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The boundary regularity of non-linear parabolic systems I. (English) Zbl 1194.35086
The paper is devoted to the study of boundary regularity for fairly general parabolic systems of the type \(u_t-\text{div}\,a(x,t,u,Du)=0\). Boundary values (given by a continuous function \(g\) with Hölder continuous spatial derivative and time derivative in some Morrey space) are prescribed at both an initial time \(t=0\) and at the spatial boundary of the domain at all times. The function \(a\) is allowed to have \(p\)-growth (\(p\geq2\)) in \(Du\) at infinity, is assumed to be uniformly elliptic, and as a function of \((x,u)\) it is merely assumed to be Hölder continuous.
The paper provides a regularity condition under which a boundary point is regular, in the sense that the spatial gradient is Hölder continuous in a relative neighborhood of such a point. More precisely, there are only two ways a boundary point can fail to be regular: Either the liminf of the mean integral of \(|D(u-g)-\overline{D(u-g)}|^p\) over parabolic (half) cylinders is positive or \(\overline{D(u-g)}\) does not stay bounded as the cylinders shrink to the boundary point.
In the proof, the weak solutions of the nonlinear system are related to nearby solutions of a some linear parabolic system derived from it. This is done using an appropriate version (including the \(L^p\) case) of what the authors call the “\(A\)-caloric approximation lemma”; a lemma that states that for every function solving the linear parabolic equation approximately, there is an exact solution within a distance that can be estimated.
The regularity criterion proved here does not guarantee the existence of even one regular boundary point, since the assumptions are fairly general. In a sequel to the paper [V. Bögelein, F. Duzaar, G. Mingione, Ann. Inst. Henry Poincaré, Anal. Non Linéaire 27, No. 1, 145–200 (2010; Zbl 1194.35085)], the authors prove dimensional reduction for the singular set under additional hypotheses which allow the conclusion that almost every boundary point is regular.

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