Superposition operator in Sobolev spaces on domains

Research Report MRR98-032

Superposition operator in Sobolev spaces on domains

Denis A. Labutin

Abstract: For an arbitrary open set $\Omega\subset { \Bbb R}^n$ we describe all functions $G$ on the real line such that $G\circ u\in W^{1,p}(\Omega)$ for all $u\in W^{1,p}(\Omega)$. New element in the proof is based on Maz'ya's capacitary criterion for the imbedding $ {W^{1,p}(\Omega)\hookrightarrow L^\infty(\Omega)}$.

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