The square root problem for second order, divergence form operators with mixed boundary conditions on $L^p$

Pascal Auscher (1) , Nadine Badr (2) , Robert Haller-Dintelmann (3) , Joachim Rehberg (4)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 ICJ - Institut Camille Jordan
3 Fachbereich Mathematik
4 WIAS - Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin]

We show that, under very general conditions on the domain $\Omega$ and the Dirichlet part $D$ of the boundary, the operator $\bigl (-\nabla \cdot \mu \nabla +1\bigr )^{1/2}$ with mixed boundary conditions provides a topological isomorphism between $W^{1,p}_D(\Omega)$ and $L^p(\Omega)$, if $p \in {]1,2]}$.

Mots clés: Kato's square root problem; Elliptic operators with bounded measurable coefficients; Interpolation in case of mixed boundary values; Hardy's inequality; Calderon-Zygmund decomposition

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

Pascal Auscher, Nadine Badr, Robert Haller-Dintelmann, Joachim Rehberg. The square root problem for second order, divergence form operators with mixed boundary conditions on $L^p$. 2013. ⟨hal-00737614v2⟩