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FINE TOPOLOGY OF VARIABLE EXPONENT ENERGY SUPERMINIMIZERS

We study the p(¢)-fine continuity in the variable exponent Sobolev spaces under the standard assumptions that p: › ! R is log-Holder continuous and 1 < p i • p + < 1. As a by-product we obtain improvements in the variational exponent capacity theory and in the non-linear potential theory based on p(¢)-Laplacian. Quasicontinuity is a central notion in the Sobolev function theory and in poten- tial theory. This is so since many crucial ideas in the theory of pde's require the use of quasicontinuous representatives of Sobolev functions. For the pointwise study of quasicontinuous functions the Euclidean topology is not relevant in general, instead one can use the fine topology, which dates back to Cartan (6) in the linear case. Nowadays it is well-known that, for any constant 1 < p < 1, the function u is p-quasicontinuous if and only if u is p-finely continuous outside a set of p-capacity zero. This result is deep, the implication from left to right requires sharp energy estimates for supersolutions of p-Laplace equation. The converse implication which is related to Choquet's property is even deeper, it was first established for general p in (24), Theorem 3; see also (12) and (2). Another proof based on the pointwise estimates of p-supersolutions of (30) can be found in (32), p. 145. The goal of this paper is to introduce the fine topology in the variable ex- ponent case and show that quasicontinuity implies fine continuity outside a set of capacity zero even in this setting (Theorem 6.5). In particular we show that each p(¢)-superharmonic function is p(¢)-finely continuous and p(¢)-quasicontinuous (Theorems 5.3 and 6.7). As a by-product we obtain several improvements related to the variational exponent capacity theory or the non-linear potential theory based on p(¢)-Laplacian. For instance we establish the existence of the capacitary extremal for arbitrary subsets compactly contained in a given bounded open set (Theorem 6.3).