相关论文: Schr\"{o}dinger operators with boundary singularit…

We consider the Schr\"{o}dinger operator $A_\l:=-\D -\l/|x|^2$, $\l\in \rr$, when the singularity is located on the boundary of a smooth domain $\Omega\subset \rr^N$, $N\geq 1$ The aim of this Note is two folded. Firstly, we justify the…

In this expository paper, we consider the Hardy-Schr\"odinger operator $-\Delta -\gamma/|x|^2$ on a smooth domain \Omega of R^n with 0\in\bar{\Omega}, and describe how the location of the singularity 0, be it in the interior of \Omega or on…

In this expository paper we survey some recent results on Dirichlet problems of the form $Lu=f(x,u)$ in $\Omega$, $u\equiv0$ in $\mathbb R^n\backslash\Omega$. We first discuss in detail the boundary regularity of solutions, stating the main…

We investigate the Hardy-Schr\"odinger operator $L_\gamma=-\Delta -\frac{\gamma}{|x|^2}$ on domains $\Omega\subset\rn$, whose boundary contain the singularity $0$. The situation is quite different from the well-studied case when $0$ is in…

Given a Lipschitz domain $\Omega $ in ${\mathbb R} ^N $ and a nonnegative potential $V$ in $\Omega $ such that $V(x)\, d(x,\partial \Omega)^2$ is bounded in $\Omega $ we study the fine regularity of boundary points with respect to the…

Using an extension of the H\"ormander product of distributions, we obtain an intrinsic formulation of one-dimensional Schr\"odinger operators with singular potentials. This formulation is entirely defined in terms of standard {\it Schwartz}…

Let $\O$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincar\'e inequalities with weight function singular at the boundary of $\O$. In particular we give sufficient conditions so that the best…

We study a class of delta-like perturbations of the Laplacian on the half-line, characterized by Robin boundary conditions at the origin. Using the formalism of nonstandard analysis, we derive a simple connection with a suitable family of…

We examine various density results related to the solutions of the non-local heat equation at a specific time slice, focusing on two distinct models: one with homogeneous Dirichlet boundary condition and the other with singular boundary…

We prove Sobolev-Poincar\'e and Poincar\'e inequalities in variable Lebesgue spaces $L^{p(\cdot)}(\Omega)$, with $\Omega\subset{\mathbb R}^n$ a bounded John domain, with weaker regularity assumptions on the exponent $p(\cdot)$ that have…

We consider eigenfunctions of a semiclassical Schr{\"o}dinger operator on an interval, with a single-well type potential and Dirichlet boundary conditions. We give upper/lower bounds on the L^2 density of the eigenfunctions that are uniform…

In this paper we prove a Pohozaev-type identity for both the problem $(-\Delta+m^2)^su=f(u)$ in $\mathbb{R}^N$ and its harmonic extension to $\mathbb{R}^{N+1}_+$ when $0<s<1$. So, our setting includes the pseudo-relativistic operator…

In this article we prove the Pohozaev identity for the semilinear Dirichlet problem of the form $-\Delta u + a(-\Delta)^s u = f(u)$ in $\Omega$, and $u=0$ in $\Omega^c$, where $a$ is a non-negative constant and $\Omega$ is a bounded $C^2$…

We establish Pohozaev identities and integration by parts type formulas for anisotropic integro-differential operators of order $2s$, with $s\in(0,1)$. These identities involve local boundary terms, in which the quantity…

The aim of this note is to discuss in more detail the Pohozaev-type identities that have been recently obtained by the author, Paul Laurain and Tristan Rivi\`ere in the framework of half-harmonic maps defined either on $R$ or on the sphere…

It is shown that an elliptic scattering operator $A$ on a compact manifold with boundary with coefficients in the bounded operators of a bundle of Banach spaces of class (HT) and Pisier's property $(\alpha)$ has maximal regularity (up to a…

We will present versions of the Rellich-Kondrachov theorem for pseudo-differential operators acting on localizable Hardy spaces. One of the techniques includes boundedness properties for pseudodifferential operators with symbols in the…

We provide a summary of the continuity properties of the boundary integral operator corresponding to the double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator…

This paper studies the local exact controllability and the local stabilization of the semilinear Schr\"odinger equation posed on a product of $n$ intervals ($n\ge 1$). Both internal and boundary controls are considered, and the results are…

Self-adjoint Schr\"odinger operators with $\delta$ and $\delta'$-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity…