Related papers: Convex duality for principal frequencies

Firstly, we derive in dimension one a new covariance inequality of $L_{1}-L_{\infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Secondly, we generalize our result to $L_{p}-L_{q}$…

In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in…

We present a fractional counterpart of a generalized Kohler-Jobin inequality, showing that, among all bounded, open sets $\Omega\subset \mathbb{R}^N$ with Lipschitz boundary, having the same fractional torsional rigidity, the first…

This work showcases level set estimates for weak solutions to the $p$-Poisson equation on a bounded domain, which we use to establish Lebesgue space inclusions for weak solutions. In particular we show that if $\Omega\subset\mathbb{R}^n$ is…

We consider some fundamental constants from the point of view of the duality symmetry. Our analysis of duality is focused on three issues: the maximum radiated power of gravitational waves, the cosmological constant, and the magnetic…

In this paper we explore the role of duality principles within the problem of rotation averaging, a fundamental task in a wide range of computer vision applications. In its conventional form, rotation averaging is stated as a minimization…

We prove Sobolev regularity for distributional solutions to the Dirichlet problem for generators of $2s$-stable processes and exterior data, inhomogeneity in weighted $L^2$-spaces. This class of operators includes the fractional Laplacian.…

We prove strong hybrid subconvex bounds simultaneously in the $q$ and $t$ aspects for $L$-functions of selfdual $\mathrm{GL}_3$ cusp forms twisted by primitive Dirichlet characters. We additionally prove analogous hybrid subconvex bounds…

Let $\Omega$ be a bounded $C^{2}$ domain in $\R^n$, and let $\Omega^{\ast}$ be the Euclidean ball centered at 0 and having the same Lebesgue measure as $\Omega$. Consider the operator $L=-\div(A\nabla)+v\cdot \nabla +V$ on $\Omega$ with…

We establish interior and up to the boundary H\"older regularity estimates for weak solutions of the Dirichlet problem for the fractional $g-$Laplacian with bounded right hand side and $g$ convex. These are the first regularity results…

We consider the torsional rigidity and the principal eigenvalue related to the $p$-Laplace operator. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in various classes of domains. The limit…

Whenever a given Poisson manifold is equipped with discrete symmetries the corresponding algebra of invariant functions or the algebra of functions twisted by the symmetry group can have new deformations, which are not captured by…

The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincar{\'e} constant. First we revisit E. Milman's result [40] on the link between weak (Poincar{\'e} or…

We consider the Dirichlet problem for second-order linear elliptic equations in divergence form \begin{equation*} -\mathrm{div }(A\nabla u)+\mathbf{b} \cdot \nabla u+\lambda u=f+\mathrm{div } \mathbf{F}\quad \text{in }…

We consider the problem of lower bounding a generalized Minkowski measure of subsets of a convex body with a log-concave probability measure, conditioned on the set size. A bound is given in terms of diameter and set size, which is sharp…

In this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise…

We study, in dimension $n\geq2$, the eigenvalue problem and the torsional rigidity for the $p$-Laplacian on convex sets with holes, with external Robin boundary conditions and internal Neumann boundary conditions. We prove that the annulus…

Let $\lambda_i(\Omega,V)$ be the $i$th eigenvalue of the Schr\"odinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset \R^n$ and with the positive potential $V$. Following the spirit of the…

We examine the duality theory for a class of non-convex functions obtained by composing a convex function with a continuous one. Using Fenchel duality, we derive a dual problem that satisfies weak duality under general assumptions. To…

This paper studies the general Lp dual curvature density equation under a group symmetry assumption. This geometric partial differential equation arises from the general Lp dual Minkowski problem of prescribing the Lp dual curvature measure…