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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.

MSC:
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
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