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

### Martin Väth

Czech Academy of Sciences, Prague, Czech Republic

## 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.

## Cite this article

Martin Väth, Continuity and Differentiability of Multivalued Superposition Operators with Atoms and Parameters II. Z. Anal. Anwend. 31 (2012), no. 2, pp. 139–160

DOI 10.4171/ZAA/1452