For a given single- or multivalued function and “atoms” , let be the set of all measurable selections of the function \( s \mapsto f(\lambda, s, x(s)) \) which are constant on each . It is discussed how this definition must be extended so that 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. 139–160DOI 10.4171/ZAA/1452