On $L^2$ Solvability of BVPs for elliptic systems

Pascal Auscher (1) , Alan Mcintosh (2) , Mihalis Mourgoglou (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 CMA - Center for Mathematics and its Applications

In this article we prove solvability results for $L^2$ boundary value problems of some elliptic systems $Lu=0$ on the upper half-space $\R^{n+1}_{+}, n\ge 1$, with transversally independent coefficients. We use the first order formalism introduced by Auscher-Axelsson-McIntosh and further developed with a better understanding of the classes of solutions in the subsequent work of Auscher-Axelsson. The interesting fact is that we prove only half of the Rellich boundary inequality without knowing the other half.

Mots clés: elliptic systems; variational solutions; Dirichlet and Neumann problems; square functions; non-tangential maximal functions; Tb theorems

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

Pascal Auscher, Alan Mcintosh, Mihalis Mourgoglou. On $L^2$ Solvability of BVPs for elliptic systems. Journal of Fourier Analysis and Applications, 2013, 19 (3), pp.478-494. ⟨hal-00765844⟩