-harmonic maps introduced by Da Lio and Riviere, which provide a new approach to the old topic of harmonic maps with free boundary. I will also state some open problems and possible generalizations.

Speaker: Ben Weinkove, Northwestern University

Title: The insulated conductivity problem and the maximum principle

Abstract: Harmonic functions in narrow regions with Neumann boundary conditions are used to model the electric potential in the presence of insulating inclusions a short distance apart. The gradient of the harmonic function - the electric field - may blow up as the inclusions approach each other. Optimal estimates for the electric field were recently obtained by Dong-Li-Yang. I will discuss a maximum principle approach to this problem, known as the insulated conductivity problem.

Speaker: Jose M. Espinar, Universidad de Cádiz (Spain)

Title: On Fraser-Li conjecture with anti-prismatic symmetry and one boundary component

Abstract: We show that an embedded free boundary minimal surface of genus g, one boundary component and anti-prismatic symmetry of order 4(g+1) satisfies that the first eigenvalue for the Steklov eigenvalue problem is one. In particular, the family constructed by Kapouleas-Wiygul satisfies a such condition.

Speaker: Shaoming Guo, University of Wisconsin at Madison

Title: Hormander-type oscillatory integral operators and related Kakeya/Nikodym set problems on manifolds

Abstract: I will talk about some recent progress on L^p bounds of Hormander-type oscillatory integral operators. These are related to the curved Kakeya problem, and Nikodym set problems on manifolds.

Speaker: Vlad Vicol, Courant Institute, New York University

Title: Formation and development of singularities for the compressible Euler equations

Abstract: We consider the compressible Euler equations of fluid dynamics, in multiple space dimensions. In this talk, we discuss our program of completely describing the formation and development of stable singularities, from smooth initial conditions. The questions we address are: given smooth initial conditions, precisely how does the first singularity arise? is the mechanism stable? how can one geometrically characterize the preshock (the boundary of the space-time set on which the solution remains smooth)? precisely how does the entropy-producing shock wave instantaneously develop from the preshock? does uniqueness hold once the shock has formed? do other singularities instantaneously arise after the preshock? In this level of detail the problem was previously open even in one space dimension. We discuss a sequence of joint works with Steve Shkoller, Tristan Buckmaster, and Theodore Drivas, in which we have developed a multidimensional theory to answer the above questions.

Speaker: Francesco Maggi, The University of Texas at Austin

Title: A stability theory for isoperimetric and minimal area problems

Abstract: We offer a non-technical, panoramic view on some old and new results concerning the quantitative description of minimizers and critical points in basic geometric variational problems involving area. In the first part of the talk we review basic results on almost-isoperimetric and almost-constant mean curvature boundaries, both in the Euclidean and in the Riemannian setting. In the second part of the talk, we introduce the approximation of possibly singular minimal surfaces by "soap films" with positve, small volume. Finally, we revisit some of these results in the more physical context of Allen-Cahn surface tension energies, and introduce a new convergence theorem for the diffused interface volume preserving mean curvature flow.

Speaker: Maria Soria Carro Rutgers University

Title: Regularity of C^{1,α} Interface Elliptic Transmission Problems

Abstract: Transmission problems describe phenomena in which a physical quantity changes behavior across some fixed surface, known as the interface. The analysis of such problems started in the 1950s with the pioneering work of Picone in elasticity. Nowadays, they have a wide range of applications in different areas, such as electromagnetic processes, composite materials, and climatology. Typically, solutions are not differentiable across the interface, and the primary interest is to study their optimal regularity from each side of this surface. In this talk, I will discuss two elliptic transmission problems where the interface has minimal regularity. The first problem is for harmonic functions, and I will explain the main ideas and techniques used to prove the optimal regularity of solutions at the boundary. The second problem is for fully nonlinear equations, and I will focus on how to obtain a maximum principle (ABP estimate) for this problem, which plays a key role in the regularity theory of viscosity solutions. These results are part of my Ph.D. dissertation, and they are in collaboration with Luis Caffarelli and Pablo Raúl Stinga.

Speaker: Jonah Duncan, Johns Hopkins University

Title: Regularity for some fully nonlinear equations in conformal geometry

Abstract: The sigma_k Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, in which one looks to prescribe symmetric functions of the eigenvalues of the Schouten tensor to be constant within a fixed conformal class. In the last 20 years or so, there has been a significant amount of progress on the sigma_k Yamabe problem in the so-called positive case, leading to many existence results for smooth solutions. In this talk, I will discuss some recent results on the regularity theory for the sigma_k Yamabe equation, with an emphasis on W^{2,p} solutions in both the positive and negative cases, and viscosity solutions to the degenerate problem. I will also mention some open problems in the field. Much of the talk will be based on joint work with Luc Nguyen.

Nonlinear Analysis Seminar ---- Spring 2022 (Organizers: Yanyan Li, Dennis Kriventsov)

Zoom Link   Meeting ID: 964 3090 5091   Passcode: 491508

Speaker: Joaquim Serra, ETH Zurich

Title: Fractional minimal surfaces: an invitation for the skeptics (and the convinced)

Abstract: Elliptic operators of fractional order were popularized, mainly thanks to Luis Caffarelli, during the early 2000's. Suddenly, we learnt that every classical PDE had a fractional counterpart (or even more than one in some cases!). Also, fractional versions of most important techniques and results in PDE were developed. In this context, the invention in the late 2000's of fractional minimal surfaces may not seem a very striking milestone. Over the years, however, the interest and depth of these new surfaces is becoming unquestionable, to the point that they may be a fundamental tool in order to better understand certain (famously delicate) questions on classical minimal sur- faces, such as Yau's conjecture. In the talk I will describe some very recent works that, I hope, may help to convince a fraction of the remaining skeptics about the beauty and usefulness of fractional minimal surfaces.

Speaker: Javier Gomez-Serrano, Brown University and Universitat de Barcelona

Title: Rigidity and flexibility of stationary solutions of the Euler equations

Abstract: In this talk, I will discuss characterizations of stationary solutions of the 2D Euler equations in two different directions under different assumptions: rigidity (is every stationary solution radial?) and flexibility (do there exist non-radial stationary solutions?). The proofs will have a calculus of variations' flavor, a new observation that finite energy, stationary solutions with simply-connected vorticity have compactly supported velocity, and an application of the Nash-Moser iteration procedure. Based on joint work with Jaemin Park, Jia Shi and Yao Yao.

Speaker: Haim Brezis, Rutgers University

Title: Some of my favorite open pbs. I

Abstract: I will present a selection of the main open problems I raised throughout my career and that have resisted so far. This is not an exhaustive list, but striking questions fairly easy to state, which might attract younger generations.

Speaker: Mikhail Korobkov, Fudan University (Shanghai) and Sobolev Institute of Mathematics (Novosibirsk)

Title: On Steady Navier--Stokes equations in 2D exterior domains

Abstract: The talk is devoted to a review of results on solutions to the steady Navier-Stokes system with a finite Dirichlet integral in the exterior plane domain (="D-solutions"). Recently, some progress has been made on this problem: the uniform boundedness in the C-norm and the uniform convergence (at infinity) of such solutions, the uniqueness of the solutions to the flow around an obstacle problem in the class of all D-solutions, the nontriviality of the Leray solutions (obtained by the method of "invading domains") in flow around an obstacle problem and their convergence to a given limit at low Reynolds numbers. These results were obtained in our joint papers with Konstantin Pileckas, Remigio Russo, and Xiao Ren.

Speaker: Haim Brezis, Rutgers University

Title: Some of my favorite open pbs. II

Abstract: I will present a selection of the main open problems I raised throughout my career and that have resisted so far. This is not an exhaustive list, but striking questions fairly easy to state, which might attract younger generations.

Speaker: Eduardo Teixeira, University of Central Florida

Title: On the differentiability of solutions to fully nonlinear degenerate elliptic PDEs

Abstract: I will discuss nonlinear diffusion processes involving diffusion resistance. Mathematically one is led to consider fully nonlinear elliptic equations whose ellipticity degenerate as a function of the gradient; the law of degeneracy of the system. For non-degenerate models, a classical theorem proven independently by Caffarelli and Trudinger in the late 80's assures that solutions are of class $C^{1,\alpha}$ and a recent theorem proven by Imbert and Silvestre yields Holder continuity of solutions "independently" of the law of degeneracy. We want to understand the difficult borderline case regarding $C^{1}$ regularity.

Speaker: Sun-Yung Alice Chang, Princeton University

Title: On bi-Lipschitz equivalence of a class of non-conformally flat spheres

Abstract: This is a report of some recent joint work with Eden Prywes and Paul Yang. The main result is a bi-Lipschitz equivalence of a class of metrics on 4-shpere under curvature con- straints. The proof involves two steps: first a construction of quasiconformal maps between two conformally related metrics in a positive Yamabe class, followed by the step of applying the Ricci flow to establish the bi-Lipschitz equivalence from such a conformal class to the standard conformal class on 4-sphere.

Speaker: Dallas Albritton, Institute for Advanced Study

Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations

Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brue and Maria Colombo.

Speaker: Hongjie Dong, Brown University

Title: Lp estimates for a class of degenerate or singular equations

Abstract: I will discuss some recent results about Sobolev type estimates for elliptic and parabolic equations, which are degenerate or singular near the boundary of the domain. This is based on joint work with Tuoc Phan (University of Tennessee) and Hung Tran (University of Wisconsin).

Speaker: Mariana Smit Vega Garcia, Western Washington University

Title: Almost-Minimizers of Two-Phase Free Boundary Problems

Abstract: In this talk, we will discuss almost minimizers of Alt-Caffarelli-Friedman type functionals. In particular, we will consider branch points in their free boundary. This is joint work with Guy David, Max Engelstein, and Tatiana Toro.

Speaker: Yilun Wu, University of Oklahoma

Title: Existence of Rotating Stars Solutions with Variable Entropy

Abstract: The Euler-Poisson equations are a classical model in the theory of stellar structure. When a star has nonzero angular momentum, it will deform from a spherical shape and settle at a steady state called a rotating star solution. An extensive literature has established the existence of such solutions for given angular velocity profile, including the case of differential rotation, where different parts of the star rotate at different speeds. However, all of the existing results only allow the angular velocity to depend on the distance to the rotation axis, but not on the distance to the equatorial plane. In this talk, I will present the first result that allows a general rotation profile, without restrictions. This enables us to model the rotation of our own Sun. I will also explain a hidden connection between the restriction on the rotation profile and the constancy of entropy across the star. Consequently, the new result is the first one that allows a genuinely changing entropy within the star. This is joint work with Juhi Jang and Walter Strauss.

Speaker: Hui Yu, National University of Singapore

Title: Rate of blow-up in the thin obstacle problem

Abstract: The thin obstacle problem is a classical free boundary problem arising from the study of an elastic membrane resting on a lower-dimensional obstacle. Concerning the behavior of the solution near a contact point between the membrane and the obstacle, many important questions remain open. In this talk, we discuss a unified method that leads to a rate of convergence to `tangent cones’ at contact points with integer frequencies. If time permits, we also discuss some partial results concerning contact points with 7/2 frequency in 3d. This talk is based on recent joint works with Ovidiu Savin (Columbia).

Speaker: Yash Jhaveri, Columbia University

Title: What is the most efficient way to fill a hole with a given pile of sand?

Abstract: The optimal transport problem, formulated by Gaspard Monge in 1781, asks whether or not it is possible to find a map minimizing the total cost of moving a distribution of mass $f$ to another $g$ given that the cost of moving from $x$ to $y$ is measured by $c = c(x,y)$. The most fundamental case is that of the quadratic cost on $\R^n$: when $c(x,y) = |x-y|^2$. In this setting, our problem is a generalization of the second boundary value problem for the Monge--Amp\`{e}re equation. In this talk, we will discuss a study of the most common image and informal description of the optimal transport problem, what is the most efficient way to fill a hole with a given pile of sand? More precisely, we will discuss some regularity results for the optimal transport problem, for quadratic cost, which include the case that $f$ and $g$ are absolutely continuous measures concentrated on bounded convex domains with densities that behave like positive powers of the distance functions to the boundaries of these domains. This is joint work with Ovidiu Savin.

Speaker: Vladimir Sverak, University of Minnesota

Title: On the long-distance asymptotics of the steady-state Navier-Stokes solutions in three dimensions

Abstract: For the linear Stokes system, the long-distance asymptotics is given by classical multipole expansions. How does non-linearity change the expansions in three dimensions? The first term in the small data expansion for the non-linear case is known to be given by the so-called Landau solution. In the talk, we explain how to obtain the next term. It turns out the non-linearity accelerates the rate of decay of some of the modes, while not slowing down the rate of decay of any of the next-order-to-Landau modes. Joint work with Hao Jia.

Nonlinear Analysis Seminar ---- Fall 2021 (Organizers: Yanyan Li, Dennis Kriventsov)

Zoom Link   Meeting ID: 964 3090 5091   Passcode: 491508

Speaker: Alexis Vasseur, UT Austin

Title: Boundary vorticity estimate for Navier-Stokes and control of layer separation in the inviscid limit

Abstract: Consider the steady solution to the incompressible Euler equation $U=Ae_1$ in the periodic tunnel $\Omega=[0,1]\times \mathbb T^2$. Consider now the family of solutions $U_\nu$ to the associated Navier-Stokes equation with no-slip condition on the flat boundaries, for small viscosities $\nu=1/Re$, and initial values close in $L^2$ to $Ae_1$. Under a conditional assumption on the energy dissipation close to the boundary, Kato showed in 1984 that $U_\nu$ converges to $Ae_1$ when the viscosity converges to 0 and the initial value converge to $A e_1$. It is still unknown whether this inviscid is unconditionally true. Actually, the convex integration method predicts the possibility of a layer separation with energy at time T up to: $$\|\bar{U}(T)-Ae_1\|^2_{L^2}\equiv A^3T$$. In this work we prove that at the double limit for the inviscid asymptotic, where both the Reynolds number $Re$ converges to infinity and the initial value $U_{\nu}$ converges to $Ae_1$ in $L^2$, the energy of layer separation cannot be more than: $$\|U_\nu(T)-Ae_1\|^2_{L^2}\lesssim A^3T\ln(Re).$$ Especially, it shows that, even if the limit is not unique, the shear flow pattern is observable up to time $1/(A\ln(Re))$. This provides a notion of stability despite the possible non-uniqueness of the limit predicted by the convex integration theory. The result relies on a new boundary vorticity estimate for the Navier-Stokes equation. This new estimate, inspired by previous work on higher regularity estimates for Navier-Stokes, provides a non-linear control scalable through the inviscid limit. This is a joint work with Jincheng Yang.

Speaker: Rupert Frank, LMU Munich

Title: Which magnetic fields support a zero mode?

Abstract: Motivated by the question from mathematical physics about the size of magnetic fields that support zero modes for the three dimensional Dirac equation, we study a certain conformally invariant spinor equation. We state some conjectures and present results in their support. Those concern, in particular, two novel Sobolev inequalities for spinors and vector fields. The talk is based on joint work with Michael Loss.

Speaker: Connor Mooney, UC Irvine

Title: Solutions to the Monge-Ampere equation with polyhedral and Y-shaped singularities

Abstract: The Monge-Ampere equation det(D^2u) = 1 arises in prescribed curvature problems and in optimal transport. An interesting feature of the equation is that it admits singular solutions. We will discuss new examples of convex functions on R^n that solve the Monge-Ampere equation away from finitely many points, but contain polyhedral and Y-shaped singular structures. Along the way we will discuss geometric and applied motivations for constructing such examples, as well as their connection to a certain obstacle problem.

Speaker: Enrico Valdinoci, The University of Western Australia

Title: Long-range phase coexistence models

Abstract: We will discuss classical and recent results concerning the Allen-Cahn equation and its long-range counterpart, especially in relation to its limit interfaces, provided by (possibly nonlocal) minimal surfaces, and to rigidity and symmetry of flat solutions.

Speaker: Xumin Jiang, Fordham University

Title: Kaehler-Einstein metrics on complex hyperbolic cusps

Abstract: Let D be a complex torus of complex dimension n-1 together with a negative holomorphic line bundle L -> D. Let V be any closed tubular neighborhood of the zero section D in L. Datar-Fu-Song proved the existence of a unique complete Kaehler-Einstein metric on V \ D if a boundary condition is assumed. We study the asymptotics of such KE metrics and prove a sharp exponentially decaying estimate. This work is joint with Xin Fu and Hans-Joachim Hein.

Speaker: Xavier Cabre, ICREA and Universitat Politecnica de Catalunya (Barcelona)

Title: Stable solutions to semilinear elliptic equations are smooth up to dimension 9

Abstract: The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist singular stable energy solutions. In this talk I will describe a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems.

Speaker: Changfeng Gui, The University of Texas at San Antonio

Title: Some New Sharp Inequalities in Analysis and Geometry

Abstract: The classical Moser-Trudinger inequality is a borderline case of Sobolev inequalities and plays an important role in geometric analysis and PDEs in general. Aubin in 1979 showed that the best constant in the Moser-Trudinger inequality can be improved by reducing to one half if the functions are restricted to the complement of a three dimensional subspace of the Sobolev space $H^1$, while Onofri in 1982 discovered an elegant optimal form of Moser-Trudinger inequality on sphere. In this talk, I will present new sharp inequalities which are variants of Aubin and Onofri inequalities on the sphere with or without mass center constraints. One such inequality, for example, incorporates the mass center deviation (from the origin) into the optimal inequality of Aubin on the sphere, which is for functions with mass centered at the origin. Efforts have also been made to show similar inequalities in higher dimensions. Among the preliminary results, we have improved Beckner's inequality for axially symmetric functions when the dimension $n=4, 6, 8$. Many questions remain open. The talk is based on several joint papers with Amir Moradifam, Sun-Yung Alice Chang, Yeyao Hu and Weihong Xie.

Speaker: Fanghua Lin, Courant Institute, NYU

Title: Boundary Harnack Principle on Nodal Domains

Abstract: The Boundary Harnack Principle for an elliptic equation asserts that two positive solutions of the same equation that vanish on a portion of the boundary of the domain, then their ratio must be locally bounded. This principle plays a very important role in classical potential analysis and has proven to be very useful, for example, in studies of harmonic measures and free boundary regularity. In this talk, we shall discuss the Harnack principle for a class of nodal domains and some related applications.

Speaker: Chao Li, Courant Institute, NYU

Title: Stable minimal hypersurfaces in R^4

Abstract: In this talk, I will discuss the Bernstein problem for minimal surfaces, and the recent solution to the stable Bernstein problem for minimal hypersurfaces in R^4. Precisely, we show that a complete, two-sided, stable minimal hypersurface in R^4 is flat. Corollaries include curvature estimates for stable minimal hypersurfaces in 4-dimensional Riemannian manifolds, and a structural theorem for minimal hypersurfaces with bounded Morse index in R^4. This is based on joint work with Otis Chodosh.

Speaker: Po Lam Yung, Australian National University

Title: Sobolev spaces revisited

Abstract: In this talk, we will describe some new ways of characterising Sobolev and BV functions, using sizes of superlevel sets of suitable difference quotients. They provide remedy in certain cases where some critical Gagliardo-Nirenberg interpolation inequalities fail, and lead us to investigate real interpolations of certain fractional Besov spaces. Some connections will be drawn to earlier work by Bourgain, Brezis and Mironescu, and an image processing application will be given. Joint work with Haim Brezis, Jean Van Schaftingen, Qingsong Gu, Andreas Seeger, Brian Street and Oscar Dominguez.

Speaker: Tianling Jin, Hong Kong University of Science and Technology

Title: Regularity of solutions to the Dirichlet problem for fast diffusion equations

Abstract: We prove global Holder gradient estimates for bounded positive weak solutions of fast diffusion equations in smooth bounded domains with homogeneous Dirichlet boundary condition, which then leads us to establish their optimal global regularity. It solves a problem raised by Berryman and Holland in 1980. This is joint work with Jingang Xiong.

Speaker: Alexander Kiselev, Duke University

Title: Small scale creation in active scalars

Abstract: An active scalar is advected by fluid velocity that is determined by the scalar itself. Active scalars appear in many situations in fluid mechanics, with the most classical example being 2D Euler equation in vorticity form. Other prominent examples are the surface quasi-geostrophic (SQG) equation that comes from atmospheric science and the incompressible porous media (IPM) equation modeling the fluid flow in porous media. Usually, active scalar equations are both nonlinear and nonlocal, and their solutions spontaneously generate small scales. In this talk, I will discuss rigorous examples of small scale formation that involves infinite in time growth of derivatives for the 2D Euler equation, the SQG equation and the 2D IPM equation.

Speaker: Yu Yuan, University of Washington

Title: Rigidity for general semiconvex entire solutions to the sigma-2 equation

Abstract: We present a rigidity result for general semiconvex entire solutions to the sigma-2 equation. Two decades ago, this result was obtained in three dimensions, as a byproduct of the work on special Lagrangian equations. A decade ago in a joint work with Chang, this result was shown for almost convex entire solutions. Warren's rare saddle entire solution confirms the necessity of the semiconvexity assumption. Recall the classical Liouville rigidity for the Laplace equation or the sigma-1 equation and Jörgens-Calabi-Pogorelov rigidity for the Monge-Ampère equation or the sigma-n equation: all convex entire solutions to those equations in general n dimensions must be quadratic. This is joint work with Ravi Shankar.

Speaker: Antonio De Rosa, The University of Maryland

Title: Regularity of anisotropic minimal surfaces

Abstract: I will present a $C^{1,alpha}$-regularity theorem for m-dimensional Lipschitz graphs with anisotropic mean curvature bounded in $L^p$, $p > m$, in every dimension and codimension. Joint work with Riccardo Tione.

Nonlinear Analysis Seminar ---- Spring 2021 (Organizers: Yanyan Li, Dennis Kriventsov)

Zoom Link   Meeting ID: 941 7571 5705   Passcode: 849396

Speaker: Camillo De Lellis, Institute for Advanced Study

Title: Flows of vector fields: classical and modern

Abstract: Consider a (possibly time-dependent) vector field $v$ on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindel\"of) Theorem states that, if the vector field $v$ is Lipschitz in space, for every initial datum $x$ there is a unique trajectory $\gamma$ starting at $x$ at time $0$ and solving the ODE $\dot{\gamma} (t) = v (t, \gamma (t))$. The theorem looses its validity as soon as $v$ is slightly less regular. However, if we bundle all trajectories into a global map allowing $x$ to vary, a celebrated theory put forward by DiPerna and Lions in the 80es show that there is a unique such flow under very reasonable conditions and for much less regular vector fields. A long-standing open question is whether this theory is the byproduct of a stronger classical result which ensures the uniqueness of trajectories for {\em almost every} initial datum. I will give a complete answer to the latter question and draw connections with partial differential equations, harmonic analysis, probability theory and Gromov's $h$-principle.

Speaker: Hector Chang-Lara, CIMAT (Centro de Investigacion en Matematicas A.C.)

Title: Eikonal vs. Brownian: Regularity for the solution of an equation with gradient constraint

Abstract: Two controllers are in charge of steering a spaceship in some domain Omega. The first controller wants to spend as much time as possible exploring Omega while the second wants to get out of it as quickly as possible. The first controller determines minute by minute whether the ship is moving by a Brownian motion or with constant speed, in which case it is the second controller who chooses the direction. Under these instructions, determining the optimal strategies for each player leads us to solve the equation min (-Delta u, |Du|) = 1 which has several interesting characteristics. Among them is the presence of a free boundary which separates the regions where a Poisson or an Eikonal equation is satisfied. In a recent collaboration with Edgard Pimentel (PUC-Rio) we showed that the solutions are Lipschitz continuous and that |Du| is continuous, even though the gradient is discontinuous in numerous examples. This problem is a simplification of interesting models in financial mathematics related with the optimal strategy for the payment of dividends from multiple insurances.

Speaker: Juncheng Wei, The University of British Columbia

Title: Recent results on Allen-Cahn equation

Abstract: I will discuss some recent results on Allen-Cahn equation, including the half space theorem and stability of saddle solutions in R^{8,10,12}.

Speaker: Hui Yu, Columbia University

Title: The N-membrane problem

Abstract: The N-membrane problem is the study of shapes of elastic membranes being pushed against each other. The main questions are the regularity of the functions modeling the membranes, and the regularity of the contact regions between consecutive membranes. These are classical questions in free boundary problems. However, very little is known when N is larger than 2. In this case, there are multiple free boundaries that cross each other, and most known techniques fail to apply. In this talk, we discuss, for general N, the optimal regularity of the solutions in arbitrary dimensions, and a classification of blow-up solutions in 2D. Then we focus on the regularity of the free boundaries when N=3. We also discuss how the techniques developed here can be applied to other problems involving multiple free boundaries. This talk is based on two recent joint works with Ovidiu Savin (Columbia University).

Speaker: Monica Musso, University of Bath

Title: Desingularization of vortices and leapfrogging phenomena for Euler equations

Abstract: In 1858, Helmholtz predicted that two vortex-rings in fluids will pass through each other and become leapfrogging. In this talk I will give a rigorous justification of Helmholtz’s conjecture for 3D Euler flow, and also discuss the dynamical desingularization of vortices for 2D Euler flows. The key ingredients are new gluing methods for Euler flows. (Joint work with Juan Davila, Manuel del Pino and Juncheng Wei.)

Speaker: Francois Hamel, Aix-Marseille University

Title: Symmetry properties for the Euler equations and semilinear elliptic equations

Abstract: In this talk, I will discuss radial and one-dimensional symmetry properties for the stationary incompressible Euler equations in dimension 2 and some related semilinear elliptic equations. I will show that a steady flow of an ideal incompressible fluid with no stagnation point and tangential boundary conditions in an annulus is a circular flow. The same conclusion holds in complements of disks as well as in punctured disks and in the punctured plane, with some suitable conditions at infinity or at the origin. I will also discuss the case of parallel flows in two-dimensional strips, in the half-plane and in the whole plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on radial and one-dimensional symmetry results for the solutions of some elliptic equations satisfied by the stream function. The talk is based on some joint works with N. Nadirashvili.

Speaker: Manuel del Pino, University of Bath

Title: Singularity formation for the Keller-Segel system in the plane

Abstract: The classical model for chemotaxis is the planar Keller-Segel system $$ u_t = \Delta u - \nabla\cdot ( u\nabla v ), \quad v(\cdot, t) = \frac 1{2\pi} \log 1{|\cdot |} * u(\cdot ,t) . $$ in $\R^2\times (0,\infty)$. Blow-up of finite mass solutions is expected to take place by aggregation, which is a concentration of bubbling type, common to many geometric flows. We build with precise profiles solutions in the critical-mass case $8\pi$, in which blow-up in infinite time takes place. We establish stability of the phenomenon detected under arbitrary mass-preserving small perturbations and present new constructions in the finite time blow-up scenario.

Speaker: Jeff Calder, Universoty of Minnesota

Title: PDE continuum limits for prediction with expert advice

Abstract: Prediction with expert advice refers to a class of machine learning problems that is concerned with how to optimally combine advice from multiple experts whose prediction qualities may vary greatly. We study a stock prediction problem with history-dependent experts and an adversarial (worst-case) market, posing the problem as a repeated two-player game. The game is a generalization of the Kohn-Serfaty two-player game for curvature motion. We prove that when the game is played for a long time, the discrete value functions converge in the continuum to the solution of a nonlinear parabolic partial differential equation (PDE) and the optimal strategies are characterized by the solution of a Poisson equation over a De Bruijn graph, arising from the history-dependence of the experts. Joint work with Nadejda Drenska (UMN).

Speaker: Siyuan Lu, McMaster University

Title: Rigidity of Riemannian Penrose inequality with corners and its implications

Abstract: Motivated by the rigidity case in the localized Riemannian Penrose inequality, we show that suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality is necessarily smooth in properly specified coordinates. If applied to hypersurfaces enclosing the horizon in a spatial Schwarzschild manifold, the result gives the rigidity of isometric hypersurfaces with the same mean curvature. This is a joint work with Pengzi Miao.

Speaker: Qi S. Zhang, University of California Riverside

Title: Time analyticity and reversibility of some parabolic equations

Abstract: We describe a concise way to prove time analyticity for solutions of parabolic equations including the heat and Navier Stokes equations. In some cases, results under sharp conditions are obtained. An application is a necessary and sufficient condition for the solvability of the backward heat equation which is ill-posed, helping to remove an old obstacle in control theory. Part of the work is joint with Hongjie Dong.

Speaker: Qing Han, University of Notre Dame

Title: Solutions of the Minimal Surface Equation and of the Monge-Ampere Equation near Infinity

Abstract: Classical results assert that, under appropriate assumptions, solutions near infinity are asymptotic to linear functions for the minimal surface equation (due to Bers and Schoen) and to quadratic polynomials for the Monge-Ampere equation (due to Caffarelli-Li) for dimension n at least 3, with an extra logarithmic term for n=2. We characterize remainders in the asymptotic expansions by a single function, which is given by a solution of some elliptic equation near the origin via the Kelvin transform. Such a function is smooth in the entire neighborhood of the origin for the minimal surface equation in every dimension and for the Monge-Ampere equation in even dimension, but only C^{n-1,\alpha} for the Monge-Ampere equation in odd dimension, for any \alpha in (0,1).

Speaker: Thomas Yizhao Hou, California Institute of Technology

Title: The Interplay between Analysis and Computation in Studying 3D Euler Singularity

Abstract: Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. We first review the numerical evidence of finite time singularity for 3D axisymmetric Euler equations by Luo and Hou. The singularity is a ring like singularity that occurs at a stagnation point in the symmetry plane located at the boundary of the cylinder. We then present a novel method of analysis and prove that the 1D HL model and the original De Gregorio model develop finite time self-similar singularity. This analysis has been generalized to prove finite time singularity of the 2D Boussinesq and 3D Euler equations with C^{1,\alpha} initial velocity and boundary, whose solutions share some essential features similar to those reported in the Luo-Hou computation. Finally, we present some recent numerical results on singularity formation of the 3D axisymmetric Navier-Stokes equations with degenerate diffusion coefficients.

Speaker: Ian Tobasco, University of Illinois at Chicago

Title: Exact solutions for the wrinkle patterns of confined elastic shells

Abstract: A basic fact of geometry is that there are no length-preserving maps from a sphere to the plane. But what happens if you confine a thin elastic shell, which prefers to be a curved surface but can deform approximately isometrically, to reside nearby a plane? It wrinkles, and forms a remarkable pattern of peaks and troughs, the arrangement of which is sometimes random, sometimes not depending on the shell. After a brief introduction to the mathematical modeling of thin elastic shells, this talk will focus on a new set of simple, geometric rules we have derived for wrinkle patterns via Gamma-convergence and convex analysis of the limit problem. Our rules govern the asymptotic layout of the wrinkle peaks and troughs --- for instance, negatively curved wrinkles tend to arrange along segments solving the minimum exit time problem, in the infinitesimally wrinkled limit. Positively curved shells can be understood more or less completely as well, through a hidden duality with their negatively curved counterparts. Our predictions for the wrinkle patterns of confined shells match the results of numerous experiments and simulations done with Eleni Katifori (U. Penn) and Joey Paulsen (Syracuse). Underlying their analysis is a certain class of interpolation inequalities of Gagliardo-Nirenberg type, whose best prefactors encode the optimal patterns.

Nonlinear Analysis Seminar ---- Fall 2020 (Organizers: Yanyan Li, Dennis Kriventsov)

Registration Link   Meeting ID: 992 8122 5008

Speaker: Luis Caffarelli, University of Texas at Austin

Title: Propagation on a domain, driven by expansion on its boundary

Abstract: Diverse models of propagation of a species that travels along a line and diffuses sideways were introduced among others by Berestycki, Coulon, Roquejoffre, Rossi I will describe work in collaboration with Roquejoffre and Tomasetti in this area of research.

Speaker: Hao Jia, University of Minnesota

Title: Long time dynamics of 2d Euler and nonlinear inviscid damping

Abstract: In this talk, we will discuss some joint work with Alexandru Ionescu on the nonlinear inviscid damping near point vortex and monotone shear flows in a finite channel. We will put these results in the context of long time behavior of 2d Euler equations and indicate further important open problems in the field.

Speaker: Yao Yao, Georgia Institute of Technology

Title: Two results on the interaction energy

Abstract: For any nonnegative density f and radially decreasing interaction potential W, the celebrated Riesz rearrangement inequality shows the interaction energy E[f] = \int f(x)f(y)W(x-y) dxdy satisfies E[f] \le E[f^*], where f^* is the radially decreasing rearrangement of f. It is a natural question to look for a quantitative version of this inequality: if its two sides almost agree, how close must f be to a translation of f^*? Previously the stability estimate was only known for characteristic functions. I will discuss a recent work with Xukai Yan, where we found a simple proof of stability estimates for general densities. I will also discuss another work with Matias Delgadino and Xukai Yan, where we constructed an interpolation curve between any two radially decreasing densities with the same mass, and show that the interaction energy is convex along this interpolation. As an application, this leads to uniqueness of steady states in aggregation-diffusion equations with any attractive interaction potential for diffusion power m\ge 2, where the threshold is sharp.

Speaker: Alexandru Ionescu, Princeton University

Title: On the global regularity for the Einstein-Klein-Gordon coupled system

Abstract: In joint work with Benoit Pausader we consider the Einstein field equations of General Relativity for self-gravitating massive scalar fields (the Einstein-Klein-Gordon system). Our main results concern the global regularity, modified scattering, and precise asymptotic analysis of solutions of this system with initial data in a small neighborhood of the Minkowski space-time.

Speaker: Weiyong He, University of Oregon

Title: Harmonic and biharmonic almost complex structures

Abstract: Harmonic almost complex structures was introduced by C. Wood in 1990s. We study the existence and regularity of weak harmonic almost complex structure from analytic point of view, as a tensor-valued version of harmonic maps. We also introduce the notion of biharmonic almost complex structures, in particular in dimension four. We prove that biharmonic almost complex structures are smooth, and there always exist an energy-minimizing biharmonic almost complex structure. Moreover, given a homotopy class, we also show existence results. We conjecture that the homotopy class of energy-minimizing biharmonic almost complex structures do not depend on a generic background metric.

Speaker: Yannick Sire, Johns Hopkins University

Title: Blow-up solutions via parabolic gluing

Abstract: We will present some recent results on the construction of blow-up solutions for critical parabolic problems of geometric flavor. Initiated in the recent years, the inner/outer parabolic gluing is a very versatile parabolic version of the well-known Lyapunov-Schmidt reduction in elliptic PDE theory. The method allows to prove rigorously some formal matching asymptotics (if any available) for several PDEs arising in porous media, geometric flows, etc….I will give an overview of the strategy and will present several applications to (variations of) the harmonic map flow, Yamabe flow and Yang-Mills flow. I will also present some open questions.

Speaker: Zhifei Zhang, Peking University

Title: Linear stability of pipe Poiseuille flow at high Reynolds number regime

Abstract: The linear stability of pipe Poiseuille flow is a long standing problem since Reynolds experiment in 1883. Joint with Qi Chen and Dongyi Wei, we solve this problem at high Reynolds regime. We first introduce a new formulation for the linearized 3-D Navier-Stokes equations around this flow. Then we establish the resolvent estimates of this new system under favorable artificial boundary conditions. Finally, we solve the original system by constructing a boundary layer corrector.

Speaker: Daniela De Silva, Barnard College

Title: On the Boundary Harnack Principle

Abstract: In this talk we discuss the classical Boundary Harnack Principle and some of its applications to Free Boundary problems. We then present a recent direct analytic proof of this result, for solutions to linear uniformly elliptic equations in either divergence or non-divergence form.

Speaker: Fanghua Lin, Courant Institute

Title: Defects in heat flows of harmonic maps

Abstract: Starting with Struve's solution of the heat flow harmonic maps from a surface, we shall discuss several model heat flow problems for maps and its associated singularities. In case there are time or energy scale separations, one can often derive dynamical laws for singularities (energy concentration sets). There are challenging open problems when such scale separation don't exist such as defect motion in 3D liquid crystal evolutions or coupled flows of sharp interfaces and phase-valued maps in models of fast reaction--slow diffusion described by Keller-Rubinstein-Steinberg.

Speaker: Alessio Figalli, ETH

Title: Generic regularity in obstacle problems

Abstract: The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is $C^\infty$ outside a set of singular points. Explicit examples show that the singular set could be in general $(n-1)$-dimensional ---that is, as large as the regular set. In a recent paper with Ros-Oton and Serra we show that, generically, the singular set has zero $\mathcal H^{n-4}$ measure (in particular, it has codimension 3 inside the free boundary), solving a conjecture of Schaeffer in dimension $n \leq 4$. The aim of this talk is to give an overview of these results.

Speaker: Shaoming Guo, University of Wisconsin

Title: Fourier restrictions and decouplings for surfaces of co-dimension two

Abstract: I will report some recent process on Fourier restriction estimates and decoupling inequalities for quadratic surfaces of co-dimension two.

Speaker: Maria Esteban, Universite Paris-Dauphine

Title: Flows and symmetry properties of positive solutions of nonlinear elliptic PDEs

Abstract: In this talk will be presented various results concerning the symmetry and symmetry breaking properties of positive solutions of nonlinear elliptic PDEs. In some cases related to the study of Caffarelli-Kohn-Nirenberg inequalities the use of flows can provide optimal results concerning symmetry or symmetry breaking. In other cases, where magnetic Laplacians are concerned, the situation becomes more complicated, and one cannot always prove optimal results. Various examples will be discussed during the talk.

Speaker: Angela Pistoia, Universita' Degli Studi Di Roma La Sapienza.

Title: Elliptic systems with critical growth

Abstract: I will present some results concerning existence and phase separation of entire solutions to a pure critical competitive elliptic system and their relation with sign-changing solutions to the Yamabe problem.

Speaker: Xinan Ma, University of Science and Technology of China

Title: Liouville theorem for a class of semilinerar elliptic problems on Heisenberg group

Abstract: We obtain an entire Liouville type theorem to the classical semilinear subcritical elliptic equation on Heisenberg group. A pointwise estimate near the isolated singularity was also proved. The soul of the proofs is an a priori integral estimate, which deduced from a generalized formula of that found by Jerison and Lee. This is a joint work with Prof. Qianzhong Ou.