Related papers: On the Hardy-Schr\"odinger operator with a boundar…

Let $\Omega\subset\BBR^N$ be a bounded $C^2$ domain and $\CL_\gk=-\Gd-\frac{\gk}{d^2}$ the Hardy operator where $d=\dist (.,\prt\Gw)$ and $0<\gk\leq\frac{1}{4}$. Let $\ga_{\pm}=1\pm\sqrt{1-4\gk}$ be the two Hardy exponents, $\gl_\gk$ the…

We establish existence of weighted Hardy and Rellich inequalities on the spaces $L_p(\Omega)$ where $\Omega= \Ri^d\backslash K$ with $K$ a closed convex subset of $\Ri^d$. Let $\Gamma=\partial\Omega$ denote the boundary of $\Omega$ and…

Let $\Omega$ be a domain in $\Ri^d$ with boundary $\Gamma$${\!,}$ $d_\Gamma$ the Euclidean distance to the boundary and $H=-\divv(C\,\nabla)$ an elliptic operator with $C=(\,c_{kl}\,)>0$ where $c_{kl}=c_{lk}$ are real, bounded, Lipschitz…

We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing $\Gamma$ for the boundary of the obstacle, the relevant integral operators map…

We study the kernel function of the operator u $\rightarrow$ L $\mu$ u = --$\Delta$u + $\mu$ |x| 2 u in a bounded smooth domain $\Omega$ $\subset$ R N + such that 0 $\in$ $\partial$$\Omega$, where $\mu$ $\ge$ -- N 2 4 is a constant. We show…

We consider a bounded domain $\Omega$ of $\mathbb{R}^N$, $N\ge3$, $h$ and $b$ continuous functions on $\Omega$. Let $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^1_0\left(\Omega\right)$…

We are looking for an optimal convex domain on which the boundary value problem $$\left\{\begin{array}{cc}(-\Delta)^2 u_\gamma-\gamma\Delta u_\gamma = f,& \mbox{ in }\Omega\\ u_\gamma=\partial_\nu u_\gamma=0,& \mbox{ on…

We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schr\"odinger operator $- \Delta + V$ with a nonnegative potential $V$ which merely belongs to $L_{\mathrm{loc}}^1(\Omega)$. More precisely, if $u…

This paper deals with the existence of positive solution for the singular quasilinear Schr\"odinger equation $-\Delta u -\Delta (u^{2})u=h(x) u^{-\gamma} + f(x,u)~\mbox{in} ~ \Omega,$ where $\gamma > 1$, $\Omega \subset \mathbb{R}^{N},…

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…

Let $\Omega\subset \R^N$ ($N\geq 3$) be an open domain (may be unbounded) with $0\in \partial\Omega$ and $\partial\Omega$ be of $C^2$ at $0$ with the negative mean curvature $H(0)$. By using variational methods, we consider the following…

We establish -among other things- existence and multiplicity of solutions for the Dirichlet problem $\sum_i\partial_{ii}u+\frac{|u|^{\crit-2}u}{|x|^s}=0$ on smooth bounded domains $\Omega$ of $ \rn$ ($n\geq 3$) involving the critical…

The purpose of this paper is to study the eigenvalues $\{\lambda_{\mu,i} \}_i$ for the Dirichlet Hardy-Leray operator, i.e. $$ -\Delta u+\mu|x|^{-2}u=\lambda u\ \ {\rm in}\ \, \Omega,\quad\quad u=0\ \ {\rm on}\ \ \partial\Omega,$$ where…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $K \subset \Omega$ be a compact, $C^2$ submanifold in $\mathbb{R}^N$ without boundary, of dimension $k$ with $0\leq k < N-2$. We consider the Schr\"odinger…

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…

For bounded domains $\Omega$ with Lipschitz boundary $\Gamma$, we investigate boundary value problems for elliptic operators with variable coefficients of fourth order subject to Wentzell (or dynamic) boundary conditions. Using form…

This article is devoted to analyze control properties for the heat equation with singular potential $-\mu/|x|^2$ arising at the boundary of a smooth domain $\Omega\subset \rr^N$, $N\geq 1$. This problem was firstly studied by Vancostenoble…

We study semilinear elliptic equations with Hardy potential $\mathrm{(E)} \; -L_\mu u+u^q=0$ in a bounded smooth domain $\Omega\subset \mathbb R^N$. Here $q>1$, $L_\mu=\Delta+\frac{\mu}{\delta_\Omega^2}$ and…

Let $\Omega$ be a Lipschitz domain in $\mathbb R^n,n\geq 3,$ and $L=\divt A\nabla$ be a second order elliptic operator in divergence form. We will establish that the solvability of the Dirichlet regularity problem for boundary data in…

We study the elliptic equation $\lambda u-L^{\Omega}u=f$ in an open convex subset $\Omega$ of an infinite dimensional separable Banach space $X$ endowed with a centered non-degenerate Gaussian measure $\gamma$, where $L^\Omega$ is the…