Schedule for: 16w5047 - Geometric and Analytic Inequalities

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk we are interested in the minimization of the so-called elastic energy of a convex planar domain (which is the $L^2$ norm of the curvature of its boundary) with a constraint on its in radius. We prove existence of a minimizer and characterize it. It turns out that the solution is not a disk but a domain whose boundary is defined by the means of elementary functions.

For the Dirichlet problem on the k- Hessian equation, Caffarelli-Nirenberg-Spruck (1986) obtained the existence of the admissible classical solution when the smooth domain is strictly k-1 convex in R^n. In this talk, we prove the existence of a classical admissible solution to a class of Neumann boundary value problems for k Hessian equations in strictly convex domain in R^n， this was asked by Prof. N. Trudinger in 1987. The methods depends upon the establishment of a priori derivative estimates up to second order. This is the joint work with Qiu Guohuan.

About the Robin-Steklov spectrum

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Let $\Sigma$ be a $k$-dimensional complete proper minimal submanifold in the Poincar\'{e} ball model $B^n$ of hyperbolic geometry. If we consider $\Sigma$ as a subset of the unit ball $B^n$ in Euclidean space, one can measure the Euclidean volumes of the given minimal submanifold $\Sigma$ and the ideal boundary $\partial_\infty \Sigma$, respectively. Using this concept, we prove an optimal linear isoperimetric inequality which gives the classical isoperimetric inequality under geometric assumption. By proving the monotonicity theorem for such $\Sigma$, we further obtain a sharp lower bound for the Euclidean volume, which is an extension of Fraser-Schoen and Brendle's recent results to hyperbolic space. Moreover we introduce the M\"{o}bius volume of $\Sigma$ in $B^n$ to prove an isoperimetric inequality via the M\"{o}bius volume for $\Sigma$. Most parts of this talk is based on the paper "Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space" which was published in Crelle's Journal). This is a joint work with Sung-Hong Min.

We consider a class of quasilinear operators on a bounded domains and address the question of optimizing the rst eigenvalue with respect to the boundary conditions, which are of the Robin-type. We describe the optimizing boundary conditions and establish upper and lower bounds on the respective maximal and minimal eigenvalue. This is a joint work with Hynek Kovark (Brescia) and Francesco Della Pietra (Napoli Federico II).

Last year, Xinan Ma and me gave a existence result about the Neumann problems for Hessian equations. In this talk, we proceed further to study classical Neumann problems for Hessian equations. We prove here the existence results of classical Neumann problems under the uniformly convex domain in R^n. Then we use the solutions of these boudary problems to give a new proof of a family of Alexandrov-Fenchel inequalities raiesd from convex geometry. This geometric application is motived from Reilly. It is a work joint with Chao Xia.

We show some sharp bounds for the optimal constant in Poincar\'e inequalities for functions having zero-mean (in a suitable sense), when we restrict to consider bounded convex subsets of $\mathbb{R}^N$. We will show that in this class, sharp estimates are possible in terms of simple geometric quantities. \par We will also present a more general result obtained by means of Optimal Transport techniques, valid for general convex sets (not necessarily bounded). \par Part of the results of this talk are contained in joint works with Carlo Nitsch and Cristina Trombetti and with Filippo Santambrogio.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In the talk I will talk about the transversal Yamabe problem: On a Riemannian foliation is there a basic conformal metric with constant transversal scalar curvature? Related to this problem there is an optimal transversal Sobolev inequality.

A class of isoperimetric problems on \(R^N\) with respect to weights that are powers of the distance to the origin is presented. For instance it is shown that, if \(k\in [0; 1]\), then among all smooth sets in \(R^N\) with fixed Lebesgue measure, \(\int_{R^N}|x|^\alpha H_{N-1}(dx)\) achieves its minimum for a ball centered at the origin. These results also imply a weighted Polya-Szego principle. In turn, radiality of optimizers in some Caarelli-Kohn-Nirenberg inequalities is established, and sharp bounds for eigenvalues of some nonlinear problems is obtained.

The classic Weyl's problem is the isometric embedding of (S^2, g) with positive Gauss curvature into Euclidean space. Motivated by quasi-local masses in Genaral Relativity, we consider the analogous problem into warped product space with natural curvature assumption. By a careful study adopting Heinz's C^2 interior estimate, we obtain the curvature estimate for the embedding. This allows us to reprove Pogorelov's isometric embedding under the condition that g\in C^2, \alpha. Together with recent work by Li-Wang, we prove some isometric embedding in Riemannian manifold.

Reilly’s formula is the integral version of Bochner's formula for manifolds with boundary. It has numerous applications when Ricci curvature is nonnegative. In this talk, I will present two kinds of generalized Reilly type formulas for manifolds with boundary which are applicable for manifolds satisfying either a sectional curvature lower bound or a sub-static condition. In particular, we use it to reprove Brendle’s result on Heintze-Karcher type inequality for warped product manifolds. Moreover, we give several new geometric inequalities using these formulas. The talk is a report of joint works with Guohuan Qiu, and separately with Junfang Li.

Let $\Omega $ be a convex, possibly unbounded, domain of $\mathbb{R}^{N}$ and denote by $\mu _{1}(\Omega )$ the first nontrivial Neumann eigenvalue of the Hermite operator in $\Omega $. It is known that $$ \mu _{1}(\Omega )\geq 1. $$ Such an inequality is sharp since equality sign holds if $\Omega $ is any $N$-dimensional strip. Some examples show that it does not hold true, in general, if one removes the assumption on the convexity of $\Omega $. We study the equality case in the plane. We show that, under some further assumptions on $\Omega $, if $\mu_{1}(\Omega )=1$ then $\Omega $ is any 2-dimensional strip. The study of the equality case requires, among other things, an asymptotic analysis of the eigenvalues of the Hermite operator in thin domains.

In this talk, by using the inverse curvature flow we first establish an optimal Sobolev type inequality for hypersurfaces in H^n. As an application, we obtain Alexandrov-Fenchel inequalities for curvature integrals. Secondly, I will talk about the Alexandrov-Fenchel inequalities with weight in H^n, which is related to the recent study of the Penrose inequality for various mass. This is a joint work with Yuxin Ge and Guofang Wang.

We consider the solution of the non linear evolution equation \(u_t =
\Delta_K u\) in \(\Omega\times(0;+\infty)\), \(u = 0\) initially and \(u = 1\) on the boundary. Here \(\Delta_
K u(x) = div (h_K(Du)Dh_K(Du))\) is the Finsler Laplacian associated with the support function of a convex body, \(h_K\). When K is the Euclidean ball it has been shown that a solution u has a timeinvariant level surface \(\Gamma\) only if \(\Omega\) is a ball. In the talk we will discuss the possibility to extend such a result to the anisotropic setting and we will show some preliminary results.

In this talk, we present some results on the critical points to the generalized Ginzburg-Landau equations in dimensions n≥ 3 which satisfy a suitable energy bound, but are not necessarily energy-minimizers. When the parameter in the equations tend to zero, such solutions are shown to converge to singular n-harmonic maps into spheres which are conformally invariant, and the convergence is strong away from a finite set consisting 1) of the infinite energy singularities of the limiting map, and 2) of points where bubbling off of finite energy n-harmonic maps takes place. The latter case is specific to dimensions greater than 2. This is a joint work with E. Sandier et P. Zhang.

We consider the problem of optimally insulating a given domain $\Omega$ of ${\mathbb R}^d$; this amounts to solve a nonlinear variational problem, where the optimal thickness of the insulator is obtained as the boundary trace of the solution. We deal with two different criteria of optimization: the first one consists in the minimization of the total energy of the system, while the second one involves the first eigenvalue of the related differential operator. Surprisingly, the second optimization problem presents a symmetry breaking in the sense that for a ball the optimal thickness is nonsymmetric when the total amount of insulator is small enough. In the last section we discuss the shape optimization problem in which $\Omega$ is allowed to vary too.

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.

In this talk we investigate the quantitative isoperimetric inequality in the plane. In particular we prove the existence of a set $\Omega$, different from a ball, which minimizes the shape functional $\delta(\Omega)/\lambda^2(\Omega)$, where $\delta$ is the isoperimetric deficit and $\lambda$ the Fraenkel asymmetry. Some new properties of the optimal set are shown.

We discuss the problem of boundedness of the Hardy-Littlewood maximal operator between weighted \(L^p\) spaces with different weights. We focus on the approach via so called "bump conditions". These conditions are strengthenings of the Muckenhoupt \(A_p\)-condition and are known to be sufficient for the two-weighted maximal inequality. We show that they are in general not necessary for this inequality to be true.

We consider the variational problem\[ \min{\left\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega)\;:\;\Omega\subset\R^d, \;|\Omega|=1\right\}}, \] where the variable is the domain \(\Omega,\, |\cdot|\) denotes the Lebesgue measure and the cost functional is the sum of the first \(k\) Dirichlet eigenvalues on \(\Omega\). We prove that the optimal sets have \(C^{1,\alpha}\) regular boundary up to a set of zero \(\mathcal{H}^{d-1}-\)measure. This is strongly related to the regularity of the free boundary \(\partial \{|U|>0\}\) of the local minima of the functional \[ H^{1}_{loc}(\R^d,\R^k)\ni W\mapsto \int |\nabla W|^2+|\{|W|>0\}|, \] on which we will focus most of our attention.

We give a survey of recent results on optimal trace embeddings for functions from Sobolev spaces built upon rearrangement-invariant spaces. We describe the optimal function space for which every function from the corresponding Sobolev space admits a trace. We establish certain reduction principle which will enable us to obtain a necessary and sufficient condition for a trace embedding in terms of an action of a one-dimensional integral operator. We give a reasonably explicit characterization of the optimal function space in a trace embedding. We discuss the applicability of techniques based on interpolation and iteration for trace problems.

A standard form of the Korn inequality amounts to an estimate for the $L^p$ norm ($1 (TCPL 201)

TBA

The Brunn-Minkowski inequality and the Minkowski problem  are centerpieces of the classical Brunn-Minkowski theory of convex bodies. The logarithmic Brunn-Minkowski inequality and the logarithmic Minkowski problem are recent proposed  objects of study. The logarithmic Brunn-Minkowski inequality  is stronger than the classical Brunn-Minkowski inequality,  and the logarithmic Minkowski problem requires solving a  more difficult Monge-Ampere equation --- one which requires  certain measure concentration for the existence of solutions.  This will be explained in the talk.

With Berck and Bernig we introduced an invariant associated to a pointed convex set which is the closest projective analogue of a valuation. Its similarity with the centro-affine area lead us to call it centro-projective area. With Deane Yang we recently proved that this invariant is bounded from above by its value on a euclidean sphere, with  equality if and only if the convex set is an ellipsoid. The links with the so-called Hilbert geometries will be explained.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.