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

Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $K \subset Ω$ be a compact, $C^2$ submanifold in $\mathbb{R}^N$ without boundary, of dimension $k$ with $0\leq k < N-2$. We consider the Schrödinger operator $L_μ= Δ+ μd_K^{-2}$ in $Ω\setminus K$, where $d_K(x) = \text{dist}(x,K)$. The optimal Hardy constant $H=(N-k-2)/2$ is deeply involved in the study of $-L_μ$. When $μ\leq H^2$, we establish sharp, two-sided estimates for Green kernel and Martin kernel of $-L_μ$. 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 $-L_μ$

No reviews yet.

No comments yet.