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

Pascal Auscher (1) , Mihalis Mourgoglou (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: Hardy spaces; Rellich inequalities; boundary layer operators; non-tangential maximal functions; elliptic systems; energy solutions; Regularity and Neumann problems; Dirichlet problem; bounded mean oscillation; tent spaces

Domaines: Equations aux dérivées partielles [math.AP]; Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]

Pascal Auscher, Mihalis Mourgoglou. Boundary layers, Rellich estimates and extrapolation of solvability for elliptic systems. Proceedings of the London Mathematical Society, 2014, 109 (3), pp.446 -- 482. ⟨hal-00835110⟩