Unilateral problems with measure data: Links and convergence. (Q1848326)

The authors study the following obstacle problem: \( \text{Find}\;u\in K\;\text{such that}: \langle -\text{div}\, a(\cdot,u,Du),v-u\rangle \;\geq\;\langle f,v-u\rangle \quad\forall v\in K, \) where \(K\) is a unilateral convex set in \({\mathcal{L}}^1_p(\Omega)\) (the closure of \(C^1_0(\Omega)\) with respect to the \(\| D\cdot\| _p\)-norm, \(1<p<n\)), \(\Omega\) is an open set in \(R^N\), \(f\) is a measure. Some existence and uniqueness results are given. Using the Mosco convergence of unilateral convex sets, the convergence of the corresponding sequence of solutions is shown. The paper extends the results of \textit{L. Boccardo}, \textit{G. R. Cirmi} [J. Convex Anal. 6, 195--206 (1999; Zbl 0958.47038)], \textit{P. Oppezzi} and \textit{A. M. Rossi} [J. Convex Anal. 2, 241--261 (1995; Zbl 0837.49008); Atti Semin. Mat. Fis. Univ. Modena 46, Suppl., 889--914 (1998; Zbl 0949.35059)].