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Zeitschrift für Analysis und ihre Anwendungen


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Volume 31, Issue 2, 2012, pp. 139–160
DOI: 10.4171/ZAA/1452

Published online: 2012-03-30

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

Martin Väth[1]

(1) Czech Academy of Sciences, Prague, Czech Republic

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.

Keywords: Superposition operator, Nemytskij operator, multivalued map, atom, parameter dependence, continuity, uniform differentiability, Sobolev space

Väth Martin: Continuity and Differentiability of Multivalued Superposition Operators with Atoms and Parameters II. Z. Anal. Anwend. 31 (2012), 139-160. doi: 10.4171/ZAA/1452