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

Second order parabolic systems, optimal regularity, and singular sets of solutions

created on 02 Nov 2004
modified by mingione on 30 Oct 2005


Published Paper

Inserted: 2 nov 2004
Last Updated: 30 oct 2005

Journal: Ann. Inst. H. Poincare Anal. Non Lineaire
Volume: 22
Number: 6
Pages: 705-751
Year: 2005


We present a new approach to the partial regularity of solutions to non-linear, second order parabolic systems of the form $$ut - div a(x,t,u,Du) =0.$$ We introduce the A-caloric approximation lemma, a parabolic analogue of the harmonic approximation lemma of De Giorgi. This allows to prove optimal partial regularity results for solutions in an elementary way, under natural assumptions and without requiring a priori regularity of solutions such as boundedness or Hölder continuity, as commonly done. After partial regulariy, we provide bounds for the parabolic Hausdorff dimension of singular sets of solutions.

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