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 $n\ge 2$. For $C^{1,\alpha }$ domains with a Reifenbeg flat interface, we obtain the optimal range $1<p<4\mathrm{/}3$, provided that $\alpha >1-1\mathrm{/}p$. 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.