Some properties of a class of nonlinear variational \(\mu\)-capacities. (English) Zbl 0657.49006
This paper deals with a class of Borel measures \(\mu\) such that \(\mu (B)=0\) for every Borel subset B belonging to the \(\sigma\)-field B(\(\Omega)\). This class of measures is denoted by \(M_ p(\Omega)\). For every \(\mu\) belonging to \(M_ p(\Omega)\) and B belonging to B(\(\Omega)\), the \(\mu\)-capacity of B, relative to a function f(x,\(\delta)\) (which is Lebesgue measurable in x and convex in \(\delta)\), is defined. The main result of this paper is an explicit formula which permits to reconstruct a measure \(\mu\) of \(M_ p(\Omega)\) from the corresponding \(\mu\)-capacity relative to f(x,\(\delta)\). The result is closely related to the study of limits of solutions of non-linear Dirichlet problems in open sets with holes.
Reviewer: M.Codegone
MSC:
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 31B15 | Potentials and capacities, extremal length and related notions in higher dimensions |
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