Green kernel and Martin kernel of Schrödinger operators with singular potential and application to the B.V.P. for linear elliptic equations

Abstract

Let $\Omega \subset {\mathbb{R}}^N$ ($N\ge 3$) 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 $0\le k<N-2$. We consider the Schrödinger operator $L_{\mu }=\Delta +\mu d_K^{-2}$ in $\Omega \setminus K$, where $d_K(x)=\text{dist}(x,K)$. The optimal Hardy constant $H=(N-k-2)/2$ is deeply involved in the study of $-L_{\mu }$. When $\mu \le H^2$, we establish sharp, two-sided estimates for Green kernel and Martin kernel of $-L_{\mu }$. We use these estimates to prove the existence, uniqueness and a priori estimates of the solution to the boundary value problem with measures for linear equations associated to $-L_{\mu }$

References in corpus (5)

• Sharp two-sided heat kernel estimates for critical Schrödinger operators on bounded domains
• Improving $L^2$ estimates to Harnack inequalities
• Boundary singularities of semilinear elliptic equations with Leray-Hardy potential
• Estimates of sub and super solutions of Schrödinger equations with very singular potentials
• Schr{ö}dinger operators with Leray-Hardy potential singular on the boundary