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Partial regularity for subquadratic parabolic systems by \(\mathcal{A}\)-caloric approximation. (English) Zbl 1235.35061

The paper is concerned with nonsingular parabolic systems of the form \(\partial_tu-\nabla\cdot a(t,x,u,Du)=B(x,t,u,Du)\). The structure function \(a\) is assumed to fulfill ellipticity and regularity conditions which are fairly general and quite natural. Moreover, it is assumed to have “polynomial” growth with growth rate \(p\in( {2n\over n+1},2)\). The assumptions on \(B\) are the usual ones.
Partial regularity of weak solutions is proven, up to a singular set of vanishing measure in spacetime. Different characterizations of the singular sets are given. In the case \(p>2\) the problem has been solved (see e.g. [F. Duzaar, G. Mingione and K. Steffen, Mem. Am. Math. Soc. 1005, i-x, 118 p. (2011; Zbl 1238.35001)] and the references therein), and even under more general assumptions. Regularity in the subquadratic case \(p<2\), however, has not been treated before the current paper, except for very special equations. The proof uses a variant of the technique of harmonic approximation.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35K40 Second-order parabolic systems
35K59 Quasilinear parabolic equations

Citations:

Zbl 1238.35001
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References:

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