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Lectures
Seminar information archive ～04/23｜Next seminar｜Future seminars 04/24～
2007/04/16
16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Francois Hamel (エクス・マルセーユ第3大学 (Universite Aix-Marseille III))
Rearrangement inequalities and isoperimetric eigenvalue problems for second-order differential operators
Francois Hamel (エクス・マルセーユ第3大学 (Universite Aix-Marseille III))
Rearrangement inequalities and isoperimetric eigenvalue problems for second-order differential operators
[ Abstract ]
The talk is concerned with various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of R^n. We show that, to each operator in a given domain, we can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types.
The results are new even for symmetric operators or in dimension 1. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new rearrangement technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ.
The talk is concerned with various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of R^n. We show that, to each operator in a given domain, we can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types.
The results are new even for symmetric operators or in dimension 1. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new rearrangement technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ.
