Elliptic Boundary Value Problems with Fractional Regularity Data: The First Order Approach

TL;DR

This paper develops a new theoretical framework for elliptic boundary value problems with fractional regularity data, extending existing results and providing a classification of solutions using advanced functional calculus and operator theory.

Contribution

It introduces a novel approach to analyze elliptic systems with fractional boundary data, including the development of BHS spaces for bisectorial operators and a classification of solutions within a specific exponent region.

Findings

BHS spaces are adapted to bisectorial operators with bounded $H^$ calculus.

Solutions can be represented as semigroup evolutions of boundary traces.

Well-posedness is characterized by boundary projection isomorphisms.

Abstract

We study well-posedness of boundary value problems of Dirichlet and Neumann type for elliptic systems on the upper half-space with coefficients independent of the transversal variable, and with boundary data in fractional Besov-Hardy-Sobolev (BHS) spaces. Our approach uses minimal assumptions on the coefficients, and in particular does not require De Giorgi-Nash-Moser estimates. Our results are completely new for the Hardy-Sobolev case, and in the Besov case they extend results recently obtained by Barton and Mayboroda. First we develop a theory of BHS spaces adapted to operators which are bisectorial on $L^2$, with bounded $H^\infty$ functional calculus on their ranges, and which satisfy $L^2$ off-diagonal estimates. In particular, this theory applies to perturbed Dirac operators $DB$. We then prove that for a nontrivial range of exponents (the identification region) the BHS spaces adapted to $DB$ are equal to those adapted to $D$ (which correspond to classical BHS spaces). Our main result is the classification of solutions of the elliptic system $\operatorname{div} A \nabla u = 0$ within a certain region of exponents. More precisely, we show that if the conormal gradient of a solution belongs to a weighted tent space (or one of their real interpolants) with exponent in the classification region, and in addition vanishes at infinity in a certain sense, then it has a trace in a BHS space, and can be represented as a semigroup evolution of this trace in the transversal direction. As a corollary, any such solution can be represented in terms of an abstract layer potential operator. Within the classification region, we show that well-posedness is equivalent to a certain boundary projection being an isomorphism. We derive various consequences of this characterisation, which are illustrated in various situations, including in particular that of the Regularity problem for real equations.