The shape of extrernal functions for Poincare-Sobolev-type inequalities in a ball

摘要

We study extremal functions for a family of Poincare-Sobolev-type inequalities. These functions minimize, for subcritical or critical p >=, 2, the quotient del u2/u(p) among all u is an element of H-1 (B) {10} with integral(B)u = 0. Here B is the unit ball in RN. We show that the minimizers are axially symmetric with respect to a line passing through the origin. We also show that they are strictly monotone in the direction of this line. In particular, they take their maximum and minimum precisely at two antipodal points on the boundary of B. We also prove that, for p close to 2, minimizers are antisymmetric with respect to the hyperplane through the origin perpendicular to the symmetry axis, and that, once the symmetry axis is fixed, they are unique (up to multiplication by a constant). In space dimension two, we prove that minimizers are not antisymmetric for large p. (c) 2006 Elsevier Inc. All rights reserved.