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The boundary regularity of non-linear parabolic systems. II. (English) Zbl 1194.35085
This is a sequel of [V. Bögelein, F. Duzaar and G. Mingione, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27, No. 1, 201–255 (2010; Zbl 1194.35086)], where the authors have proved a rather general boundary regularity criterion for nonlinear parabolic systems of the form \(u_t-\text{div}\,a(x,t,u,Du)=0\).
The boundary data are of Cauchy-Dirichlet type, given by a function \(g\) on the initial time slice and the lateral boundary. The function \(g\) is assumed to be continuous with Hölder continuous spatial derivative and time derivative in some appropriate Morrey space. The growth, regularity and ellipticity assumptions on \(a\) are somewhat stronger than in the first paper, but completely natural. The authors prove existence of regular initial and lateral boundary points for weak solutions, which has been open except for very special cases.
More precisely, they estimate the parabolic Hausdorff dimension of the singular set in terms of the Hölder exponents from the assumptions, finding that, once these exponents are not too small, the singular sets have lower dimension than the boundary components they are subsets of. Which means that almost every boundary point must be regular.
The proofs involve technically involved continuations of methods many of which have been developped by the authors in earlier work, including fractional difference quotients and the relations between fractional Sobolev and Nikolskii spaces.

MSC:
35B65 Smoothness and regularity of solutions to PDEs
35K51 Initial-boundary value problems for second-order parabolic systems
35K59 Quasilinear parabolic equations
35K65 Degenerate parabolic equations
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