Abstract Besov-Hardy-Sobolev spaces and elliptic boundary value problems with complex bounded measurable coefficients

International audienceIn this monograph our main goal is to study the well-posedness of boundary value problems of Dirichlet and Neumann type for elliptic systems div A∇u = 0 on the upper half-space with coefficients independent of the transversal variable, and with boundary data in fractional Hardy–Sobolev and Besov spaces. Our approach uses minimal assumptions on the coefficients A, 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 corresponding results recently obtained by Barton and Mayboroda.First we develop a theory of Besov–Hardy–Sobolev spaces adapted to operators which are bisectorial on L^2, with bounded H^∞ 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 Besov–Hardy–Sobolev spaces adapted to DB are equal to those adapted to the unperturbed Dirac operator D (which correspond to classical Besov–Hardy–Sobolev spaces).Our main result is the classification of solutions of the elliptic system div A∇u = 0 within a certain classification region of exponents (defined in terms of the identification regions for certain perturbed Dirac operators associated with A). 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 Hardy–Sobolev (or Besov) space, and can be represented as a semigroup evolution of this trace in the transversaldirection. As a particular corollary, any such solution can be represented in terms of an abstract layer potential operator.Within the classification region, we show that well-posedness of the boundary value problems under consideration is equivalent to a certain boundary projection being an isomorphism. We derive various consequences of this characterisation, including interpolation, extrapolation, and duality properties of well-posedness, as well as some results on stability of well-posedness under coefficient perturbation. These consequences are illustrated in various situations, including in particular that of the Regularity problem for real equations