Revista Matemática Iberoamericana

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Volume 27, Issue 3, 2011, pp. 751–801
DOI: 10.4171/RMI/652

Published online: 2011-12-06

Partial regularity for subquadratic parabolic systems by $\mathcal{A}$-caloric approximation

Christoph Scheven[1]

(1) Friedrich-Alexander-Universität Erlangen, Germany

We establish a partial regularity result for weak solutions of nonsingular parabolic systems with subquadratic growth of the type $$ \partial_t u - \mathrm{div} a(x,t,u,Du) = B(x,t,u,Du), $$ where the structure function $a$ satisfies ellipticity and growth conditions with growth rate $\frac{2n}{n+2} < p < 2$. We prove Hölder continuity of the spatial gradient of solutions away from a negligible set. The proof is based on a variant of a harmonic type approximation lemma adapted to parabolic systems with subquadratic growth.

Keywords: Parabolic systems, partial regularity, harmonic approximation, singular set, subquadratic growth

Scheven Christoph: Partial regularity for subquadratic parabolic systems by $\mathcal{A}$-caloric approximation. Rev. Mat. Iberoamericana 27 (2011), 751-801. doi: 10.4171/RMI/652