Related papers: On a reverse Kohler-Jobin inequality

We prove that the largest convex shape that can be placed inside a given convex shape $Q \subset \mathbb{R}^{d}$ in any desired orientation is the largest inscribed ball of $Q$. The statement is true both when "largest" means "largest…

In this short note we consider an unconventional overdetermined problem for the torsion function: let $n\geq 2$ and $\Omega$ be a bounded open set in $\mathbb{R}^n$ whose torsion function $u$ (i.e. the solution to $\Delta u=-1$ in $\Omega$,…

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider $T(\Omega)/(M(\Omega)|\Omega|)$ and $M(\Omega)\lambda_1(\Omega) $, where $\Omega$ is a…

For a given bounded Lipschitz set $\Omega$, we consider a Steklov--type eigenvalue problem for the Laplacian operator whose solutions provide extremal functions for the compact embedding $H^1(\Omega)\hookrightarrow L^2(\partial \Omega)$. We…

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…

We focus here on the analysis of the regularity or singularity of solutions $\Om_{0}$ to shape optimization problems among convex planar sets, namely: $$ J(\Om_{0})=\min\{J(\Om),\ \Om\ \textrm{convex},\ \Omega\in\mathcal S_{ad}\}, $$ where…

For every given $\beta<0$, we study the problem of maximizing the first Robin eigenvalue of the Laplacian $\lambda_\beta(\Omega)$ among convex (not necessarily smooth) sets $\Omega\subset\mathbb{S}^{n}$ with fixed perimeter. In particular,…

We investigate a shape optimization problem of the Polya-Szego type and prove some results on symmetry and asymmetry of extremal domains.

In this paper, we consider a maximizing problem associated with the Sobolev type embedding on the space of bounded variation. We show that, although the maximizing problem suffers from both of the non-compactness of vanishing and…

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality $\lambda_1(\Omega)\|u\|_{L^1(\partial\Omega)} \le \|u\|_{W^{1,1}(\Omega)}$ that are independent of $\Omega$. This estimates generalize those…

We consider the problem of the maximum concentration in a fixed measurable subset $\Omega\subset\mathbb{R}^{2d}$ of the time-frequency space for functions $f\in L^2(\mathbb{R}^{d})$. The notion of concentration can be made mathematically…

For all functions on an arbitrary open set $\Omega\subset\R^3$ with zero boundary values, we prove the optimal bound \[ \sup_{\Omega}|u| \leq (2\pi)^{-1/2} \left(\int_{\Omega}|\nabla u|^2 \,dx\, \int_{\Omega}|\Delta u|^2 \,dx\right)^{1/4}.…

We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two…

The second eigenvalue of the Robin Laplacian is shown to be maximal for the ball among domains of fixed volume, for negative values of the Robin parameter $\alpha$ in the regime connecting the first nontrivial Neumann and Steklov…

For $d\geq 2$ and $\frac{2d+2}{d+2} < p < \infty $, we prove a strict Faber-Krahn type inequality for the first eigenvalue $\lambda _1(\Omega )$ of the $p$-Laplace operator on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ (with…

In this paper we study the regularity of the optimal sets for the shape optimization problem \[ \min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega)\ :\ \Omega\subset\mathbb{R}^d,\ \text{open}\ ,\ |\Omega|=1\Big\}, \] where…

Motivated by a long-standing conjecture of Polya and Szeg\"o about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the…

In this paper, we are interested in the analysis of a well-known free boundary/shape optimization problem motivated by some issues arising in population dynamics. The question is to determine optimal spatial arrangements of favorable and…

A symmetrization inequality of Rogers and of Brascamp-Lieb-Luttinger states that for a certain class of multilinear integral expressions, among tuples of sets of prescribed Lebesgue measures, tuples of balls centered at the origin are among…

Payne-Weinberger showed that \textit{`among the class of membranes with given area $A$, free along the interior boundaries and fixed along the outer boundary of given length $L_0$, the annulus $\Omega^\#$ has the highest fundamental…