Let f be a Lipschitz function of N real variables with values into R^k. Let Ω be a bounded domain in $R^h$,and take $1<p<\infty$. The Nemitsky operator $T$ associated with $f$ is defined, for $u$ in the Sobolev space $W^{1,p}(Ω,R^N)$, by $Tu=f o u$. When $N=1$, $T$ is continuous from $W^{1,p}(Ω,R^N)$ into $W^{1,p}(Ω,R^k)$. In this paper a counterexample is given to show that the above is not true when $N=2$. However, additional conditions on $f$ are provided which ensure that $T$ does map $W^{1,p}(Ω,R^N)$ continuously into $W^{1,p}(Ω,R^k)$. In particular this is so when $f$ is Lipschitz with first order derivatives which are continuous outside a closed singular set with empty interior.

ON THE CONTINUITY OF THE NEMITSKY OPERATOR INDUCED BY A LIPSCHITZ CONTINUOUS MAP

Musina, Roberta
1991

Abstract

Let f be a Lipschitz function of N real variables with values into R^k. Let Ω be a bounded domain in $R^h$,and take $1
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http://www.ams.org/journals/proc/1991-111-04/S0002-9939-1991-1039260-X/home.html?pagingLink=%3Ca+href%3D%22%2Fepubsearch%2Fservlet%2FPubSearch%3Fco1%3Dand%26co2%3Dand%26co3%3Dand%26endmo%3D00%26f1%3Dmsc%26f2%3Dtitle%26f3%3Danywhere%26f4%3Dauthor%26pubname%3Dall%26sendit22%3DSearch%26sperpage%3D30%26ssort%3Dd%26startmo%3D00%26v4%3Dmusina%26startRec%3D1%22%3E
Musina, Roberta
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/32718
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