Geometrical Aspects of Symmetrization

Abstract

Symmetrization is one of the most powerful mathematical tools with several applications both in Analysis and Geometry. Probably the most remarkable application of Steiner symmetrization of sets is the De Giorgi proof (see [14], [25]) of the isoperimetric property of the sphere, while the spherical symmetrization of functions has several applications to PDEs and Calculus of Variations and to integral inequalities of Poincare and Sobolev type (see for instance [23], [24], [19], [20]). The two model functionals that we shall consider in the sequel are: the perimeter of a set E in IR and the Dirichlet integral of a scalar function u. It is well known that on replacing E or u by its Steiner symmetral or its spherical symmetrization, respectively, both these quantities decrease. This fact is classical when E is a smooth open set and u is a C function ([22], [21]). Moreover, on approximating a set of finite perimeter with smooth open sets or a Sobolev function by C functions, these inequalities can be easily extended by lower semicontinuity to the general setting ([19], [25], [2], [4]). However, an approximation argument gives no information about the equality case. Thus, if one is interested in understanding when equality occurs, one has to carry on a deeper analysis, based on fine properties of sets of finite perimeter and Sobolev functions. Let us start by recalling what the Steiner symmetrization of a measurable set E is. For simplicity, and without loss of generality, in the sequel we shall always consider the symmetrization of E in the vertical direction. To this aim, it is convenient to denote the points x in IR also by (x′, y), where x′ ∈ IRn−1 and y ∈ IR. Thus, given x′ ∈ IRn−1, we shall denote by Ex′ the corresponding one-dimensional section of E