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On the continuity of the Nemitsky operator induced by a Lipschitz continuous map
1991
Proceedings of the American Mathematical Society
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Let / € R -> R be a Lipschitz continuous function, and let fi be a bounded domain in the Euclidean space R" . For every exponent p e [1, +oo[, the composite map T, = fou maps the Sobolev space W 'p{ÇI,Hl ) into W 'p(Cl, R ). In the scalar case, namely, when A^ = 1 , the operator Tf is continuous from W' 'p(f2, R*) into Wx 'p(il, Uk). In this paper we illustrate a counterexample to the continuity of the operator T, in the case where N > 1 . In the last part of the paper we give some sufficient
doi:10.1090/s0002-9939-1991-1039260-x
fatcat:kjsrfwlaz5bvzhmuyygi7mdwom