Critical Perturbations for Second Order Elliptic Operators. Part II: Non-tangential maximal function estimates

TL;DR

This paper proves the stability of boundary value problem solutions for complex elliptic operators under perturbations, introducing a new method to establish non-tangential maximal function estimates without classical theory.

Contribution

It develops a novel approach to obtain non-tangential maximal function estimates for perturbed elliptic operators, extending solvability results to unbounded domains with magnetic Schrödinger operators.

Findings

Stability of $L^2$ well-posedness under perturbations.

First $L^p$ solvability results for magnetic Schrödinger operators in unbounded domains.

Introduction of a new weak-$L^p$ estimate coupled with square function bounds.

Abstract

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the $L^2$ well-posedness (with natural non-tangential maximal function estimates) of the Dirichlet, Neumann and regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. Due to the lack of the classical De Giorgi-Nash-Moser theory in our setting, our method to prove the non-tangential maximal function estimates relies on a completely new argument: We obtain a certain weak-$L^p$ ''$N<S$'' estimate, which we eventually couple with square function bounds, weighted extrapolation theory, and a bootstrapping argument to recover the full $L^2$ bound. Finally, we show the existence and uniqueness of solutions in a relatively broad class. As a corollary, we claim the first results in an unbounded domain concerning the $L^p$-solvability of boundary value problems for the magnetic Schr\"odinger operator $-(\nabla-i{\bf a})^2+V$ when the magnetic potential ${\bf a}$ and the electric potential $V$ are accordingly small in the norm of a scale-invariant Lebesgue space.