Related papers: Isoperimetric estimates for the first Neumann eige…

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain $\Omega $ in ${\mathbb{R}}^N$. We consider deformations $\phi (\Omega)$ of $\Omega $ obtained by means of a locally Lipschitz…

For a semifinite von Neumann algebra M, individual convergence of subsequential, \mathcal{Z}(M) (center of M) valued weighted ergodic averages are studied in noncommutative Orlicz spaces. In the process, we also derive a maximal ergodic…

We prove explicit and sharp eigenvalue estimates for Neumann $p$-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if $\gamma$ denotes a non-closed curve in $\mathbb{R}^2$ symmetric with…

We investigate the geometric behavior of $\tau(E)$ for bounded finite-perimeter sets $E \subset \mathbb R^n$, where $\tau(E)$ is the trace constant introduced by Figalli--Maggi--Pratelli [Invent. Math. 2010]. This quantity is a key…

Let $\Omega \subset \mathbb{R}^n$ be a convex domain and let $f:\Omega \rightarrow \mathbb{R}$ be a positive, subharmonic function (i.e. $\Delta f \geq 0$). Then $$ \frac{1}{|\Omega|} \int_{\Omega}{f dx} \leq \frac{c_n}{ |\partial \Omega| }…

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: \begin{equation*} \lambda_1(\beta,\Omega)=\min_{\psi\in W^{1,p}(\Omega)\setminus\{0\} }…

Let $\Omega \subset \mathbb{R}^2$ be a bounded, convex domain and let $-\Delta \phi_1 = \mu_1 \phi_1$ be the first nontrivial Laplacian eigenfunction with Neumann boundary conditions. The Hot Spots conjecture claims that the maximum and…

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, we can measure the Euclidean…

Let $A$ be an $n\times n$ matrix with mutually independent centered Gaussian entries. Define \begin{align*} \sigma^*:=\max\limits_{i,j\leq n}\sqrt{{\mathbb E}\,|A_{i,j}|^2}, \quad \sigma:=\max\bigg(\max\limits_{j\leq n}\sqrt{{\mathbb…

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following anisotropic elliptic Neumann problem with Hardy-H\'{e}non weight $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=a(x)|x-q|^{2\alpha}u^p,\,\,\,\,…

We prove a quantitative isoperimetric inequality for the nearly spherical subset of the Bergman ball in $\mathbb{C}^n$. We prove the Fuglede theorem for such sets. This result is a counterpart of a similar result obtained for the hyperbolic…

Let $\hat \Omega \subset \mathbb R^2$ be a bounded domain with smooth boundary and $\hat \sigma$ a smooth anisotropic conductivity on $\hat \Omega$. Starting from the Dirichlet-to-Neumann operator $\Lambda_{\hat \sigma}$ on $\partial \hat…

We consider large non-Hermitian $N\times N$ matrices with an additive independent, identically distributed (i.i.d.) noise for each matrix elements. We show that already a small noise of variance $1/N$ completely thermalises the bulk…

Given a bounded open set $\Omega\subseteq{\mathbb{R}}^n$, we consider the eigenvalue problem of a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of $\Omega$. We prove that the second eigenvalue…

We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell-Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a…

We prove an inequality with applications to solutions of the Schr\"odinger equation. There is a universal constant $c>0$, such that if $\Omega \subset \mathbb{R}^2$ is simply connected, $u:\Omega \rightarrow \mathbb{R}$ vanishes on the…

We consider the monomial weight $x^{A}=\vert x_{1}\vert^{a_{1}}\ldots\vert x_{N}\vert^{a_{N}}$, where $a_{i}$ is a nonnegative real number for each $i\in\{1,\ldots,N\}$, and we establish the existence and nonexistence of isoperimetric…

Using the localization technique, we prove a sharp upper bound on the first Dirichlet eigenvalue of metric balls in essentially non-branching $\mathsf{CD}^{\star}(K,N)$ spaces. This extends a celebrated result of Cheng to the non-smooth…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ is a $C^2$ compact boundaryless submanifold in $\mathbb{R}^N$ of dimension $k$, $0\leq k < N-2$. For $\mu\leq (\frac{N-k-2}{2})^2$, put…

We say that a bounded domain $\Omega$ is optimal for the first positive curl eigenvalue $\mu_1(\Omega)$ if $\mu_1(\Omega)\leq \mu_1(\Omega')$ for any domain $\Omega'$ with the same volume. In spite of the fact that $\mu_1(\Omega)$ is…