Solvability of boundary value problem for Schr\"odinger Equations with Reverse H\"older Potentials on $L^p$ and endpoint spaces

TL;DR

This paper investigates the solvability of boundary value problems for Schr"odinger equations with Reverse H"older potentials, using layer potential methods and perturbation techniques, and provides norm estimates for related potentials.

Contribution

It establishes solvability conditions for boundary value problems with Reverse H"older potentials and offers Campanato norm estimates for layer potentials, extending existing theories.

Findings

Solvability proven for small $L^ abla$ perturbations of matrices satisfying De Giorgi-Nash-Moser bounds.

Boundary data in specific Hardy and Campanato spaces are handled.

Campanato norm estimates for double layer potentials are provided.

Abstract

In this paper we discuss the solvability of the Neumann and Regularity boundary value problem of elliptic Schr\"odinger-type equation $-\DIV(A(x)\nabla u(x,t))+V(x)u(x,t)=0$ with bounded measurable uniformly elliptic coefficinets $A(x)$ independent of $t$ and $V$ in Reverse H\"older class $\mathcal{B}_q$, and Neumann boundary data $\partial_{\nu_A}u(x,0)=f(x)\in H^p_{\mathcal{L}}(\rn)$, or Regularity data $u(x,0)=g\in H^{1,p}_V(\rn)$, utilizing the method of layer potential. We prove the solvability when $A$ is a small $L^\infty$ perturbation of a matrix satisfying De Giorgi-Nash-Moser bounds. Besides we also give the Campanato norm estimate of the double layer potential related to the Dirichlet problem with boundary data in certain Campanato-type spaces.