Optimal Elliptic Sobolev Regularity Near Three-Dimensional Multi-Material Neumann Vertices

We study the optimal elliptic regularity (within the scale of Sobolev spaces) of anisotropic div–grad operators in three dimensions at a multi-material vertex on the Neumann part of the boundary of a 3D polyhedral domain. The gradient of any solution of the corresponding elliptic partial differential equation (in a neighborhood of the vertex) is $p$-integrable with $p>3$.