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Solvability of boundary value problem for Schrödinger Equations with Reverse Hölder Potentials on $L^p$ and endpoint spaces

Botian Xiao, Lin Tang

Abstract

In this paper we discuss the solvability of the Neumann and Regularity boundary value problem of elliptic Schrödinger-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 Hölder class $\mathcal{B}_q$, and Neumann boundary data $\partial_{ν_A}u(x,0)=f(x)\in H^p_{\mathcal{L}}(\rn)$, 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.

Solvability of boundary value problem for Schrödinger Equations with Reverse Hölder Potentials on $L^p$ and endpoint spaces

Abstract

In this paper we discuss the solvability of the Neumann and Regularity boundary value problem of elliptic Schrödinger-type equation with bounded measurable uniformly elliptic coefficinets independent of and in Reverse Hölder class , and Neumann boundary data , or Regularity data , utilizing the method of layer potential. We prove the solvability when is a small 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.

Paper Structure

This paper contains 15 sections, 19 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Consequences of Reverse Hölder Inequalities
  4. Estimates on Fundamental Solutions and Green Functions
  5. Function Spaces
  6. Estimates for Layer Potentials
  7. $L^p$ Maximal Bounds for Layer Potentials
  8. Single Layer Potential when $1-\varepsilon^\prime <p\le 2$
  9. Single Layer Potential when $2\le p < 2+\varepsilon$
  10. Double Layer Potential for $\Lambda^\beta_\mathcal{L}$ Data
  11. Rellich Estimates on the Boundary
  12. Solution of the Equation
  13. Existence and Uniqueness
  14. Boundary Values of Solutions
  15. Regularity of Complex Solutions

Key Result

Theorem 1.1

Given $A_0(x)$ a uniformly elliptic matrix and the potential $V\in \mathcal{B}_q(\mathbb{R}^n)$ with $q\ge\frac{n+1}{2}$. Suppose $A_0$ is real symmetric, or $A_0$ is of the block form and any all the weak solutions to both $\,\textup{div}\,(A_0\nabla u)=0$ and $\,\textup{div}\,(A_0^\ast\nabla u)=0$

Theorems & Definitions (39)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Definition 2.1
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 29 more