Symmetrization Of Functions Research Articles

Symmetrization of functions and principal eigenvalues of elliptic operators

In this paper, we consider shape optimization problems for the principal eigen-values 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 −∆ + v · ∇, for which the minimization problem is still well posed. Next, we deal with more general elliptic operators −div(A∇) + v · ∇ + 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.

Quantitative Pólya-Szegö principle for convex symmetrization

We study Polya-Szego inequality involving convex symmetrization of functions and anisotropic Dirichlet integrals. A quantitative estimate of the deviation of a function u from its convex symmetral in terms of the gap between their Dirichlet integrals is given.

Symmetrization of functions in Sobolev spaces and the isoperimetric inequality

A positive measurable function f on Rd can be symmetrized to a function f* depending only on the distance r, and with the same distribution function as f. If the distribution derivatives of f are Radon measures then we have the inequality ∥∇f*∥≤∥∇f∥, where ∥∇f∥ is the total mass of the gradient. This inequality is a generalisation of the classical isoperimetric inequality for sets. Furthermore, and this is important for applications, if f belongs to the Sobolev space H1,P then f* belongs to H1,P and ∥∇f*∥p≤∥∇f∥p.