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.
Year manuscript completed
Year degree awarded
Eigenvalues, elliptic equations, Sobolev spaces, partial differential equations
Miyazaki, Kohei, "W1,p Regularity of Eigenfunctions for the Mixed Problem with Nonhomogeneous Neumann Data" (2018). Murray State Theses and Dissertations. 100.