Gradient continuity estimates

Duzaar F, Mingione G (2010)

Publication Type: Journal article

Publication year: 2010

Journal

Calculus of Variations and Partial Differential Equations Springer Verlag (Germany)

Publisher: Springer Verlag (Germany)

Book Volume: 39

Pages Range: 379-418

Journal Issue: 3-4

DOI: 10.1007/s00526-010-0314-6

Abstract

We consider degenerate elliptic equations of p-Laplacean type 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 Ln-norm on small balls, are identified as well as corollaries of the main non-linear potential criterium.

Authors with CRIS profile

Frank Duzaar Lehrstuhl für Partial Differential Equations

Involved external institutions

University of Parma / Università degli Studi di Parma IT Italy (IT)

How to cite

APA:

Duzaar, F., & Mingione, G. (2010). Gradient continuity estimates. Calculus of Variations and Partial Differential Equations, 39(3-4), 379-418. https://dx.doi.org/10.1007/s00526-010-0314-6

MLA:

Duzaar, Frank, and Giuseppe Mingione. "Gradient continuity estimates." Calculus of Variations and Partial Differential Equations 39.3-4 (2010): 379-418.

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