Optimal Poincaré type trace inequalities on the Euclidean ball (Analysis on Shapes of Solutions to Partial Differential Equations)

Abstract

The shape of extremal functions in Poincaré type trace inequalities for functions of bounded variation in the unit ball mathrm{B}^{n} of the n-dimensional Euclidean space mathbb{R}^{n} is discussed. Both customary and less standard normalization conditions are considered. The extremals in question tum out to take a different form, depending on the condition imposed. A key step in our analysis is a characterization of the sharp constants in the relevant trace inequalities in any admissible domain $Omega$subset mathbb{R}^{n}, in terms of isoperimetric inequalities for subsets of $Omega$.