The best constant in a trace inequality

Abstract

The best possible constant in the trace inequality ∥v∥ L 1(Γ 1) ⩽ 0 · ∥▽v∥ 2(Ω) , ∀v ϵ H 1 Γ 2 (Ω) is shown to be given by a quantity in terms of the solution of an elliptic boundary value problem, where, Γ 1, Γ 2 are disjoint subsets of ∂Ω with positive measures, and the Sobolev space H 1 Γ 2 (Ω) = {v ϵ H 1 (Ω) ¦v = 0 on Γ 2} . Some examples are presented with either an analytic expression or a numerical value for the best constant. As an application, the best constant c 0 is used to give an upper bound for the first eigenvalue of a modified Steklov eigenvalue problem.