Continuity and Differentiability of Multivalued Superposition Operators with Atoms and Parameters II

Abstract

For a given single- or multivalued function $f$ and “atoms” $S_i$, let $S_f(\lambda ,x)$ be the set of all measurable selections of the function $s\mapsto f(\lambda ,s,x(s))$ which are constant on each $S_i$. It is discussed how this definition must be extended so that $S_f$ can serve as a right-hand side for PDEs when one is looking for weak solutions in Sobolev spaces. Continuity and differentiability of the corresponding operators are studied.