Related papers: Quantitative Resolvent and Eigenfunction Stability…

This paper is dedicated to the spectral optimization problem $$ \mathrm{min}\left\{\lambda_1^s(\Omega)+\cdots+\lambda_m^s(\Omega) + \Lambda \mathcal{L}_n(\Omega)\colon \Omega\subset D \mbox{ s-quasi-open}\right\} $$ where $\Lambda>0,…

In this paper, we obtain sharp Faber-Krahn inequalities for the first Dirichlet eigenvalue of the combinatorial $p$-Laplacian on connected graphs with a fixed number of vertices or with a fixed number of edges. More precisely, we show that…

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=…

We consider a quantity $\kappa(\Omega)$ -- the distance to the origin from the null variety of the Fourier transform of the characteristic function of $\Omega$. We conjecture, firstly, that $\kappa(\Omega)$ is maximized, among all convex…

We consider a general class of metric measure spaces equipped with a regular Dirichlet form and then provide a lower bound on the hitting time probabilities of the associated Hunt process. Using these estimates we establish (i) a…

In this work we prove that given an open bounded set $\Omega \subset \mathbb{R}^2$ with a $C^2$ boundary, there exists $\epsilon := \epsilon(\Omega)$ small enough such that for all $0 < \delta < \epsilon$ the maximum of $\{\lambda_1(\Omega…

In this paper we study the $\Gamma$-limit, as $p\to 1$, of the functional $$ J_{p}(u)=\frac{\displaystyle\int_\Omega |\nabla u|^p + \beta\int_{ \partial \Omega} |u|^p}{\displaystyle \int_\Omega |u|^p}, $$ where $\Omega$ is a smooth bounded…

In this article we prove that the first eigenvalue of the $\infty-$Laplacian $$ \left\{ \begin{array}{rclcl} \min\{ -\Delta_\infty v,\, |\nabla v|-\lambda_{1, \infty}(\Omega) v \} & = & 0 & \text{in} & \Omega v & = & 0 & \text{on} &…

In this paper we study a Steklov-Robin eigenvalue problem for the Laplacian in annular domains. More precisely, we consider $\Omega=\Omega_0 \setminus \overline{B}_{r}$, where $B_{r}$ is the ball centered at the origin with radius $r>0$ and…

We exploit the connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians. First, we find a graphical relation between their smallest positive eigenvalues, which allows us to deduce a recipe for translating…

For a hyponormal operator, C. R. Putnam's inequality gives an upper bound on the norm of its self-commutator. In the special case of a Toeplitz operator with analytic symbol in the Smirnov space of a domain, there is also a geometric lower…

We prove explicit upper and lower bounds for the Poisson hierarchy, the averaged $L^1$-moment spectra $\{\dfrac{\mathcal{A}_k\left(B_R^M\right)}{\text{vol}\left(S_R^M\right)}\}_{k=1}^\infty$, and the torsional rigidity…

We study the eigenvalue problem for the Dirichlet Laplacian in bounded simply connected plane domains $\Omega\subset\mathbb{C}$ using conformal transformations of the original problem to the weighted eigenvalue problem for the Dirichlet…

In 1960, Payne and Weinberger proved that among all domains that lie within a wedge (an angle whose measure is less than or equal to $\pi$), and have a given value of a certain integral the circular sector has the lowest fundamental…

In this paper we prove the existence of an optimal domain $\Omega_{opt}$ for the shape optimization problem $$\max\Big\{\lambda_q(\Omega)\ :\ \Omega\subset D,\ \lambda_p(\Omega)=1\Big\},$$ where $q<p$ and $D$ is a prescribed bounded subset…

We consider the Dirichlet Laplacian $A_q=-\Delta+q$ in a bounded domain $\Omega \subset \mathbb{R}^d$, $d \ge 3$, with real-valued perturbation $q \in L^{\max(2 , 3 d / 5)}(\Omega)$. We examine the stability issue in the inverse problem of…

In this paper, we consider the bifurcation problem for fractional Laplace equation \begin{eqnarray*} \begin{array}{ll} (-\Delta)^{s} u = \lambda u + f(\lambda,\,x,\,u)& \mbox{in }\Omega, u = 0 &\mbox{in }\mathbb{R}^n\backslash \Omega,…

We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body $\Omega \subset \mathbb{R}^n$, not necessarily vanishing on the boundary $\partial \Omega$. This reduces the study of the Neumann…

We consider the shape optimization problems for the quantities $\lambda(\Omega)T^q(\Omega)$, where $\Omega$ varies among open sets of $\mathbb{R}^d$ with a prescribed Lebesgue measure. While the characterization of the infimum is completely…

In this paper, we investigate eigenvalues of the Dirichlet problem and the closed eigenvalue problem of drifting Laplacian on the complete metric measure spaces and establish the corresponding general formulas. By using those general…