Inclusion relations among fine topologies in nonlinear potential theory.

*(English)*Zbl 0545.31011Here the nonlinear potential theory in question is that version of potential theory that extends the classical results of Wiener-Brelot- Cartan-Choquet et al. from the usual \(L^ 2\) or Hilbert space setting to the \(L^ p\) setting. This version orginated in the works of V. G. Maz’ya, V. P. Havin, N. G. Meyers, and Ju. G. Rešetnjak in the late 1960’s and early 1970’s. What happens is that the usual linear potentials of measures (e.g. Newtonian potentials of charges on conductors) now get replaced by some rather badly nonlinear potentials of such measures. And as a result of this nonlinearity many of the obvious advantages of the classical approach are now lost. However, there are several analogies to the classical theory that can be preserved. These have been developed lately with moderate to good success. For example, one of the principal motivations has been a desire to treat questions of existence and regularity for certain nonlinear second order partial differential equations, paralleling the classical potential theoretic treatment of the Laplace equation and its linear variants. The nonlinear prototype here is the Euler equation for: \(\min \int | \text{grad} u(x)|^ p dx,\) \(p\neq 2\). Some key questions have been: removable singularities, boundary regularity, the Wiener test, and the Kellogg property. For a recent treatment of the potential theory see L. I. Hedberg and Th. H. Wolff, Ann. Inst. Fourier 33, No.4, 161-187 (1983; Zbl 0508.31008), and for the p.d.e. side, R. Gariepy and W. P. Ziemer, Arch. Ration. Mech. Anal. 67, 25-39 (1977; Zbl 0389.35023). Also, quite unexpectedly, this nonlinear theory can be successfully applied to answer removable singularity questions for the semilinear equation \(-\Delta u=u^ p+f\), \(f\geq 0\); see M. Pierre and P. Baras, C. R. Acad. Sci., Paris, Sér. I 295, 519-522 (1982; Zbl 0517.35033).

The present article is yet another addition to this general literature. It deals with the fine topologies generated by the nonlinear potentials (the analogue of the classical fine topology of H. Cartan - the smallest topology making the super harmonic functions continuous). Now there is such a topology for every pair (\(\alpha\),p), where p is as before and \(\alpha\) corresponds to the number of derivatives inherent in the problem. All the inclusion relations between these topologies, for various choices of \(\alpha\) and p, are found. Recently, the reviewer together with J. L. Lewis has continued this study especially with regard to the fine connectivity of sets in N-space [see the article reviewed below]. The case \(\alpha =1\), \(p=2\) is, of course, the classical one where the method of balayage is the force behind the results. In the \(L^ p\) case, geometric measure theoretic methods still, however, produce quite remarkably similar nonlinear analogues of the classical results; see B. Fuglede, Ann. Inst. Fourier 21, No.3, 227-244 (1971; Zbl 0208.138).

The present article is yet another addition to this general literature. It deals with the fine topologies generated by the nonlinear potentials (the analogue of the classical fine topology of H. Cartan - the smallest topology making the super harmonic functions continuous). Now there is such a topology for every pair (\(\alpha\),p), where p is as before and \(\alpha\) corresponds to the number of derivatives inherent in the problem. All the inclusion relations between these topologies, for various choices of \(\alpha\) and p, are found. Recently, the reviewer together with J. L. Lewis has continued this study especially with regard to the fine connectivity of sets in N-space [see the article reviewed below]. The case \(\alpha =1\), \(p=2\) is, of course, the classical one where the method of balayage is the force behind the results. In the \(L^ p\) case, geometric measure theoretic methods still, however, produce quite remarkably similar nonlinear analogues of the classical results; see B. Fuglede, Ann. Inst. Fourier 21, No.3, 227-244 (1971; Zbl 0208.138).

##### MSC:

31D05 | Axiomatic potential theory |