Continuity and differentiability of multivalued superposition operators with atoms and parameters. II (Q415182)

The article is a continuation of [Part I, {ibid., No.~1, 93--124 (2012; Zbl 1237.47065)]. It deals with the continuity and differentiability properties of the (multivalued) superposition operator NEWLINE\[NEWLINE\begin{multlined} S_f(\lambda,u,v) = \{y: y \;\text{measurable}, \;y(s) \in f(\lambda,s,u(s),v(s)) \text{ a.e.}\\ \text{ and } y\big|_{S_i} \text{ a.e. constant for every } i\}\end{multlined}NEWLINE\]NEWLINE in Sobolev spaces. The following notations are used here: \(S\) is a domain with atoms \(S_i\) in a finite-dimensional space, \(s \in S\), the functions \(u, v, y\) take values in \({\mathbb R}\), \(\lambda\) is a parameter from \(\Lambda\), \(f(\cdot,\cdot,\cdot,\cdot):\Lambda \times S \times {\mathbb R} \times {\mathbb R} \to {\mathbb R}\) is a single-valued function possibly containing some ``jumps'' on surfaces of type \(u = u_0(\lambda,s,v)\). The necessity of studying continuity and differentiability properties of this operator appears, in particular, in connection with weak solutions to the problem NEWLINE\[NEWLINE-\Delta u(s) \in f(\lambda,s,u(s),\nabla u(s)) \;\text{on} \;S, \quad u\big|_{\partial S} = 0,NEWLINE\]NEWLINE for which the Sobolev space \(W_0^{1,2}(S)\) is natural. The passage from the study of this problem to the study of the operator \(S_f\) is based on the analysis of the operator \(F = T \circ S_f\), where NEWLINE\[NEWLINE(Tv)w = \int_{\Omega_0} v(s)w(s) \, ds + \int_\Gamma v(s)w(s) \, dsNEWLINE\]NEWLINE (\(\Omega_0\) is a subset of \(N\)-dimensional finite measure, \(\Gamma\) is a subset of \((N-1)\)-dimensional finite Hausdorff measure, \(S = \Omega_0 \cup \Gamma\)). It should be noted that the notion of ``atom'' in this article is different from the usual one (an atom \(S_i\) is a subset of \(S\) of positive measure on which any function under consideration is equivalent to a constant).}