Symmetrization of functions and principal eigenvalues of elliptic operators

Hamel, François; Nadirashvili, Nikolai; Russ, Emmanuel

Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 24, 15 p.

In this paper, we consider shape optimization problems for the principal eigenvalues of second order uniformly elliptic operators in bounded domains of ${ℝ}^n$. We first recall the classical Rayleigh-Faber-Krahn problem, that is the minimization of the principal eigenvalue of the Dirichlet Laplacian in a domain with fixed Lebesgue measure. We then consider the case of the Laplacian with a bounded drift, that is the operator $-\Delta +v·\nabla$, for which the minimization problem is still well posed. Next, we deal with more general elliptic operators $-\mathrm{div}\left(A\nabla \right)+v·\nabla +V$ , for which the coefficients fulfill various pointwise, integral or geometric constraints. In all cases, some operators with radially symmetric coefficients in an equimeasurable ball are shown to have smaller principal eigenvalues. Whereas the Faber-Krahn proof relies on the classical Schwarz symmetrization, another type of symmetrization is defined to handle the case of general (possibly non-symmetric) operators.

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