Choquet integrals in potential theory. (English) Zbl 0923.31006

Adams, David R.

Publ. Mat., Barc. 42, No. 1, 3-66 (1998).

This article is a survey about the applications of the Choquet integral to questions in potential theory. The topics include the notion of the Choquet integral as the integral of a function with respect to capacity. A study of the Choquet space \(L^q(C)\) for various choices of the capacity \(C\) is included. The special role of the Hausdorff capacity is given a thorough presentation and the dual spaces and the embeddings of the spaces into more familiar ones is discussed for the case of the Bessel capacity. The author also discusses a general stability criterion for solutions to an obstacle problem for the \(p\)-Laplace operator on a given bounded domain of the \(n\)-space using Choquet integral techniques. Finally the use of the Choquet integral in refining the usual pointwise differentiation theory for functions is discussed.
The importance of the article is twofold. Firstly, a large amount of material is reorganized in a most readable form, and secondly, as also emphasized by the author, many of the results discussed in the survey are quite recent so that the presentation makes them better available to the mathematical community.

Reviewer: O.-P.Piirilä (Helsinki)

MSC:

31C15	Potentials and capacities on other spaces
28A25	Integration with respect to measures and other set functions

Keywords:

Choquet integral; Choquet space; Hausdorff capacity; Bessel capacity; obstacle problem for the \(p\)-Laplace operator; differentiation