Calculus of Variations and Geometric Measure Theory
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F. Duzaar - G. Mingione

Gradient estimates via linear and nonlinear potential

created by mingione on 29 Sep 2012
modified on 30 Sep 2012


Inserted: 29 sep 2012
Last Updated: 30 sep 2012

Journal: J. Funct. Anal.
Volume: 259
Pages: 2961-2998
Year: 2010


We prove new potential and nonlinear potential pointwise gradient estimates for solutions to measure data problems, involving possibly degenerate quasilinear operators whose prototype is given by $-\triangle_p u =\mu$. In particular, no matter the nonlinearity of the equations considered, we show that in the case $p\leq 2$ a pointwise gradient estimate is possible using standard, linear Riesz potentials. The proof is based on the identification of a natural quantity that on one hand respects the natural scaling of the problem, and on the other allows to encode the weaker coercivity properties of the operators considered, in the case $p\leq 2$. In the case $p> 2$ we prove a new gradient estimate employing nonlinear potentials of Wolff type.

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