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

Let f∈R N →R k be a lipschitz continuous function, and let Ω be a bounded domain in the Euclidean space R n . For every exponent p∈[1, +∞], the composite map T f =f○u maps the Sobolev space Ws1 ,p (Ω,R N ) into W 1,p (Ω,R k ). In the scalar case, namely, when N=1, the operator T f is continuous from W 1,p (Ω,R N ) into W 1,p (Ω,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