Schedule for: 17w5047 - Geometric Properties of Local and non-Local PDEs

A welcome drink will be served at the hotel.

We derive the Moser-Trudinger-Onofri inequalities on the 2-sphere and the 4-sphere as the limiting cases of the fractional power Sobolev inequalities on the same spaces, and justify our approach as the dimensional continuation argument initiated by Thomas P. Branson.

We study singularity formation in the harmonic map flow from a two dimensional domain into S2 and show that for convenient initial conditions the flow develops a type II singularity at multiple points in finite time. We also discuss the stability of this phenomenon. This is joint work with Manuel del Pino (Universidad de Chile) and Juncheng Wei (University of British Columbia).

The Willmore energy of a surface is a conformal measure of its failure to be conformally spherical. In physics the energy is variously called the bending energy, or rigid string action. In both geometric analysis and physics it has been the subject of great recent interest. We explain that its Euler-Lagrange equation is an extremely interesting equation in conformal geometry: the energy gradient is a fundamental curvature that is a scalar-valued hypersurface analogue of the Bach tensor (of dimension 4) of intrinsic conformal geometry. We next show that that these surface conformal invariants, i.e. the Willmore energy and its gradient (the Willmore invariant), are the lowest dimensional examples in a family of similar invariants in higher dimensions. A generalising analogue of the Willmore invariant arises directly in the asymptotics associated with a singular Yamabe problem on conformally compact manifolds. A result of Graham shows that an energy giving this (as gradient with respect to variation of hypersurface embedding) arises in a different way as a so-called "anomaly term" in a related renormalised volume expansion. We show that this anomaly term is, in turn, the integral of a local Q-curvature quantity for hypersurfaces that generalises Branson's Q-curvature by including coupling to the (extrinsic curvature) data of the embedding. This is associated to concormally invariant Laplacian power operators related to the celbrated GJMS operators, but which are coupled to the extrinsic curvature data of the embedding. This is joint work with Andrew Waldron arXiv:1506.02723, arXiv:1603.07367, arXiv:1611.08345

We deal with functions satisfying \begin{equation}\label{P_C} \begin{cases} (-\Delta)^s u_s=0 & \mathrm{in}\quad C, \\ u_s=0 & \mathrm{in}\quad \mathbb{R}^n\setminus C, \end{cases} \end{equation} where $s\in(0,1)$ and $C$ is a given cone on $\mathbb R^n$ with vertex at zero. We are mainly concerned with the case when $s$ approaches $1$. These functions are involved in the study of the nodal regions in the case of optimal partitions and other free boundary problems and play a crucial role in the extension of the Alt-Caffarelli-Friedman monotonicity formula to the case of fractional diffusions. This is a joint work with Giorgio Tortone and Stefano Vita.

The $k$-Hessian operator $\sigma_k$ is the $k$-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the $k$-Hessian equation $\sigma_k(D^2 u)=f$ with Dirichlet boundary condition $u=0$ is variational; indeed, this problem can be studied by means of the $k$-Hessian energy $\int -u \sigma_k(D^2 u)$. We construct a natural boundary functional which, when added to the $k$-Hessian energy, yields as its critical points solutions of $k$-Hessian equations with general non-vanishing boundary data. As a consequence, we prove a sharp Sobolev trace inequality for $k$-admissible functions $u$ which estimates the $k$-Hessian energy in terms of the boundary values of $u$. This is joint work with Jeffrey Case.

I will report on recent work with V. Millot and K. Wang on the singular limit for a fractional Allen-chan equation leading to stationary nonlocal minimal surfaces. I will introduce these latter concepts and will prove the convergence result, based on a deep Geometric Theory argument from Marstrand.

Abstract: I will begin with a brief overview of the existence question for conformally compact Einstein manifolds with prescribed conformal infinity. After stating the seminal result of Graham-Lee, I will mention a recent non-existence result (joint with Qing Han) for certain conformal classes on the 7-dimensional sphere. I will then discuss some ongoing work (with Gabor Szekelyhidi) on a version of "local existence" of Poincare-Einstein metrics.

In 4-d conformal geometry there is the notion of Q-curvature whose integral is a global invariant, similarly in CR 3-d, there is the analogous notion of Q-prime curvature. I will explain an elementary argument behind the result that under the positivity assumption of the Yamabe invariant, these curvature integrals are extremized by the standard sphere.

bout 15 years ago, Bourgain and Brezis discovered astonishing Sobolev style inequalities at the end-point where the classical Sobolev embedding theorem fails. In this talk we will extend these inequalities to Riemannian symmetric spaces of non-compact type of any rank and also present applications to Strichartz inequalities for wave and Schrodinger equations, incompressible Navier-Stokes flow in 2D with prescribed vorticity and the Maxwell equations for Electromagnetism. These results have been obtained with Jean Van Schaftingen and Po-lam Yung.

The aim of this talk is to define conformal operators that arise from an extension problem of co-dimension two. To this end we interpret and extend results of representation theory from a purely analytic point of view. In the first part of the talk I will give definitions and interpretations of the fractional Laplacian and the conformal fractional Laplacian in the general framework of representation theory on symmetric spaces and also from the point of view of scattering operators in conformal geometry. In the second part of the talk I will show constructions of boundary operators with good conformal properties that generalise the fractional Laplacian in $\mathbb R^n$ using an extension problem in which the boundary is of co-dimension two. Then we extend these results to more general manifolds that are not necessarily symmetric space

The question addressed here is how fast a front will propagate when a line, having a strong diffusion of its own, exchanges mass with a reactive medium. More precisely,we wish to know how much the diffusion on the line will affect the overall front propagation. This setting was proposed (collaboration with H. Berestycki and L. Rossi) as a model of how biological invasions can be enhanced by network transportations. In a previous series of works, we were able to show that the line could speed up propagation indefinitely with its diffusivity. For that, we used a special type of nonlinearity that allowed the reduction of the problem to explicit computations. In the work presented here, the reactive medium is governed by nonlinearity that does not allow explicit computations anymore. We will explain how propagation speed-up still holds. In doing so, we will discuss a new transition phenomenon between two speeds of different orders of magnitude. Joint work with L. Dietrich.

We discuss a classification result for entire solutions of a quasi-linear Liouville equation in $R^n$ involving the n-Laplace operator and an exponential nonlinearity. We first review the semilinear case in $R^2$ where three alternative approaches are available, and we then discuss the quasi-linear case $n>2$.

We construct some solutions for the fractional Yamabe problem with isolated singularities, problem which arises in conformal geometry, $$ (-\Delta)^\gamma u= c_{n, {\gamma}}u^{\frac{n+2\gamma}{n-2\gamma}}, u>0 \ \mbox{in}\ {\mathbb R}^n \backslash \Sigma.$$ The fractional curvature, a generalization of the usual scalar curvature, is defined from the conformal fractional Laplacian, which is a non-local operator constructed on the conformal infinity of a conformally compact Einstein manifold. When the singular set $\Sigma$ is composed by one point, some new tools for fractional order ODE can be applied to show that a generalization of the usual Delaunay solves the fractional Yamabe problem with an isolated singularity at $\Sigma$. When the set $\Sigma$ is a finite number of points, using gluing methods, we will provide a solution for the fractional Yamabe problem with singularities at $\Sigma$. In order to preserve the non-locality of the problem, we need to glue infinitely many bubbles per point removed. This seems to be the first time that a gluing method is successfully applied to a non-local problem. This is a joint work with Weiwei Ao, Mar\'ia del Mar Gonz\'alez and Juncheng Wei.

We consider the problem of the existence of positive solutions with prescribed isolated singularities of the fractional Yamabe problem. Near each singular point, these solutions are approximated by the Delaunay-type singular solution which has been studied recently by De la Torre, Del Pino, Mar Gonzalez and J.C. Wei. Away from the singular points, these solutions are approximated by the summation of the Green's function. This result is the analogous result for the classical Yamabe problem studied by Mazzeo and Pacard (1999). This is a joint work with De la Torre, Mar Gonzalez and J.C. Wei.

Let $(X^{n+1}, g^+)$ be an $(n+1)$-dimensional asymptotically hyperbolic manifold with a conformal infinity $(M^n, [h])$. The fractional Yamabe problem consists in finding a metric in the conformal class $[h]$ whose fractional scalar curvature is constant. In this talk, I will present some recent results concerning existence of solutions to the fractional Yamabe problem, and also properties of compactness and non compactness of its solution set, in comparison with what is known in the classical case. These results are in collaboration with Seunghyeok Kim and Juncheng Wei.

In this talk, I will discuss the existence and multiplicity of blowing-up solutions for linear perturbation of Yamabe problem.

We consider the remaining unsettled cases in the problem of existence of positive solutions for the Dirichlet value problem $L_\gamma u-\lambda u=\frac{u^{2^*(s)-1}}{|x|^s}$ on a smooth bounded domain $\Omega$ in $\mathbb{R}^n$ ($n\geq 3$) having the singularity $0$ in its interior. Here $\gamma <\frac{(n-2)^2}{4}$, $0\leq s <2$, $2^*(s):=\frac{2(n-s)}{n-2}$ and $0\leq \lambda <\lambda_1(L_\gamma)$, the latter being the first eigenvalue of the Hardy-Schr\"odinger operator $L_\gamma:=-\Delta -\frac{\gamma}{|x|^2}$. The higher dimensional case (i.e., when $\gamma \leq \frac{(n-2)^2}{4}-1$) has been settled sometime ago. In this paper we deal with the case when $ \frac{(n-2)^2}{4}-1<\gamma <\frac{(n-2)^2}{4}$. If either $s>0$ or $\{s=0$ and $\gamma > 0\}$, we show that a solution is guaranteed by the positivity of the ``Hardy-singular internal mass" of $\Omega$, a notion that we introduce herein. On the other hand, the classical positive mass theorem is needed for when $s=0$, $\gamma \leq 0$ and $n=3$, which in this case is the critical dimension. We will also discuss some extensions of this work in the nonlocal setting of the fractional Laplacian. This is joint work with Nassif Ghoussoub (UBC, Vancouver).

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^{N}$ which contains a ball of radius $R$ centered at the origin, $N\geq3.$ Under suitable symmetry assumptions, for each $\delta\in(0,R),$ we establish the existence of a sequence $(u_{m,\delta})$ of nodal solutions to the critical problem% \[ \left\{ \begin{array} [c]{ll}% -\Delta u=|u|^{2^{*}-2}u & \text{in }\Omega_{\delta}:=\{x\in\Omega :\left\vert x\right\vert >\delta\},\\ u=0 & \text{on }\partial\Omega_{\delta}, \end{array} \right. \] where $2^{*}:=\frac{2N}{N-2}$ is the critical Sobolev exponent. We show that, if $\Omega$ is strictly starshaped then, for each $m\in\mathbb{N},$ the solutions $u_{m,\delta}$ concentrate and blow up at $0,$ as $\delta \rightarrow0,$ and their limit profile is a tower of nodal bubbles, i.e., it is a sum of rescaled nonradial sign-changing solutions to the limit problem% \[ \left\{ \begin{array} [c]{c}% -\Delta u=|u|^{2^{*}-2}u,\\ u\in D^{1,2}(\mathbb{R}^{N}), \end{array} \right. \] centered at the origin. This is joint work with Jorge Faya (Universidad de Chile) and Filomena Pacella (Universit\`{a} "La Sapienza" di Roma).

Abstract: In this talk we discuss a geometric variational problem with inhibitory long range interaction. It is a ternary system originally proposed to model triblock copolymers. There exists a morphological phase of a double bubble assembly as a stable stationary point of the variational problem. While the locations of the double bubbles in the assembly are determined in an earlier analytical result, the directions of the double bubbles are studied by a recent numerical computation. One of the conditions for the existence of the double bubble assembly is that the two by two nonlocal interaction matrix parameter is positive definite and a bound is assumed on the ratio of the two eigenvalues of the matrix. A more complete study of the interaction matrix shows that the double bubble assembly may lose to a disc assembly as the two eigenvalues become less comparable. When the matrix becomes indefinite, there appears a stationary disc assembly whose primary structure is the microscopic disc. It also has a secondary structure that the discs of one type are separated from discs of the second type by a macroscopic interface.

In joint work with David Jerison we study the compactness of the space of solutions to the one-phase free boundary problem in the disk, whose positive phase is of fixed genus. We describe the local structure of the free boundary and obtain rigidity estimates on its shape. Via a correspondence due to Traizet, our results are direct counterparts to theorems by Colding and Minicozzi for minimal surfaces.

In this talk, we will look at the question of existence of blowing-up solutions for smooth perturbations of critical elliptic second-order equations on a closed manifold. I will present new existence results in situations where the limit equation is different from the Yamabe equation. I will also discuss the special role played by the dimension 6 in this case. This is a joint work with Frederic Robert.

The KP-I equation \[ \partial_{x}\left( \partial_{t}u+\partial_{x}^{3}u+3\partial_{x}\left( u^{2}\right) \right) -\partial_{y}^{2}u=0 \] has a lump solution of the form $Q\left( x-t,y\right) ,$ where \[ Q\left( x,y\right) =Q\left( x,y\right) =4\frac{y^{2}-x^{2}+3}{\left( x^{2}+y^{2}+3\right)^{2}}. \] We show that $Q$ is nondegenerate in the following sense: Suppose $\phi$ is a smooth solution to the equation \[ \partial_{x}^{2}\left( \partial_{x}^{2}\phi-\phi+6Q\phi\right) -\partial _{y}^{2}\phi=0. \] Assume \[ \phi\left( x,y\right) \rightarrow0,\text{ as }x^{2}+y^{2}\rightarrow+\infty. \] Then $\phi=c_{1}\partial_{x}Q+c_{2}\partial_{y}Q,$ for certain constants $c_{1},c_{2}.$

In this talk, we focus on the evolution of graphs in the curvature flows and existence of translating solitons. First we discuss preservation of the graph, curvature estimates, long time existence, and the evolution of the support of the height function. And then we also discuss the Hamilton's question on the existence of cigar-type translating solitons with flat spots.

Given a compact Riemannian manifold $(M; g)$ without boundary of dimension $m\geq 3$ and under some symmetry assumptions, we establish the existence and multiplicity of positive and sign changing solutions to the following Yamabe type equation $$ -div_g(a \nabla u) + bu = c|u|^{2^{*}-2} u \ \mbox{ on} \ M $$ where $div_g$ denotes the divergence operator on $(M; g)$, $a; b $ and $c$ are smooth functions with a and $c$ positive, and $2^{*}= \frac{2m}{m-2}$ denotes the critical Sobolev exponent. In particular, if $R_g$ denotes the scalar curvature, we give some examples where the Yamabe equation $$ -\frac{4 (m-1)}{m-2} \Delta_g u+ R_g u= \kappa u^{2^{*}-2} \ \mbox{on} \ M$$ admits an infinite number of sign changing solutions. We also study the lack of compactness of these problems in a symmetric setting and how the symmetries restore it at some energy levels. This allows us to use a suitable variational principle to show the existence and multiplicity of such solutions. This is joint work with Monica Clapp.