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

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.