Singular sets of Sobolev functions

Abstract

We are interested in finding Sobolev functions with “large” singular sets. Given N,k∈ N , 1< p<∞, kp< N, for any compact subset A of R N , such that its upper box dimension is less than N− kp, we construct a Sobolev function u∈ W k,p( R N) which is singular precisely on A. We introduce the notions of lower and upper singular dimensions of Sobolev space, and show that both are equal to N− kp. To cite this article: D. Žubrinić, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 539–544.