Boundary layers, Rellich estimates and extrapolation of solvability for elliptic systems

TL;DR

This paper develops a method to extrapolate solvability of boundary value problems for elliptic systems in divergence form, using Rellich estimates, Hardy spaces, and duality principles, under De Giorgi conditions.

Contribution

It introduces a novel approach to treat boundary value problems independently, extending duality principles and providing a new framework for solvability extrapolation.

Findings

Reproved the Regularity-Dirichlet duality principle.

Extended duality principles to H^1-BMO.

Established a method for extrapolating solvability using atomic Hardy spaces.

Abstract

The purpose of this article is to study extrapolation of solvability for boundary value problems of elliptic systems in divergence form on the upper half-space assuming De Giorgi type conditions. We develop a method allowing to treat each boundary value problem independently of the others. We shall base our study on solvability for energy solutions, estimates for boundary layers, equivalence of certain boundary estimates with interior control so that solvability reduces to a one-sided Rellich inequality. Our method then amounts to extrapolating this Rellich inequality using atomic Hardy spaces, interpolation and duality. In the way, we reprove the Regularity-Dirichlet duality principle between dual systems and extend it to $H^1-BMO$. We also exhibit and use a similar Neumann-Neumann duality principle.