Nonlinear Analysis and its Applications in Geometry

2023.07.16-07.22
Beijing, China & Zoom

July 16 - July 22, 2023

organized by

Shibing Chen, Guohuan Qiu, Yi Zhang

All times are local to Beijing

Titles and abstract:

●  Karoly J. Boroczky (Renyi Institute of Mathematics, Budapest, Hungary)

Title: Lp-Minkowski problem - Old and New results

Abstract: Lutwak’s Lp Minkowski problem as a Monge-Ampere equation on the n-dimensional sphere for real p has been in the center of attention the last couple of decades. The talk surveys the state of art (like the recent resolution of the case p < -n-1, or stability versions strengthening Brendle, Choi and Daskalopoulos’ celebrated result about the uniqueness of the unit ball as a solution for the suitable equation for p >- n- 1), and points out some major open problems.

●  Xavier Cabre (ICREA and 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.

●  Zhijie Chen (Tsinghua University)

Title: Asymptotic behaviors of low energy nodal solutions for Lane Emden problems

Abstract: In this talk, I will introduce our recent work about asymptotic behaviors of low energy nodal solutions for Lane-Emden equations when the exponent goes to infinity. Aymptotics of positive solutions have been studied well, but asymptotics of nodal solutions is more difficult to study and not much is known. Here we can obtain some results for low energy nodal solutions. This is based on joint work with my students Zetao Cheng and Hanqing Zhao.

●  Alessio Figalli (ETH Zurich)

Title: Quantitative stability in geometric and functional inequalities

Abstract: Geometric and functional inequalities play a crucial role in several problems arising in analysis and geometry. Proving the validity of such inequalities, and understanding the structure of minimizers, is a classical and important question. In this talk, I will overview this beautiful topic and discuss some recent results.

●  Nicola Fusco (Universita di Napoli "Federico II")

Title:Local and global minimizers for a capillarity type problem

Abstract:I will present a model for vapor-liquid-solid growth of nanowires where liquid drops are described as local or global volume-constrained minimizers of the capillarity energy outside a semi-infinite convex obstacle modeling the nanowire. I will first discuss global existence of minimizers and then, in the case of rotationally symmetric nanowires, I will explain how the presence of a sharp edge affects the shape of local minimizers and the validity of Young’s law. Finally, I will present some recent regularity results for local minimizers and the connections of this problem with an isoperimetric inequality outside convex sets.

●  Jose A. Galvez (Universidad de Granada)

Title: Linearity of homogeneous solutions to elliptic equations in dimension three

Abstract:An old conjecture by Alexandrov, Koutrofiotis and Nirenberg states that every 1-homogeneous solution to a linear elliptic equation in Euclidean 3-space must be linear. A striking counterexample to this claim was found by Martinez-Maure in 2001. In it, the Hessian of the solution vanishes exactly at 4 disjoint geodesic semicircles of the unit sphere, and along them the equation is not uniformly elliptic. In this talk we prove the converse of this result: for any (non-linear) homogeneous solution of a linear elliptic equation in Euclidean 3-space, there must exist four disjoint geodesic semicircles in the unit sphere along which the Hessian of u vanishes, and the uniform ellipticity of the equation is lost. The result is sharp, by Martinez-Maure’s example. Joint work with Pablo Mira.

●  Yuxin Ge (University of toulouse 3)

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 (X^4 , g+) such that r ^2 g+|M = h on the boundary M =？ X for some defining function r on X^4 ? 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 manifolds 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 works with Alice Chang.

●  Genggeng Huang (Fudan University)

Title:Monge-Ampere equation with Guillemin boundary condition

Abstract: We will talk about the following boundary value problem of Monge-Ampere equation

det D^{2}u = h(x)/ II^N_{i=1}^{N} l_{i}(x) , in P \in R^ n , (1)

u(x)-∑_{i=1}^{N} l_{i}(x) log l_{i}(x) ∈ C^ ∞(\bar{P}) (2)

Here

0 < h(x) ∈ C^ ∞(\bar{P}), P = ∩ _{ i=1}^N{l_i(x) > 0}

is a simple convex polytope in R ^n. li(x) are affine functions i = 1, · · · , N. Under suitable conditions, we will show (1) and (2) are solvable. This is a joint work with Weiming Shen.

●  Qirui Li (Zhejiang University)

Title: On the Lp Minkowski problem with super critical exponents.

Abstract: The Lp-Minkowski problem deals with the existence of closed convex hypersurface with prescribed p-area measure. The problem has been solved in the sub-critical case p > -n- 1, but remains widely open 4 in the super-critical case p < -n-1. In this talk, we introduce new ideas to solve the problem for all super-critical exponents. A crucial ingredient in the proof is a topological method based on the calculation of the homology of a topological space of ellipsoids. The talk is based on recent joint work with Qiang Guang and Xu-Jia Wang.

●  Emanuel Milman (Technion-Israel Institute of Technology)

Title: Multi-Bubble Isoperimetric Problems - Old and New

Abstract:The classical isoperimetric inequality in Euclidean space R n states that among all sets of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems for more general metric-measure spaces, such as on the n-sphere S n and on n dimensional Gaussian space G^n (i.e. R ^n endowed with the standard Gaussian measure). Furthermore, one may consider the “multi-bubble" isoperimetric problem, in which one prescribes the volume of p ≥ 2 bubbles (possibly disconnected) and minimizes their total surface area – as any mutual interface will only be counted once, the bubbles are now incentivized to clump together. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to p = 1; the case p = 2 is called the double-bubble problem, and so on. In 2000, Hutchings, Morgan, Ritoré and Ros resolved the double-bubble conjecture in Euclidean space R 3 (and this was subsequently resolved in R n as well) – the boundary of a minimizing double-bubble is given by three spherical caps meeting at 120 -degree angles. A more general conjecture of J. Sullivan from the 1990’s asserts that when p ≤ n + 1, the optimal multi-bubble in R ^n (as well as in S ^n) is obtained by taking the Voronoi cells of p + 1 equidistant points in S n and applying appropriate stereographic projections to R n (and backwards). In 2018, together with Joe Neeman, we resolved the analogous multi bubble conjecture for p ≤ n bubbles in Gaussian space Gn – the unique partition which minimizes the total Gaussian surface area is given by the Voronoi cells of (appropriately translated) p + 1 equidistant points.

In the present talk, we describe our recent progress with Neeman on the multi-bubble problem on R n and S n. In particular, we show that minimizing bubbles in R n and S n are always spherical when p ≤ n, and we resolve the latter conjectures when in addition p ≤ 5 (e.g. the triple bubble conjectures when n ≥ 3 and the quadruple-bubble conjectures when n ≥ 4).

●  Xavier Ros Oton (Universitat de Barcelona)

Title: The singular set in the Stefan problem

Abstract: The Stefan problem, dating back to the XIXth century, is probably the most classical and important free boundary problem. The regularity of free boundaries in the Stefan problem was developed in the groundbreaking paper (Caffarelli, Acta Math. 1977). The main result therein establishes that the free boundary is C∞ in space and time, outside a certain set of singular points. The fine understanding of singularities is of central importance in a number of areas related to nonlinear PDEs and Geometric Analysis. In particular, a major question in such a context is to establish estimates for the size of the singular set. The goal of this talk is to present some recent results in this direction for the Stefan problem. This is a joint work with A. Figalli and J. Serra.

●  Yannick Sire (Johns Hopkins University)

Title: Geometric variational problems: regularity vs singularity formation

Abstract: I will describe in a very informal way some techniques to deal with the existence ( and more qualitatively regularity vs singularity formation) in different geometric problems and their heat flows motivated by (variations of) the harmonic map problem, the construction of Yang-Mills connections or nematic liquid crystals. I will emphasize in particular on recent results on the construction of very fine asymptotics of blow up solutions via a new gluing method designed for parabolic flows. I’ll describe several open problems and many possible generalizations, since the techniques are rather flexible.

●  Guofang Wang (University of Freiburg)

Title: Optimal geometric inequalities for capillary hypersurfaces

Abstract: In the talk I will first review our previous work on hypersurfaces with free boundary supported on the unit sphere. Then I will introduce suitable geometric quantities, quermassintegrals, for capillary hypersurfaces supported on a hyperplane and consider the corresponding Alexandrov-Fenchel inequalities by introducing a suitable curvature flow. I will also talk about a corresponding Heintze-Karcher-Ros inequality and a Minkowski problem. The talk bases on the joint work with Chao Xia and other collaborators.

●  Kelei Wang (Wuhan University)

Title: Nondegeneracy for stable solutions to one phase free boundary problem

Abstract: Since the seminal work of Alt-Caffarelli in 1981, the one phase free boundary problem has been studied by many people. To study the regularity and singularity of free boundaries, the blow up analysis is a standard method. It turns out for this free boundary problem, the nondegeneracy condition is crucial for the application of this method. Although the nondegeneracy condition has been known for energy minimizers for a long time, it's not true for general solutions. In this talk, I will discuss a proof of the nondegeneracy for stable solutions. This is based on a joint work with N. Kamburov.

●  Xu-jia Wang (Australian National University)

Title: Free boundary problems in the Monge-Ampere equation

Abstract: In this talk we consider the regularity of free boundary in the Monge- Ampere obstacle problem, and the regularity of free boundary in the Gauss curvature flow of convex hypersurface with flat side. By the Legendre transform, these problems are equivalent to the regularity of solutions to Monge-Ampere type equations with a singular point in polar coordinates. By analysing the geometric profile carefully near the singular point, we prove the C^ 2,α regularity for the free boundary in all dimensions.

●  Zhizhang Wang (Fudan University)

Title: Hessian equations on exterior domains in hyperbolic space

Abstract: Suppose Ω is some domain in the hyperbolic space Hn. In this talk, we will consider the homogenous k-Hessian equations on Hn \Ω with constant -1 on the boundary of  Ω and asymptotic to zero at the infinity. We will give the existence of this equation. This is a joint work with Ling Xiao.

●  Yong Wei (University of Science and Technology of China)

Title: Curvature measures and volume preserving curvature flows

Abstract: Volume preserving mean curvature flow was introduced by Huisken in 1987 and it was proved that the flow deforms convex initial hypersurface smoothly to a round sphere. This was generalized later by McCoy in 2005 and 2017 to volume preserving flows driven by a large class

of 1-homogeneous symmetric curvature functions. In this talk, we discuss the flows with higher homogeneity and describe the convergence result for volume preserving curvature flows in Euclidean space by arbitrary positive powers of k-th mean curvature for all k = 1, · · · , n. As key ingredients, the monotonicity of a generalized isoperimetric ratio will be used to control the geometry of the evolving hypersurfaces and the curvature measure theory will be used to prove the Hausdorff convergence to a sphere. We also discuss some generalizations including the flows in the anisotropic setting, and the flows in the hyperbolic setting. The talk is based on joint work with Ben Andrews (ANU), Yitao Lei (ANU), Changwei Xiong (Sichuan Univ.), Bo Yang (CAS) and Tailong Zhou (USTC).

●  Jingang Xiong (Beijing Normal University)

Title: Harmonic maps with finite hyperbolic distances to the Extreme Kerr

Abstract: Motivated by stationary vacuum solutions of the Einstein field equations, we study singular harmonic maps from domains of 3-dimensional Euclidean space to the hyperbolic plane having bounded hyperbolic distance to Kerr harmonic maps. In the degenerate case, we prove that every such harmonic map admits a unique tangent harmonic map at the extreme black hole horizon. The possible tangent maps are classified and rates of convergence to the tangent map are established. Similarly, expansions in the asymptotically flat end are presented. These results, together with those of Li-Tian 1992 and Weinstein 1989, provide a complete regularity theory for such singular harmonic maps. This is joint with Q. Han, M. Khuri and G. Weinstein.

●  Jie Zhou (Capital Normal University)

Title: Regularity for varifolds with critical Allard conditions

Abstract: The classical Allard regularity theorem says, for a rectifiable n varifold in the unit ball of the Euclidean space passing through the origin with density not less than one, if its the mass in the unit ball is close to the volume of a flat n-dimensional unit disk and the L ^p norm of the generalized mean curvature is small enough for some supercritical index p > n, then the support of the varifold is a C ^{1,α=1-n /p} graph near the origin. In this talk, we will present some regularity result in the critical case. In dimension two, we show the support of the varifold is (locally) bi-Lipschitz homeomorphic to the unit disk. In dimension n > 2, we discuss the W^1,p regularity for p < ∞. The presentation is based on joint works with Dr. Yuchen Bi.