On the continuity of the Nemitsky operator induced by a Lipschitz continuous map

Abstract:

Let $f \in {\mathbb {R}^N} \to {\mathbb {R}^k}$ be a Lipschitz continuous function, and let $\Omega$ be a bounded domain in the Euclidean space ${\mathbb {R}^n}$. For every exponent $p \in [1, + \infty [$ the composite map ${T_f} = f \circ u$ maps the Sobolev space ${W^{1,p}}(\Omega ,{\mathbb {R}^N})$) into ${W^{1,p}}(\Omega ,{\mathbb {R}^k})$). In the scalar case, namely, when $N = 1$, the operator ${T_f}$ is continuous from ${W^{1,p}}(\Omega ,{\mathbb {R}^N})$ into ${W^{1,p}}(\Omega ,{\mathbb {R}^k})$. In this paper we illustrate a counterexample to the continuity of the operator ${T_f}$ in the case where $N > 1$. In the last part of the paper we give some sufficient conditions for the continuity of ${T_f}$, and we conclude with some examples.