The shape of extremal functions for Poincar\'e-Sobolev-type inequalities in a ball

TL;DR

This paper investigates the shape and symmetry properties of extremal functions for Poincaré-Sobolev inequalities in a unit ball, revealing their axial symmetry, monotonicity, and how these properties change with the parameter p.

Contribution

It provides a detailed analysis of the symmetry, monotonicity, and uniqueness of extremal functions for Poincaré-Sobolev inequalities in a ball, including new results on symmetry breaking as p varies.

Findings

Minimizers are axially symmetric and strictly monotone along the symmetry axis.

For p close to 2, minimizers are antisymmetric with respect to a hyperplane.

In 2D, minimizers are not antisymmetric for large p.

Abstract

We study extremal functions for a family of Poincar\'e-Sobolev-type inequalities. These functions minimize, for subcritical or critical $p\geq 2$, the quotient ${\|\nabla u\|_2}/{\|u\|_p}$ among all $u \in H^1(B)\setminus\{0\}$ with $\int_{B}u=0$. Here $B$ is the unit ball in $\mathbb{R}^N$. 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 $p$.