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)-2}*Du*) = \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
*C*^{1}-regularity of the solution, also
in terms of the Wolff potential. The *C*^{1}-regularity criterion is also
related to the density of \mu
and the decay rate of its *L*^{n}-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.

Full document as PDF file

doi:10.5186/aasfm.2010.3541

Copyright © 2010 by Academia Scientiarum Fennica