Zeitschrift für Analysis und ihre Anwendungen
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Published online: 2012-03-30
Continuity and Differentiability of Multivalued Superposition Operators with Atoms and Parameters IIMartin Väth (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