Gradient continuity estimates

Duzaar, Frank; Mingione, Giuseppe
November 2010
Calculus of Variations & Partial Differential Equations;Nov2010, Vol. 39 Issue 3/4, p379
Academic Journal
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 L-norm on small balls, are identified as well as corollaries of the main non-linear potential criterium.


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