Some Imbedding Theorems for Sobolev Spaces

Source abstract

We shall be concerned throughout this paper with the Sobolev space W m,p ( G ) and the existence and compactness (or lack of it) of its imbeddings (i.e. continuous inclusions) into various L P spaces over G , where G is an open, not necessarily bounded subset of n -dimensional Euclidean space E n . For each positive integer m and each real p ≧ 1 the space W m,p ( G ) consists of all u in L P ( G ) whose distributional partial derivatives of all orders up to and including m are also in L P ( G ). With respect to the norm 1.1 W m,p ( G ) is a Banach space. It has been shown by Meyers and Serrin [ 9 ] that the set of functions in C m ( G ) which, together with their partial derivatives of orders up to and including m , are in L P ( G ) forms a dense subspace of W m,p ( G ).