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

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