A Poincaré-Sobolev type inequality on compact Riemannian manifolds with boundary

Abstract

In this paper we study a Poincare-Sobolev type inequality on compact Riemannian n-manifolds with boundary where the exponent growth is critical. Two constants have to be determined. We show that, contrary to the classical Sobolev inequality, the first best constant in this inequality does not depend on the dimension only, but depends on the geometry. It can be represented as the minimum \(\mu\) of a given energy functional. We study the nonlinear PDE associated to this functional which involves the geometry of the boundary. For a star-shaped domain D in \({\mathbb R}^n\) whose boundary has positive Ricci curvature, we give explicitly two Sobolev constants corresponding to the embedding \(H^1(D)\) in \(L^{\frac{2(n-1)}{n-2}}(\partial D)\). This result is used to obtain an explicit geometrical lower bound for \(\mu\).