A Rellich-Kondrachov theorem for spiky domains

Suppose that Omega subset of R-n is a bounded domain whose boundary can be locally represented as the graph of a (necessarily lower semicontinuous) fu nction with Omega lying below its boundary. We show that if such boundary f unctions have everywhere defined limits, then, for 1 less than or equal to p < infinity, the Sobolev embedding iota : W-1,W-p(Omega) hooked right arro w L-p(Omega) is compact. If n = 2, then it is sufficient for the boundary f unctions to have left and right limits everywhere defined; in particular, f or n = 2 it is sufficient for partial derivative Omega to be locally connec ted or to have finite length. The same results hold for bounded domains loc ally diffeomorphic to such graphical domains, and thus in particular for st arlike domains defined by radial functions with the appropriate properties. For n = 3 these results, in greater generality, follow easily from the lite rature. For n = 2 the key property of the above domains to be established i s that locally, after the application of interior diffeomorphisms, their bo undaries can be written as the graphs of continuous functions; compactness then follows from standard theorems.