Density results in sobolev spaces whose elements vanish on a part of the boundary

Abstract

This paper is devoted to the study of the subspace ofW ^m,r of functions that vanish on a part γ ₀ of the boundary. The author gives a crucial estimate of the Poincaré constant in balls centered on the boundary of γ ₀. Then, the convolution-translation method, a variant of the standard mollifier technique, can be used to prove the density of smooth functions that vanish in a neighborhood of γ ₀, in this subspace. The result is first proved for m = 1, then generalized to the case where m ≥ 1, in any dimension, in the framework of Lipschitz-continuous domain. However, as may be expected, it is needed to make additional assumptions on the boundary of γ ₀, namely that it is locally the graph of some Lipschitz-continuous function.