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Second order parabolic systems, optimal regularity, and singular sets of solutions. (English) Zbl 1099.35042
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 \(u\): \[ | A(x,t,u,p)-A(x_0,t_0,u_0,p)| \leq K(| u| +| u_0| ) (| x-x_0| +| t-t_0| ^{1/2}+| u-u_0| )^\beta (1+| p| ), \] \(K: [0,\infty)\to (1,\infty)\) is a nondecreasing function. If \(K\) is a constant, the estimate of the parabolic Hausdorff dimension \(\dim \Sigma\leq n+2-\delta\) is given;
if \(A\) does not depend \(u\) (system \(u_t - \text{div} A(x,t,Du)=0\) ), the estimate \(\dim \Sigma\leq n+2-2\beta -\delta\) is given.

MSC:
35K55 Nonlinear parabolic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35A20 Analyticity in context of PDEs
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