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2008 Regularity results for non smooth parabolic problems
Giovanni Pisante, Anna Verde
Adv. Differential Equations 13(3-4): 367-398 (2008).


In this paper we deal with the study of some regularity properties of weak solutions to non-linear, second-order parabolic equations and systems of the type \[ u_{t}-{\operatorname{div}} A(Du)=0 \;,\;\;\; (x,t)\in \Omega \times (-T,0)=\Omega_{T}, \] where $\Omega \subset {\mathbb{R}}^{n}$ is a bounded domain, $T>0$, $A:{\mathbb{R}}^{nN}\to {\mathbb{R}}^{N}$ satisfies a $p$-growth condition and $u:\Omega_{T}\to {\mathbb{R}}^{N}$. In particular, we focus our attention on local regularity of the spatial gradient of solutions of problems characterized by weak differentiability and ellipticity assumptions on the vector field $A(z)$. We prove the local Lipschitz continuity of solutions in the scalar case ($N=1$). We extend this result in some vectorial cases under an additional structure condition. Finally, we prove higher integrability and differentiability of the spatial gradient of solutions for general systems.


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Giovanni Pisante. Anna Verde. "Regularity results for non smooth parabolic problems." Adv. Differential Equations 13 (3-4) 367 - 398, 2008.


Published: 2008
First available in Project Euclid: 18 December 2012

zbMATH: 1160.35019
MathSciNet: MR2482422

Primary: 35K55
Secondary: 35B65, 35D10

Rights: Copyright © 2008 Khayyam Publishing, Inc.


Vol.13 • No. 3-4 • 2008
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