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

Abstract

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.

submitted, 41 pages

References in corpus (4)

• The Green function estimates for strongly elliptic systems of second order
• Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II
• Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains
• Global estimates for Green's matrix of second order parabolic systems with application to elliptic systems in two dimensional domains

Cited by in corpus (1)

• Bounds on layer potentials with rough inputs for higher order elliptic equations