Mathematica
Volumen 33, 2008, 491-510

# FINE TOPOLOGY OF VARIABLE EXPONENT ENERGY SUPERMINIMIZERS

## Petteri Harjulehto and Visa Latvala

University of Helsinki, Department of Mathematics and Statistics
P.O. Box 68, FI-00014 University of Helsinki, Finland; petteri.harjulehto 'at' helsinki.fi

University of Joensuu, Department of Physics and Mathematics
P.O. Box 111, FI-80101 Joensuu, Finland; visa.latvala 'at' joensuu.fi

Abstract. We study the p(\cdot)-fine continuity in the variable exponent Sobolev spaces under the standard assumptions that p : \Omega \to R is \log-Hölder continuous and 1 < p- \le p+ < \infty. As a by-product we obtain improvements in the variational exponent capacity theory and in the non-linear potential theory based on p(\cdot)-Laplacian.

2000 Mathematics Subject Classification: Primary 31C05; Secondary 31C45, 46E35, 49N60.

Key words: Non-standard growth, variable exponent, Laplace equation, supersolution, fine topology.

Reference to this article: P. Harjulehto and V. Latvala: Fine topology of variable exponent energy superminimizers. Ann. Acad. Sci. Fenn. Math. 33 (2008), 491-510.