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Zbl 1099.35042
Duzaar, Frank; Mingione, Giuseppe
Second order parabolic systems, optimal regularity, and singular sets of solutions. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22, No. 6, 705-751 (2005). ISSN 0294-1449

The paper presents a new approach to the partial regularity of solutions to nonlinear parabolic systems of type $$u_t - \text{div} A(x,t,u,Du)=0.$$ The method is based on an approximation, the ``$A$-caloric approximation lemma", that is a parabolic analog of the classical harmonic approximation lemma of De Giorgi. This allows to prove optimal partial regularity results for solutions in an elementary way, and under minimal, only natural, assumptions. The assumptions on $\partial_p A(x,t,u,p)$ are standard, but minimal assumptions on the continuity of $A(x,t,u,p)$ in $x,t,u$ are posed. E.g., the Hölder continuity $Du\in C^{\beta,\beta/2}$ of a closed subset $\Sigma$ of Lebesgue measure zero, is proved under the Hölder continuity of $A$ with the same $\beta$, and not uniform in $u$: $$ \vert A(x,t,u,p)-A(x_0,t_0,u_0,p)\vert \le K(\vert u\vert +\vert u_0\vert ) (\vert x-x_0\vert +\vert t-t_0\vert ^{1/2}+\vert u-u_0\vert )^\beta (1+\vert p\vert ), $$ $K: [0,\infty)\to (1,\infty)$ is a nondecreasing function. If $K$ is a constant, the estimate of the parabolic Hausdorff dimension $\dim \Sigma\le n+2-\delta$ is given; \par if $A$ does not depend $u$ (system $u_t - \text{div} A(x,t,Du)=0$ ), the estimate $\dim \Sigma\le n+2-2\beta -\delta$ is given.
[Evgeniy A. Kalita (Donetsk)]

MSC 2000:: *35K55 Nonlinear parabolic equations
35D10 Regularity of generalized solutions of PDE
35B65 Smoothness of solutions of PDE
35K50 Systems of parabolic equations, boundary value problems
35A20 Analytic methods (PDE)

Keywords: partial regularity; caloric approximation lemma; parabolic Hausdorff dimension

Cited in: Zbl 1248.35035 Zbl 1183.35158 Zbl 1097.35070

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