On two problems related to the Laplace operator

FARINA, MARIA ANTONIETTA

2015-04-16

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Abstract

We investigate maximization of the functional Ω → ε(Ω) where Ω runs in the set of compact domains of fixed volume v in any Riemannian manifold (M; g) and where ε(Ω) is the mean exit time from of the Brownian motion. Concerning this functional, we study its critical points and prove that they are harmonic domains. We analyze the special case of the Coarea formula when we take a Morse function. We investigate minimization and maximization of the principal eigenvalue of the Laplacian under mixed boundary conditions in case the weight has indefinite sign and varies in the class of rearrangements of a fixed function g0 defined on a smooth and bounded domain Ω in Rn. We prove existence and uniqueness results, and in special cases, we prove results of symmetry and results of symmetry breaking for the minimizer.