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Seminar

This is a Weekly Online Seminar in the areas of geometric and functional inequalities and closely related areas of partial differential equations, geometric analysis and etc. It is usually held between 9:00-11:00am US Eastern Time on Mondays. The researchseminars.org page may be found here.

A unique Zoom link for each talk is sent out in a mailing list each week. Please subscribe below to join the mailing list or email geometricinequalitiesandpdes@gmail.com. If you wish to attend a talk without subscribing to the mailing list, you may email geometricinequalitiesandpdes@gmail.com to request the Zoom link for a given talk.

If you have any other questions or comments, please email one of the organizers.

Organizers

Joshua Flynn
University of Connecticut
(joshua.flynn@uconn.edu)
Nguyen Lam
Memorial University of Newfoundland Grenfell Campus
(nlam@grenfell.mun.ca)

Jungang Li
Brown University
(jungang_li@brown.edu)
Guozhen Lu
University of Connecticut
(guozhen.lu@uconn.edu)

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**Note: The Zoom link will typically be sent Sunday evening ET.
If you do not register with an academic email, you may be asked to provide some sort of verification (e.g., an academic website with your identity and the given email).

Next Talk

Luis Silvestre
(University of Chicago)
Sep 13 10:00 EDT

Regularity estimates for the Boltzmann equation without cutoff.

We study the regularization effect of the inhomogeneous Boltzmann equation without cutoff. We obtain a priori estimates for all derivatives of the solution depending only on bounds of its hydrodynamic quantities: mass density, energy density and entropy density. As a consequence, a classical solution to the equation may fail to exist after a certain time T only if at least one of these hydrodynamic quantities blows up. Our analysis applies to the case of moderately soft and hard potentials. We use methods that originated in the study of nonlocal elliptic and parabolic equations: a weak Harnack inequality in the style of De Giorgi, and a Schauder-type estimate..

Scheduled Talks

Jill Pipher
(Brown University)
Sep 20 10:00 EDT

TBA.

TBA.

Yoshikazu Giga
(University of Tokyo)
Sep 27 09:00 EDT

TBA.

TBA.

Previous Talks

Lorenzo D’Ambrosio
(Universita di Bari)
Jul 05 09:00 EDT

Liouville theorems for semilinear biharmonic equations and inequalities.

We study nonexistence results for a coercive semilinear biharmonic equation on the whole R^N. The analysis is made for general solutions without any assumption on their sign nor on their behaviour at infinity. A relevant role is played by some extensions of the Hardy-Rellich inequalities for general functions (not necessarily compactly supported).

Susanna Terracini
(Universitá di Torino)
Jun 28 09:00 EDT

Free boundaries in segregation problems.

We first consider classes of variational problems for densities that repel each other at distance. Examples are given by the minimizers of Dirichlet functional or the Rayleigh quotient D({\bf u}) =  \sum_{i=1}^k \int_{\Omega} |\nabla u_i|^2 \quad \text{or} \quad R({\bf u}) =  \sum_{i=1}^k \frac{\int_{\Omega} |\nabla u_i|^2}{\int_{\Omega} u_i^2} over the class of H^1(\Omega,\mathbb{R}^k) functions attaining some boundary conditions on \partial \Omega, and subjected to the constraint dist (\{u_i > 0\}, \{u_j > 0\}) \ge 1 \qquad \forall i \neq j. As second class of problems, we consider energy minimizers of Dirichlet energies with different metrics D({\bf u}) =  \sum_{i=1}^k \int_{\Omega} \langle A_i\nabla u_i, \nabla u_i\rangle with constraint u_i(x)\cdot u_j(x)=0, \qquad \forall x\in \Omega\;, \forall i \neq j. For these problems, we investigate the optimal regularity of the solutions, prove a free-boundary extremality condition, and derive some preliminary results characterising the emerging free boundary.

Roger Moser
(University of Bath)
Jun 21 09:00 EDT

The infinity-elastica problem.

The Euler elastica problem seeks to minimise the L^2-norm of the curvature of curves under certain boundary conditions. If we replace the L^2-norm with the L^\infty-norm, then we obtain a variational problem with quite different properties. Nevertheless, even though the underlying functional is not differentiable, it turns out that the solutions of the problem can still be described by differential equations. An analysis of these equations then gives a classification of the solutions.

Svitlana Mayboroda 
(University of Minnesota)
Jun 14 10:00 EDT

Green Function vs. Geometry.

In this talk we will discuss connections between the geometric and PDE properties of sets. The emphasis is on quantifiable, global results which yield true equivalence between the geometric and PDE notions in very rough scenarios, including domains and equations with singularities and structural complexity. The main result establishes that in all dimensions d < n, a d-dimensional set in \mathbb{R}^n is regular (rectifiable) if and only if the Green function for elliptic operators is well approximated by affine functions (distance to the hyperplanes). To the best of our knowledge, this is the first free boundary result of this type for lower dimensional sets and the first free boundary result in the classical case d=n-1 without restrictions on the coefficients of the equation.

Sun-Yung Alice Chang 
(Princeton University)
Jun 07 10:00 EDT

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

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 constraints. 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.

Xiaojun Huang
(Rutgers University)
May 17 10:00 EDT

Revisit to a non-degeneracy property for extremal mappings.

I will discuss a generalization of my previous result on the localization of extremal maps near a strongly pseudo-convex point.

Michael Struwe
(ETH Zürich)
May 10 10:00 EDT

Normalized harmonic map flow.

Finding non-constant harmonic 3-spheres for a closed target manifold N is a prototype of a super-critical variational problem. In fact, the direct method fails, as the infimum of Dirichlet energy in any homotopy class of maps from the 3-sphere to any closed N is zero; moreover, the harmonic map heat flow may blow up in finite time, and even the identity map from the 3-sphere to itself is not stable under this flow. To overcome these difficulties, we propose the normalized harmonic map heat flow as a new tool, and we show that for this flow the identity map from the 3-sphere to itself now, indeed, is stable; moreover, the flow converges to a harmonic 3-sphere also when we perturb the target geometry. While our results are strongest in the perturbative setting, we also outline a possible global theory.

Jungang Li 
(Brown University)
May 03 10:00 EDT

Sharp critical and subcritical Moser-Trudinger inequalities on complete and noncompact Riemannian manifolds.

TBA.

Carlos Kenig
(University of Chicago)
Apr 26 10:00 EDT

Wave maps into the sphere.

We will introduce wave maps, an important geometric flow, and discuss, for the case when the target is the sphere, the asymptotic behavior near the ground state (without symmetry) and recent results in the general case (under co-rotational symmetry) in joint work with Duyckaerts, Martel and Merle.

Yanyan Li
(Rutgers University)
Apr 19 09:00 EDT

Regular solutions of the stationary Navier-Stokes equations on high dimensional Euclidean space.

We study the existence of regular solutions of the incompressible stationary Navier-Stokes equations in n-dimensional Euclidean space with a given bounded external force of compact support. In dimensions n\le 5, the existence of such solutions was known. In this paper, we extend it to dimensions n\le 15. This is a joint work with Zhuolun Yang.

Jingzhi Tie
(University of Georgia)
Apr 05 10:00 EDT

CR analogue of Yau’s Conjecture on pseudo harmonic functions of polynomial growth.

Cheng and Yau derived the well-known gradient estimate for positive harmonic functions and obtained the classical Liouville theorem, which states that any bounded harmonic function is constant in complete noncompact Riemannian manifolds with nonnegative Ricci curvature. I will talk about the CR analogue of Yau’s conjecture. We need to derive the CR volume doubling property, CR Sobolev inequality, and mean value inequality. Then we can apply them to prove the CR analogue of Yau’s conjecture on the space consisting of all pseudoharmonic functions of polynomial growth of degree at most d in a complete noncompact pseudohermitian (2n+1)-manifold. As a by-product, we obtain the CR analogue of volume growth estimate and Gromov precompactness theorem.

Brian Street
(University of Wisconsin-Madison)
Mar 29 09:00 EDT

Maximal Hypoellipticity

In 1974, Folland and Stein introduced a generalization of ellipticity known as maximal hypoellipticity. This talk will be an introduction to this concept and some of the ways it generalizes ellipticity.

Wenxiong Chen 
(Yeshiva University)
Mar 22 10:00 EDT

Asymptotic radial symmetry, monotonicity, non-existence for solutions to fractional parabolic equations.

In this talk, we will consider nonlinear parabolic fractional equations We develop a systematical approach in applying an asymptotic method of moving planes to investigate qualitative properties of positive solutions for fractional parabolic equations. To this end, we derive a series of needed key ingredients such as narrow region principles, and various asymptotic maximum and strong maximum principles for antisymmetric functions in both bounded and unbounded domains. Then we illustrate how these new methods can be employed to obtain asymptotic radial symmetry and monotonicity of positive solutions in a unit ball and on the whole space. Namely, we show that no matter what the initial data are, the solutions will eventually approach to radially symmetric functions. We will also consider the entire positive solutions on a half space, in the whole space, and with indefinite nonlinearity. Monotonicity and nonexistence of solutions are obtained. This is joint work with P. Wang, Y. Niu, Y. Hu and L. Wu.

Man Wah Wong 
(York University)
Mar 15 10:00 EDT

Spectral Theory and Number Theory of the Twisted Bi-Laplacian.

We begin with the sub-Laplacian on the Heisenberg group and then the twisted Laplacian by taking its inverse Fourier transform with respect to the center of the group. The eigenvalues and the eigenfunctions of the twisted Laplacian are computed explicitly. Then we turn our attention to the product of the twisted Laplacian and its transpose, thus obtaining a fourth order partial differential operator dubbed the twisted bi-Laplacian. The connections between the spectral analysis of the twisted bi-Laplacian and Dirichlet divisors, the Riemann zeta function and the Dixmier trace are explained.

Enrique Zuazua
(Friedrich-Alexander-Universität)
Mar 08 09:00 EST

UNILATERAL BOUNDS FOR NONLINEAR SEMIGROUPS AND TIME-INVERSION.

Abstract: Some classical nonlinear semigroups arising in mechanics induce unilateral bounds on solutions. Hamilton–Jacobi equations and 1-d scalar conservation laws are classical examples of such nonlinear effects: solutions spontaneously develop one-sided Lipschitz or semi-concavity conditions. When this occurs the range of the semigroup is unilaterally bounded by a threshold. On the other hand, in practical applications, one is led to consider the problem of time-inversion, so to identify the initial sources that have led to the observed dynamics at the final time. In this lecture we shall discuss this problem answering to the following two questions: On one hand, to identify the range of the semigroup and, given a target, to characterize and reconstruct the ensemble of initial data leading to it. Illustrative numerical simulations will be presented, and a complete geometric interpretation will also be provided. We shall also present a number of open problems arising in this area and the possible link with reinforcement learning.

Manuel Del Pino
(University of Bath)
Mar 01 09:00 EST

Dynamics of concentrated vorticities in 2d and 3d Euler flows.

Abstract: A classical problem that traces back to Helmholtz and Kirchoff is the understanding of the dynamics of solutions to the 2d and 3d Euler equations of an inviscid incompressible fluid, when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent result on existence and asymptotic behaviour of these solutions. We describe, with precise asymptotics, interacting vortices and travelling helices. We rigorously establish the law of of motion of of “leapfrogging vortex rings”, originally conjectured by Helmholtz in 1858. This is joint work with Juan Davila, Monica Musso and Juncheng Wei.

Saikat Mazumdar
(Indian Institute of Technology Bombay)
Feb 22 09:00 EST

EXISTENCE RESULTS FOR THE HIGHER-ORDER Q-CURVATURE EQUATION.

Abstract: In this talk, we will obtain some existence results for the Q-curvature equation of arbitrary 2k-th order, where k \geq 1 is an integer, on a compact Riemannian manifold of dimension n \geq 2k + 1. This amounts to solving a nonlinear elliptic PDE involving the powers of Laplacian called the GJMS operator. The difficulty in determining the explicit form of this GJMS operator together with a lack of maximum principle complicates the issues of existence. This is a joint work with Jérôme Vètois (McGill University).

Yehuda Pinchover
(Technion -Israel Institute of Technology)
Feb 15 09:00 EST

On families of optimal Hardy-weights for linear second-order elliptic operators.

Abstract: We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator using a one-dimensional reduction. More precisely, we first characterize all optimal Hardy-weights with respect to one-dimensional subcritical Sturm-Liouville operators on (a,b), \infty  \leq  a < b  \leq  \infty, and then apply this result to obtain families of optimal Hardy inequalities for general linear second-order elliptic operators in higher dimensions. This is a joint work with Idan Versano.

Martin Dindos
(The University of Edinburgh)
Feb 08 09:00 EST

On p-ellipticity and connections to solvability of elliptic complex valued PDEs.

Abstract: The notion of an elliptic partial differential equation (PDE) goes back at least to 1908, when it appeared in a paper J. Hadamard. In this talk we present a recently discovered structural condition, called p-ellipticity, which generalizes classical ellipticity. It was co-discovered independently by Carbonaro and Dragicevic on one hand, and Pipher and myself on the other, and plays a fundamental role in many seemingly mutually unrelated aspects of the L^p theory of elliptic complex-valued PDE. So far, p-ellipticity has proven to be the key condition for: (i) convexity of power functions (Bellman functions) (ii) dimension-free bilinear embeddings, (iii) L^p-contractivity and boundedness of semigroups (P_t^A)_{t>0} associated with elliptic operators, (iv) holomorphic functional calculus, (v) multilinear analysis, (vi) regularity theory of elliptic PDE with complex coefficients. During the talk, I will describe my contribution to this development, in particular to (vi).

Yunfeng Zhang
(University of Connecticut)
Feb 01 09:00 EST

Schrödinger equations on compact globally symmetric spaces.

Abstract: Let M be a compact manifold of dimension d. Scale-invariant Strichartz estimates of the form \|e^{it\Delta}f\|_{L^p(I\times M)}\lesssim \|f\|_{H^{d/2-(d+2)/p}(M)} have only been proved for a few model cases of M, most of which are compact globally symmetric spaces. In this talk, we report that the above estimate holds true on an arbitrary compact globally symmetric space M equipped with the canonical Killing metric, for all p\geq 2+8/r, where r denotes the rank of M. As an immediate application, we provide local well-posedness results for nonlinear Schrödinger equations of polynomial nonlinearities of degree \beta\geq 4 on any compact globally symmetric space of large enough rank, in all subcritical spaces. We also discuss bilinear Strichartz estimates on compact globally symmetric spaces, and critical and subcritical local well-posedness results for the cubic nonlinearity.

Zhen-Qing Chen
(University of Washington)
Jan 25 10:00 EST

Stability of Elliptic Harnack Inequality.

Abstract: Harnack inequality, if it holds, is a useful tool in analysis and probability theory. In this talk, I will discuss scale invariant elliptic Harnack inequality for general diffusions, or equivalently, for general differential operators on metric measure spaces, and show that it is stable under form-comparable perturbations for strongly local Dirichlet forms on complete locally compact separable metric spaces that satisfy metric doubling property. Based on Joint work with Martin Barlow and Mathav Murugan.

Matthew Gursky
(University of Notre Dame)
Jan 18 09:00 EST

Extremal Eigenvalues of the conformal laplacian.

Abstract: I will report on joint work with Samuel Perez-Ayala in which we consider the problem of extremizing eigenvalues of the conformal laplacian in a fixed conformal class. This generalizes the problem of extremizing the eigenvalues of the laplacian on a compact surface. I will explain the connection of this problem to the existence of harmonic maps, and to nodal solutions of the Yamabe problem (first noticed by Ammann-Humbert).

Gilles Carron
(University of Nantes)
Jan 11 09:00 EST

Euclidean heat kernel rigidity.

Abstract: This is joint work with David Tewodrose (Bruxelles). I will explain that a metric measure space with Euclidean heat kernel are Euclidean. An almost rigidity result comes then for free, and this can be used to give another proof of Colding’s almost rigidity for complete manifold with non negative Ricci curvature and almost Euclidean growth.

Betsy Stovall
(University of Wisconsin-Madison)
Dec 14 10:00 EST

Fourier restriction to degenerate hypersurfaces.

Abstract: In this talk, we will describe various open questions and recent progress on the Fourier restriction problem associated to hypersurfaces with varying or vanishing curvature.

Almut Burchard
(University of Toronto)
Dec 14 09:00 EST

Rearrangement inequalities on spaces of bounded mean oscillation.

Abstract: Spaces of bounded mean oscillation (BMO) are relatively large function spaces that are often used in place of L^\infinity to do basic Fourier analysis. It is not well-understood how geometric properties of the underlying point space enters into the functional analysis of BMO. I will describe recent work with Galia Dafni and Ryan Gibara, where we take some steps towards geometric inequalities. Specifically, we show that the symmetric decreasing rearrangement in n-dimensions is bounded, but not continuous in BMO. The question of sharp bounds remains open.

Giovanna Citti
(University of Bologna)
Dec 07 09:00 EST

Degree preserving variational formulas for submanifolds.

Abstract: I present a joint work with M. Ritoré and G. Giovannardi related to an area functional for submanifolds of fixed degree immersed in a graded manifold. The expression of this area functional strictly depends on the degree of the manifold, so that, while computing the first variation, we need to keep fixed its degree. We will show that there are isolated surfaces, for which this type of degree preserving variations do not exist: they can be considered higher dimensional extension of the subriemannian abnormal geodesics.

Annalisa Baldi
(University of Bologna)
Nov 30 09:00 EST

Poincaré and Sobolev inequalities for differential forms in Euclidean spaces and Heisenberg groups.

Abstract: In this talk I present some recent results obtained in collaboration with B. Franchi and P. Pansu about Poincaré and Sobolev inequalities for differential forms in Heisenberg groups (some results are new also for Euclidean spaces). For L^p, p>1, the estimates are consequence of singular integral estimates. In the limiting case L^1, the singular integral estimates are replaced with inequalities which go back to Bourgain-Brezis and Lanzani-Stein in Euclidean spaces, and to Chanillo-Van Schaftingen and Baldi-Franchi-Pansu in Heisenberg groups. Also the case p=Q (Q is the homogeneous dimension of the Heisenberg group ) is considered.

Emmanuel Hebey
(Université de Cergy-Pontoise)
Nov 16 09:00 EST

Schrödinger-Proca constructions in the closed setting

Abstract: We discuss Schrödinger-Proca constructions in the context of closed manifolds leading to the Bopp-Podolsky-Schrödinger-Proca and the Schrödinger-Poisson-Proca systems. The goal is to present an introduction to these equations (how we build them, what do they represent) and then to present the result we got on these systems about the convergence of (BPSP) to (SPP) as the Bopp-Podolsky parameter goes to zero.

Ling Xiao
(University of Connecticut)
Nov 09 09:00 EST

Entire spacelike constant \sigma_{n-1} curvature in Minkowski space.

Abstract: We prove that, in the Minkowski space, if a spacelike, (n-1)-convex hypersurface M with constant \sigma_{n-1} curvature has bounded principal curvatures, then M is convex. Moreover, if M is not strictly convex, after an R^{n,1} rigid motion, M splits as a product M^{n-1}\times R. We also construct nontrivial examples of strictly convex, spacelike hypersurface M with constant \sigma_{n-1}curvature and bounded principal curvatures. This is a joint work with Changyu Ren and Zhizhang Wang.

Joshua Flynn
(University of Connecticut)
Nov 02 10:00 EST

Sharp Caffarelli-Kohn-Nirenberg Inequalities for Grushin Vector Fields and Iwasawa Groups.

Abstract: Caffarelli-Kohn-Nirenberg inequalities are established for the Grushin vector fields and for Iwasawa groups (i.e., the boundary group of a real rank one noncompact symmetric space). For all but one parameter case, this is done by introducing a generalized Kelvin transform which is shown to be an isometry of certain weighted Sobolev spaces. For the exceptional parameter case, the best constant is found for the Grushin vector fields by introducing Grushin cylindrical coordinates and studying the transformed Euler-Lagrange equation.

William Minicozzi
(MIT)
Oct 26 10:00 EDT

Mean curvature flow in high codimension.

Abstract: Mean curvature flow (MCF) is a geometric heat equation where a submanifold evolves to minimize its area. A central problem is to understand the singularities that form and what these imply for the flow. I will talk about joint work with Toby Colding on higher codimension MCF, where the flow becomes a complicated system of equations and much less is known.

Jiuyi Zhu
(Louisiana State University)
Oct 19 10:00 EDT

The bounds of nodal sets of eigenfunctions.

Abstract: Motivated by Yau’s conjecture, the study of the measure of nodal sets (Zero level sets) for eigenfunctions is interesting. We investigate the measure of nodal sets for Steklov, Dirichlet and Neumann eigenfunctions in the domain and on the boundary of the domain. For Dirichlet or Neumann eigenfunctions in the analytic domains, we show some sharp upper bounds of nodal sets which touch the boundary. We will also discuss some upper bounds of nodal sets for eigenfunctions of general eigenvalue problems. Furthermore, some sharp doubling inequalities and vanishing order are obtained. Part of the talk is based on joint work with Fanghua Lin.

Cristian Cazacu
(University of Bucharest)
Oct 12 09:00 EDT

Optimal constants in Hardy and Hardy-Rellich type inequalities.

Abstract: In this talk we discuss Hardy and Hardy-Rellich type inequalities, so important in establishing useful properties for differential operators with singular potentials and their PDEs. We recall some well-known and recent results and present some new extensions. We analyze singular potentials with one or various singularities. The tools of our proofs are mainly based on the method of supersolutions, proper transformations and spherical harmonics decomposition. We also focus on the best constants and the existence/nonexistence of minimizers in the energy space. This presentation is partially supported by CNCS-UEFISCDI Grant No. PN-III-P1-1.1-TE-2016-2233.

Laurent Saloff-Coste
(Cornell University)
Oct 5 10:00 EDT

Heat kernel on manifolds with finitely many ends.

Abstract: For over twenty years A. Grigor’yan and the speaker have studied heat kernel estimates on manifolds with a finite number of nice ends. Despite these efforts, question remains. In this talk, after giving an overview of what the problem is and what we know, the main difficulty will be explained and recent progresses involving joint work with Grigor’yan and Ishiwata will be explained. They provide results concerning Poincaré inequality in large central balls on such manifold.

Jungang Li
(Brown University)
Sep 28 09:00 EDT

Higher order Brezis-Nirenberg problems on hyperbolic spaces.

Abstract: The Brezis-Nirenberg problem considers elliptic equations whose nonlinearity is associated with critical Sobolev exponents. In this talk we will discuss a recent progress on higher order Brezis-Nirenberg problem on hyperbolic spaces. The existence of solutions relates closely to the study of higher order sharp Hardy-Sobolev-Maz’ya inequalities, which is due to G. Lu and Q. Yang. On the other hand, we obtain a nonexistence result on star-shaped domains. In addition, with the help of Green’s function estimates, we apply moving plane method to establish the symmetry of positive solutions. This is a joint work with Guozhen Lu and Qiaohua Yang..

Phan Thành Nam
(LMU Munich)
Sep 21 09:00 EDT

Lieb-Thirring inequality with optimal constant and gradient error term.

Abstract: In 1975, Lieb and Thirring conjectured that the kinetic energy of fermions is not smaller than its Thomas-Fermi (semiclassical) approximation, at least in three or higher dimensions. I will discuss a rigorous lower bound with the sharp semiclassical constant and a gradient error term which is normally of lower order in applications. The proof is based on a microlocal analysis and a variant of the Berezin-Li-Yau inequality. This approach can be extended to derive an improved Lieb-Thirring inequality for interacting systems, where the Gagliardo-Nirenberg constant appears in the strong coupling limit.

Jean Dolbeault
(Université Paris-Dauphine)
Sep 14 09:00 EDT

Stability in Gagliardo-Nirenberg inequalities.

Abstract: Optimal constants and optimal functions are known in some functional inequalities. The next question is the stability issue: is the difference of the two terms controlling a distance to the set of optimal functions ? A famous example is provided by Sobolev’s inequalities: in 1991, G. Bianchi and H. Egnell proved that the difference of the two terms is bounded from below by a distance to the manifold of the Aubin-Talenti functions. They argued by contradiction and gave a very elegant although not constructive proof. Since then, estimating the stability constant and giving a constructive proof has been a challenge. This lecture will focus mostly on subcritical inequalities, for which explicit constants can be provided. The main tool is based on entropy methods and nonlinear flows. Proving stability amounts to establish, under some constraints, a version of the entropy – entropy production inequality with an improved constant. In simple cases, for instance on the sphere, rather explicit results have been obtained by the « carré du champ » method introduced by D. Bakry and M. Emery. In the Euclidean space, results based on constructive regularity estimates for the solutions of the nonlinear flow and corresponding to a joint research project with Matteo Bonforte, Bruno Nazaret, and Nikita Simonov will be presented.

Yi Wang
(Johns Hopkins University)
Sep 07 09:00 EDT

Rigidity of local minimizers of the \sigma_k functional.

Abstract: In this talk, I will present a result on the rigidity of local minimizers of the functional \int \sigma_2+ \oint H_2 among all conformally flat metrics in the Euclidean (n + 1)-ball. We prove the metric is flat up to a conformal transformation in some (noncritical) dimensions. We also prove the analogous result in the critical dimension n + 1 = 4. The main method is Frank-Lieb’s rearrangement-free argument. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. I will also discuss a nonsharp Sobolev trace inequality. This is joint work with Jeffrey Case.

Pengfei Guan
(McGill University)
Aug 31 10:00 EDT

A mean curvature type flow and isoperimetric problem in warped product spaces.

Abstract: We will discuss a mean curvature type flow with the goal to solve isoperimetric problem. The flow is induced from the variational properties associated to conformal Killing fields. Such flow was first introduced in space forms in a previous joint work with Junfang Li, where we provided a flow approach to the classical isoperimetric inequality in space form. Later, jointly with Junfang Li and Mu-Tao Wang, we considered the similar flow in warped product spaces with general base. Under some natural conditions, the flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. Furthermore, the regularity and convergence of the flow can be established, thereby the isoperimetric problem in warped product spaces can be solved. The flow serves as an interesting way to achieve the optimal solution to the isoperimetric problem.

Rupert Frank
(CalTech)
Aug 24 10:30

REVERSE HARDY–LITTLEWOOD–SOBOLEV INEQUALITIES.

Abstract: We are interested in a new family of reverse Hardy–Littlewood–Sobolev inequalities which involve a power law kernel with positive exponent and a Lebesgue exponent <1. We characterize the range of parameters for which the inequality holds and present results about the existence of optimizers. A striking open question is the possibility of concentration of a minimizing sequence. This talk is based on joint work with J. Carrillo, M. Delgadino, J. Dolbeault and F. Hoffmann.

Juncheng Wei
University of British Columbia)
Aug 17 10:00 EDT

Rigidity Results for Allen-Cahn Equation.

Abstract: I will discuss two recent rigidity results for Allen-Cahn: the first is Half Space Theorem which states that if the nodal set lies above a half space then it must be one-dimensional. The second result is the stability of Cabre-Terra saddle solutions in R^8, R^{10} and R^{12}.

Fengbo Hang
(New York University)
Aug 10 10:00 EDT

Concentration compactness principle in critical dimensions revisited.

Abstract: Concentration compactness principle for functions in W^{1,n}_0 on a n-dimensional domain was introduced by Lions in 1985 with the Moser-Trudinger inequality in mind. We will discuss some further refinements after Cerny-Cianchi-Hencl’s improvement in 2013. These refinements unifiy the approach for n=2 and n>2 cases and work for higher order or fractional order Sobolev spaces as well. They are motivated by and closely related to the recent derivation of Aubin’s Moser-Trudinger inequality for functions with vanishing higher order moments on the standard 2-sphere (one may see math.sjtu.edu.cn/conference/2020p&g/videos/20200707_FengboHang_M1.html for that part).

Dongmeng Xi
(NYU)
Aug 03 10:00 EDT

An isoperimetric type inequality via a modified Steiner symmetrization scheme.

Abstract: We establish an affine isoperimetric inequality using a symmetrization scheme that involves a total of 2n elaborately chosen Steiner symmetrizations at a time. The necessity of this scheme, as opposed to the usual Steiner symmetrization, will be demonstrated with an example. This is a joint work with Dr. Yiming Zhao.

Yiming Zhao
(MIT)
Aug 03 09:00 EDT

Reconstruction of convex bodies via Gauss map.

Abstract: In this talk, we will discuss the Gauss image problem, a problem that reconstructs the shape of a convex body using partial data regarding its Gauss map. In the smooth category, this problem reduces to a Monge-Ampere type equation on the sphere. But, we will use a variational argument that works with generic convex bodies. This is joint work with Károly Böröczky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang.

Xavier Cabre
(ICREA and UPC (Barcelona))
Jul 27 09:00 EDT

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.

Andrea Malchiodi
(Scuola Normale Superiore)
Jul 20 10:00 EDT

On the Sobolev quotient in sub-Riemannian geometry.

Abstract: We consider three-dimensional CR manifolds, which are modelled on the Heisenberg group. We introduce a natural concept of “mass” and prove its positivity under the condition that the scalar curvature is positive and in relation to their (holomorphic) embeddability properties. We apply this result to the CR Yamabe problem, and we discuss extremality of Sobolev-type quotients, giving some counterexamples for “Rossi spheres”. This is joint work with J.H.Cheng and P.Yang.

William Beckner
(University of Texas at Austin)
Jul 20 09:00 EDT

Symmetry in Fourier Analysis – Heisenberg to Stein-Weiss.

Abstract: Embedded symmetry within the Heisenberg group is used to couple geometric insight and analytic calculation to obtain a new sharp Stein-Weiss inequality with mixed homogeneity on the line of duality. SL(2,R) invariance and Riesz potentials define a natural bridge for encoded information that connects distinct geometric structures. Insight for Stein-Weiss integrals is gained from vortex dynamics, embedding on hyperbolic space, and conformal geometry. The intrinsic character of the Heisenberg group makes it the natural playing field on which to explore the laws of symmetry and the interplay between analysis and geometry on a manifold.