W1,p Regularity of Eigenfunctions for the Mixed Problem with Nonhomogeneous Neumann Data

We consider an eigenvalue problem with a mixed boundary condition, where a second-order differential operator is given in divergence form and satisfies a uniform ellipticity condition. We show that if a function u in the Sobolev space W1,pD is a weak solution to the eigenvalue problem, then u also belongs to W1,pD for some p>2. To do so, we show a reverse Hölder inequality for the gradient of u. The decomposition of the boundary is assumed to be such that we get both Poincaré and Sobolev-type inequalities up to the boundary.