On the $W^{2,p}$ solvability for mixed boundary value problems

Abstract: We establish global $W<sup>{2,p}$ solvability for the Poisson equation with mixed Dirichlet--Neumann boundary conditions in two classes of domains in all dimensions $n\geq 2$. For $C<sup>{1,α}$ domains with a Reifenbeg flat interface, we obtain the optimal range $1&lt;p\&lt;4/3$, provided that $$α\&gt;1-1/p$$. For Lipschitz polyhedra with facewise boundary decompositions, we obtain solvability for $p$ close to $1$. In both settings, we establish endpoint $W<sup>{2,1}$ solvability for data in an adapted atomic Hardy space. Applications of the methods to Lamé systems are also discussed.