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Gradient continuity estimates

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Abstract

We consider degenerate elliptic equations of p-Laplacean type

$$-{\rm{div}}\, (\gamma(x)|Du|^{p-2}Du)=\mu\,,$$

and give a sufficient condition for the continuity of Du in terms of a natural non-linear Wolff potential of the right-hand side measure μ. As a corollary we identify borderline condition for the continuity of Du in terms of the data: namely μ belongs to the Lorentz space L(n, 1/(p − 1)), and γ(x) is a Dini continuous elliptic coefficient. This last result, together with pointwise gradient bounds via non-linear potentials, extends to the non homogeneous p-Laplacean system, thereby giving a positive answer in the vectorial case to a conjecture of Verbitsky. Continuity conditions related to the density of μ, or to the decay rate of its L n-norm on small balls, are identified as well as corollaries of the main non-linear potential criterium.

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Authors and Affiliations

  1. Department of Mathematics, Erlangen-Nürnberg, Bismarckstrasse 1 1/2, 91054, Erlangen, Germany

    Frank Duzaar

  2. Dipartimento di Matematica, Università di Parma, Viale G. P. Usberti 53/a, Campus, 43100, Parma, Italy

    Giuseppe Mingione

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  1. Frank Duzaar
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  2. Giuseppe Mingione
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Correspondence to Giuseppe Mingione.

Additional information

Communicated by A. Malchiodi.

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Duzaar, F., Mingione, G. Gradient continuity estimates. Calc. Var. 39, 379–418 (2010). https://doi.org/10.1007/s00526-010-0314-6

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  • DOI: https://doi.org/10.1007/s00526-010-0314-6

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