Related papers: Dirichlet Fractional Laplacian in multi-tubes
We consider the Dirichlet Laplacian in a waveguide of uniform width and infinite length which is ideally divided into three parts: a "vertex region", compactly supported and with non zero curvature, and two "edge regions" which are…
It is well known that the spectrum of the Dirichlet Laplacian for a compact perturbation of a three-dimensional, periodically twisted tube is unstable with respect to domain deformations. This means that if the periodically twisted tube is…
We study Dirichlet Laplacian in a screw-shaped region, i.e. a straight twisted tube of a non-circular cross section. It is shown that a local perturbation which consists of "slowing down" the twisting in the mean gives rise to a non-empty…
We derive spectral estimates of the Lieb-Thirring type for eigenvalues of Dirichlet Laplacians on strictly shrinking spiral-shaped domains.
We investigate the spectrum of the three-dimensional Dirichlet Laplacian in a prototypal infinite polyhedral layer, that is formed by three perpendicular quarter-plane walls of constant width joining each other. Alternatively, this domain…
We prove lower bounds for the Dirichlet Laplacian on possibly unbounded domains in terms of natural geometric conditions. This is used to derive uncertainty principles for low energy functions of general elliptic second order divergence…
We propose a new fractional Laplacian for bounded domains, expressed as a conservation law and thus particularly suited to finite-volume schemes. Our approach permits the direct prescription of no-flux boundary conditions. We first show the…
The Dirichlet Laplacian in curved tubes of arbitrary cross-section rotating with respect to the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension is investigated. If the reference curve is not straight and its…
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 spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. This problem arises in the study of the spectrum of the Dirichlet Laplacian in thick polyhedral domains having some…
We consider the divergent fractional Laplace operator presented in [Dipierro-Savin-Valdinoci, Rev. Mat. Iberoam.] and we prove three types of results. Firstly, we show that any given function can be locally shadowed by a solution of a…
In this paper, we study the spectral fractional Laplacian with inhomogeneous Dirichlet boundary data. Our contributions are twofold: first we introduce a Dirichlet-to-Neumann map for this operator and analyze an associated inverse problem;…
Consider a bounded domain with the Dirichlet condition on a part of the boundary and the Neumann condition on its complement. Does the spectrum of the Laplacian determine uniquely which condition is imposed on which part? We present some…
The Dirichlet Laplacian in curved tubes of arbitrary constant cross-section rotating together with the Tang frame along a bounded curve in Euclidean spaces of arbitrary dimension is investigated in the limit when the volume of the…
We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for a class of very degenerate elliptic operators, with the aim to show that, at least for square type domains having fixed volume, the symmetry of the domain…
We discuss several geometric conditions guaranteeing the finiteness or the infiniteness of the discrete spectrum for Robin Laplacians on conical domains.
Let $\Omega \subset \mathbb R^3$ be a waveguide which is obtained by translating a cross-section in a constant direction along an unbounded spatial curve. Consider $-\Delta_{\Omega}^D$ the Dirichlet Laplacian operator in $\Omega$. Under the…
For the scattering system given by the Laplacian in a half-space with a periodic boundary condition, we derive resolvent expansions at embedded thresholds, we prove the continuity of the scattering matrix, and we establish new formulas for…
We describe the Dirichlet spectrum structure for the Fichera layers and crosses in any dimension $n\ge3$. Also the application of the obtained results to the classical Brownian exit times problem in these domains.
We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We characterize completely the range of parameters for which solutions of the…