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The eigenvalue problem for the Laplacian on bounded, planar, convex domainswith mixed boundary conditions is considered, where a Dirichlet boundarycondition is imposed on a part of the boundary and a Neumann boundary conditionon its complement. Given two different such choices of boundary conditions forthe same domain, we prove inequalities between their lowest eigenvalues. As aspecial case, we prove parts of a conjecture on the order of mixed eigenvaluesof triangles.

Inequalities between the lowest eigenvalues of Laplacians with mixed boundary conditions