Related papers: P\'olya's conjecture for thin products

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schr\"odinger operator with convex potential and Dirichlet boundary data on a convex domain is…

We study the existence and regularity of weak solutions to the following quasilinear elliptic system: \[ -\mathrm{div}(A_k(x, u_k) |\nabla u_k|^{p_k - 2} \nabla u_k) + \dfrac{1}{p_k} D_s A_k(x, u_k) |\nabla u_k|^{p_k} = g_k(x, u) \quad…

Let $\Omega$ be a bounded domain in R d with Lipschitz boundary $\Gamma$. We define the Dirichlet-to-Neumann operator N on L 2 ($\Gamma$) associated with a second order elliptic operator A = -- d k,j=1 $\partial$ k (c kl $\partial$ l) + d…

It is widely known that the spectrum of the Dirichlet Laplacian is stable under small perturbations of a domain, while in the case of the Neumann or mixed boundary conditions the spectrum may abruptly change. In this work we discuss an…

We investigate the overdetermined problem given by \begin{equation*} \Delta u=0 \text{ in } \Omega,\quad \frac{\partial u}{\partial\nu} =\sigma_1 u \text{ on } \partial \Omega, \quad |\nabla u|=\text{constant on } \partial \Omega,…

We consider modulation space and spaces of Schatten-von Neumann symbols where corresponding pseudo-differential operators map one Hilbert space to another. We prove H\"older-Young and Young type results for such spaces under dilated…

Classical boundary Hardy inequality, that goes back to 1988, states that if $1 < p < \infty, \ ~\Omega$ is bounded Lipschitz domain, then for all $u \in C^{\infty}_{c}(\Omega)$, $$\int_{\Omega} \frac{|u(x)|^{p}}{\delta^{p}_{\Omega}(x)} dx…

Let $\mu_2(\Omega)$ be the first positive eigenvalue of the Neumann Laplacian in a bounded domain $\Omega\subset\mathbb{R}^N$. It was proved by Szeg\H{o} for $N=2$ and by Weinberger for $N \geq 2$ that among all equimeasurable domains…

We prove universal inequalities for Laplacian eigenvalues with Dirichlet boundary conditions on subsets of certain discrete groups. The study of universal inequalities on Riemannian manifolds was initiated by Weyl, Polya, Yau, and others.…

Given a compact Riemannian manifold $(M,g)$ with smooth boundary $\partial M$, we give an explicit expression for full symbol of the thermoelastic Dirichlet-to-Neumann map $\Lambda_g$ with variable coefficients $\lambda,\mu,\alpha,\beta \in…

Let $(M^n,g)$ be a complete simply connected $n$-dimensional Riemannian manifold with curvature bounds $\operatorname{Sect}_g\leq \kappa$ for $\kappa\leq 0$ and $\operatorname{Ric}_g\geq(n-1)Kg$ for $K\leq 0$. We prove that for any bounded…

We compute the whole spectrum of the Dirichlet-to-Neumann operator acting on differential p-forms on the unit Euclidean ball. Then, we prove a new upper bound for its first eigenvalue on a domain $\Omega$ in Euclidean space in terms of the…

In these notes we study the Dirichlet problem for critical points of a convex functional of the form \[ F(u)=\int_{\Omega}\phi\left( \left\vert \nabla u\right\vert \right) , \] where $\Omega$ is a bounded domain of a complete Riemannian…

For a bounded Lipschitz domain $\Sigma$ in a Riemannian surface $M$ satisfying certain curvature condition, we prove that $$\mu_{3-\beta_1} \leq \lambda_{1},$$ where $\mu_k$ ($\lambda_k$ resp.) is the $k$-th Neumann (Dirichlet resp.)…

In this paper we consider the first eigenvalue $\lambda_1(\Omega)$ of the Grushin operator $\Delta_G:=\Delta_{x_1}+|x_1|^{2s}\Delta_{x_2}$ with Dirichlet boundary conditions on a bounded domain $\Omega$ of $\mathbb{R}^d=…

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…

Let $\mathcal{M}$ be a smooth manifold of positive dimension $n$ equipped with a smooth density $d\mu_{\mathcal{M}}$. Let $A$ be a polyhomogeneous elliptic pseudo-differential operator of positive order $m$ on $\mathcal{M}$ which is…

We deduce eigenvalue asymptotics of the Neumann--Poincar\'e operators in three dimensions. The region $\Omega$ is $C^{2, \alpha}$ ($\alpha>0$) bounded in ${\mathbf R}^3$ and the Neumann--Poincar\'e operator ${\mathcal K}_{\partial\Omega} :…

Let $\Omega $ be a bounded ${\mathcal{C}}^{\infty}$-smoothly bounded domain in ${\mathbb{C}}^{n}.$ For such a domain we define a new notion between strict pseudo-convexity and pseudo-convexity: the size of the set $W$ of weakly…

Let $\Omega\subset{\mathbb R}^2$ be a bounded domain on which Hardy's inequality holds. We prove that $[\exp(u^2)-1]/\delta^2\in L^1(\Omega)$ if $u\in H^1_0(\Omega)$, where $\delta$ denotes the distance to $\partial\Omega$. The…