Advances in Differential Equations

Regularity results for non smooth parabolic problems

Giovanni Pisante and Anna Verde

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Abstract

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.

Article information

Source
Adv. Differential Equations Volume 13, Number 3-4 (2008), 367-398.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867354

Mathematical Reviews number (MathSciNet)
MR2482422

Zentralblatt MATH identifier
1160.35019

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B65: Smoothness and regularity of solutions 35D10

Citation

Pisante, Giovanni; Verde, Anna. Regularity results for non smooth parabolic problems. Adv. Differential Equations 13 (2008), no. 3-4, 367--398. https://projecteuclid.org/euclid.ade/1355867354.


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