- Max-to-Mean Ratio Bounds for the Fundamental Eigenfunction of the Dirichlet Laplacian
Max-to-Mean Ratio Bounds for the Fundamental Eigenfunction of the Dirichlet Laplacian
Abstract: Max-to-Mean ratio for the fundamental eigenfunction of the Dirichlet Laplacian was developed by Payne-Stakgold, and Payne-Rayner in the early 1970s in the context of shape optimization, with applications in mind. It is a means of gauging various norms of this eigenfunction, and falls in a class known as ``reverse H"older inequalities''. A body of literature has developed around this concept and is now part of core results in Schwarz symmetrization. We review some of its salient features and prove some new results for wedge-like membranes. This is joint work with N. Gamara and A. Hasnaoui (University of Tunis, El-Manar).