Boundary value problems and Hardy spaces for elliptic systems with block structure

TL;DR

This paper advances the understanding of boundary value problems for elliptic systems with block structure in higher dimensions, establishing well-posedness, uniqueness, and optimal data regularity ranges using modern harmonic analysis techniques.

Contribution

It proves the well-posedness and uniqueness of boundary value problems for elliptic systems with block structure in higher dimensions, extending prior two-dimensional results and introducing new methods.

Findings

Well-posedness of Dirichlet, regularity, and Neumann problems in optimal exponent ranges

Complete uniqueness results for these boundary value problems

Identification of optimal ranges for fractional regularity data

Abstract

For elliptic systems with block structure in the upper half-space and t-independent coefficients, we settle the study of boundary value problems by proving compatible well-posedness of Dirichlet, regularity and Neumann problems in optimal ranges of exponents. Prior to this work, only the two-dimensional situation was fully understood. In higher dimensions, partial results for existence in smaller ranges of exponents and for a subclass of such systems had been established. The presented uniqueness results are completely new. We also elucidate optimal ranges for problems with fractional regularity data. Methods use and improve, with some new results, all the machinery developed over the last two decades to study such problems: the Kato square root estimates and Riesz transforms, Hardy spaces associated to operators, off-diagonal estimates, non-tangential estimates and square functions and abstract layer potentials to replace fundamental solutions in the absence of local regularity of solutions. This self-contained monograph provides a comprehensive overview on the field and unifies many earlier results that have been obtained by a variety of methods.