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Parabolic Systems with Polynomial Growth and Regularity
About this Title
Frank Duzaar, Department Mathematik, Universität Erlangen–Nürnberg, Bismarckstrasse 1 1/2, 91054 Erlangen, Giuseppe Mingione, Dipartimento di Matematica, Università di Parma, Viale Usberti 53/a, Campus, 43100 Parma, Italy and Klaus Steffen, Mathematisches Institut, Heinrich Heine Universität Düsseldorf, Universitätstr.1 D-40225, Düsseldorf, Germany
Publication: Memoirs of the American Mathematical Society
Publication Year:
2011; Volume 214, Number 1005
ISBNs: 978-0-8218-4967-5 (print); 978-1-4704-0622-6 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00614-3
Published electronically: March 10, 2011
Keywords: Parabolic systems,
regularity,
higher integrability,
singular sets
MSC: Primary 35D10, 35K92
Table of Contents
Chapters
- Acknowledgments
- Introduction
- 1. Results
- 2. Basic material, assumptions
- 3. The $A$-caloric approximation lemma
- 4. Partial regularity
- 5. Some basic regularity results and a priori estimates
- 6. Dimension estimates
- 7. Hölder continuity of $u$
- 8. Non-linear Calderón-Zygmund theory
Abstract
We establish a series of optimal regularity results for solutions to general non-linear parabolic systems \[ u_t- \mathrm {div} \ a(x,t,u,Du)+H=0\,, \] under the main assumption of polynomial growth at rate $p$ i.e. \[ |a(x,t,u,Du)|\leq L(1+|Du|^{p-1})\,,\qquad p \geq 2 \;. \] We give a unified treatment of various interconnected aspects of the regularity theory: optimal partial regularity results for the spatial gradient of solutions, the first estimates on the (parabolic) Hausdorff dimension of the related singular set, and the first Calderón-Zygmund estimates for non-homogeneous problems are here achieved.- Emilio Acerbi and Giuseppe Mingione, Gradient estimates for the $p(x)$-Laplacean system, J. Reine Angew. Math. 584 (2005), 117–148. MR 2155087, DOI 10.1515/crll.2005.2005.584.117
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