Calculus of Variations and Geometric Measure Theory

L. Beck - B. Stroffolini

Regularity results for differential forms solving degenerate elliptic systems

created by beck on 23 Nov 2010
modified on 27 May 2013


Published Paper

Inserted: 23 nov 2010
Last Updated: 27 may 2013

Journal: Calc. Var. Partial Differential Equations
Volume: 46
Number: 3-4
Pages: 769-808
Year: 2013
Doi: 10.1007/s00526-012-0503-6


We present a partial Hölder regularity result for differential forms $\omega$ solving degenerate systems of the form \[ d^* A(\cdot,\omega) = 0 \quad \rm{and} \quad d\omega = 0 \] on bounded domains in the weak sense. Here certain continuity, monotonicity, growth and structure condition are imposed on the coefficients, including an asymptotic Uhlenbeck behavior close to the origin. Pursuing an approach of Duzaar and Mingione, we combine non-degenerate and degenerate harmonic-type approximation lemmas for the proof of the partial regularity result, giving several extensions and simplifications. In particular, we benefit from a direct proof of the approximation lemma Diening, Stroffolini and Verde that simplifies and unifies the proof in the power growth case. Moreover, we give the dimension reduction for the set of singular points.