Polar sets for supersolutions of degenerate elliptic equations.

Fine topology and quasilinear elliptic equations

“…The function ^ is e^-superharmonic in ^ and ^^(xo) = 1 if E is not p-thin at XQ e Q; for these results see [HK2] and [HK3]. and by Lemma 3.7 the set {x e B: ^.…”

“…Recall that a set E in IR" is s^-polar if there is an j^-superharmonic function u in R", u ^ oo, such that u = oo in E. It was shown in [HK2] (see also [K]) that E is j^-polar if and only if E is of p-capacity zero, i.e. capp(£nQ,Q) = 0 for each open set 0.…”

“…On the other hand, T^ does not coincide with the euclidean topology since there are discontinuous j^-superharmonic functions in H\ see [HK2], [K].…”

Nonlinear balayage on metric spaces

“…This makes it possible for us to develop the theory of balayage in a way different from Heinonen-Kilpeläinen-Martio [24], where a substantial part of the balayage theory was developed before proving that the infimum only needs to be regularized on a set of capacity zero. We generalize the balayage results from [22], [23] and [24] to metric spaces, but in most cases our proofs are different.…”

“…Heinonen, Kilpeläinen and Martio were the first to use nonlinear balayage for studying A-harmonic functions on R n in [22], [23] and [24]. The purpose of this paper is to develop the nonlinear balayage theory on metric spaces.…”