Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 35, 2010, 641-678

GRADIENT ESTIMATES VIA NON STANDARD POTENTIALS AND CONTINUITY

Verena Bögelein and Jens Habermann

Università di Parma, Dipartimento di Matematica
Viale Usberti 53/a, Campus, 43100 Parma, Italy; boegelein 'at' mi.uni-erlangen.de

Università di Parma, Dipartimento di Matematica
Viale Usberti 53/a, Campus, 43100 Parma, Italy; habermann 'at' mi.uni-erlangen.de

Abstract. We consider elliptic problems with non standard growth conditions whose most prominent model example is the p(x)-Laplacean equation

- div (|Du|^p(x)-2Du) = \mu,

with a measure data right-hand side \mu. We prove pointwise gradient estimates in terms of a non standard version of the non-linear Wolff potential of the right-hand side measure, and moreover a characterization for C¹-regularity of the solution, also in terms of the Wolff potential. The C¹-regularity criterion is also related to the density of \mu and the decay rate of its Lⁿ-norm on small balls. Moreover, from the pointwise gradient estimates the Calderón and Zygmund theory and several types of local estimates follow as a consequence.

2000 Mathematics Subject Classification: Primary 35D10, 35J60, 35J70.

Key words: Pointwise gradient estimates, non standard Wolff potential, partial differential equations with non standard growth, measure data problems.

Reference to this article: V. Bögelein and J. Habermann: Gradient estimates via non standard potentials and continuity. Ann. Acad. Sci. Fenn. Math. 35 (2010), 641-678.

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doi:10.5186/aasfm.2010.3541