This thesis consists of three papers, all concerned with the eigenvalue problem for the Schrödinger operator -Δ+V, and in particular the Laplacian -Δ, on bounded, connected, Lipschitz domains with mixed boundary conditions, where a Dirichlet boundary condition is imposed on a subset of the boundary and a Neumann boundary condition on its complement. Given different such choices of boundary conditions on the same domain, we compare the resulting mixed Dirichlet-Neumann eigenvalues by establishing inequalities between them, and prove a variant of the hot spots conjecture for the lowest mixed Dirichlet-Neumann eigenfunction of the Laplacian. Our approach is purely variational and relies on both classical and novel variational principles; the geometric features of the underlying domain, such as convexity or curvature of the boundary, play a crucial role in our results.

In Paper I we consider the Laplacian on planar, convex domains and compare the lowest eigenvalues corresponding to different choices of mixed boundary conditions in the case in which the boundary contains a straight line segment. The proof relies on estimating the Rayleigh quotient of the derivative of a certain eigenfunction in the unique direction normal to this segment; as a result the established inequalities depend on the geometry of the boundary with respect to this direction, as well as on the convexity of the domain.

In Paper II we also compare the lowest mixed eigenvalues of the Laplacian on simply connected planar domains, but instead rely on a novel variational principle where the minimizers are gradients of eigenfunctions. To the best of our knowledge, this variational principle has not appeared in the literature before. This allows to replace the convexity assumption with a more general assumption regulating the normal directions to the boundary, and to drop the assumption that the boundary contains a straight line segment. Using this novel variational principle we also prove a version of the hot spots conjecture for mixed Dirichlet-Neumann boundary conditions.

In Paper III we extend the eigenvalue inequalities of Paper I to Schrödinger operators on both planar and higher-dimensional domains by generalizing the variational approach therein established; in this case we require the boundary to contain a subset of a hyperplane. The inequalities rely again on the convexity of the domain and on the geometry of both the boundary and the potential V with respect to the unique direction normal to this hyperplane. Further, we prove an inequality between higher order mixed Dirichlet-Neumann eigenvalues and pure Dirichlet eigenvalues of Schrödinger operators.