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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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##### References:
  Acerbi, E.; Mingione, G., Gradient estimates for a class of parabolic systems, Duke math. J., 136, 285-320, (2007) · Zbl 1113.35105  Acerbi, E.; Mingione, G.; Seregin, G.A., Regularity results for parabolic systems related to a class of non-Newtonian fluids, Ann. inst. H. Poincaré anal. non linéaire, 21, 25-60, (2004) · Zbl 1052.76004  Arkhipova, A.A., On a partial regularity up to the boundary of weak solutions to quasilinear parabolic systems with quadratic growth, Zap. nauchn. sem. S.-peterburg. otdel. mat. inst. Steklov. (POMI), 249, 5, 20-39, (1997) · Zbl 0969.35032  Beck, L., Partial regularity for weak solutions of nonlinear elliptic systems: the subquadratic case, Manuscripta math., 123, 4, 453-491, (2007) · Zbl 1151.35023  Bögelein, V., Partial regularity and singular sets of solutions of higher order parabolic systems, Ann. mat. pura appl. (4), 188, 61-122, (2009) · Zbl 1183.35158  Bögelein, V.; Duzaar, F.; Mingione, G., The boundary regularity of non-linear parabolic systems II, Ann. inst. H. Poincaré anal. non linéaire, 27, 1, 145-200, (2010), (in this issue) · Zbl 1194.35085  Campanato, S., Equazioni paraboliche del secondo ordine e spazi $$\mathfrak{L}^{2, \theta}(\Omega, \delta)$$, Ann. mat. pura appl. (4), 73, 55-102, (1966) · Zbl 0144.14101  Campanato, S., On the nonlinear parabolic systems in divergence form. Hölder continuity and partial Hölder continuity of the solutions, Ann. mat. pura appl. (4), 137, 83-122, (1984) · Zbl 0704.35024  De Giorgi, E., Frontiere orientate di misura minima, (1961), Sem. Scuola Normale Superiore Pisa · Zbl 0296.49031  De Giorgi, E., Un esempio di estremali discontinue per un problema variazionale di tipo ellitico, Boll. unione mat. ital., 4, 135-137, (1968) · Zbl 0155.17603  DiBenedetto, E., Degenerate parabolic equations, Universitext, (1993), Springer-Verlag New York · Zbl 0794.35090  Chen, Y.Z.; DiBenedetto, E., Boundary estimates for solutions of nonlinear degenerate parabolic systems, J. reine angew. math., 395, 102-131, (1989) · Zbl 0661.35052  Duzaar, F.; Gastel, A.; Mingione, G., Elliptic systems, singular sets and dini continuity, Comm. partial differential equations, 29, 1215-1240, (2004) · Zbl 1140.35415  Duzaar, F.; Grotowski, J.F., Optimal interior partial regularity for nonlinear elliptic systems: the method of a-harmonic approximation, Manuscripta math., 103, 267-298, (2000) · Zbl 0971.35025  Duzaar, F.; Grotowski, J.F.; Kronz, M., Partial and full boundary regularity for minimizers of functionals with nonquadratic growth, J. convex anal., 11, 437-476, (2004) · Zbl 1066.49022  Duzaar, F.; Kristensen, J.; Mingione, G., The existence of regular boundary points for non-linear elliptic systems, J. reine angew. math., 602, 17-58, (2007) · Zbl 1214.35021  Duzaar, F.; Mingione, G., The p-harmonic approximation and the regularity of p-harmonic maps, Calc. var. partial differential equations, 20, 235-256, (2004) · Zbl 1142.35433  Duzaar, F.; Mingione, G., Second order parabolic systems, optimal regularity, and singular sets of solutions, Ann. inst. H. Poincaré anal. non linéaire, 22, 705-751, (2005) · Zbl 1099.35042  Duzaar, F.; Mingione, G., Harmonic type approximation lemmas, J. math. anal. appl., 352, 301-335, (2009) · Zbl 1172.35002  F. Duzaar, G. Mingione, K. Steffen, Parabolic systems with polynomial growth and regularity, Mem. Amer. Math. Soc., in press · Zbl 1238.35001  Duzaar, F.G.; Steffen, K., Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. reine angew. math., 546, (2002) · Zbl 0999.49024  Giaquinta, M., A counter-example to the boundary regularity of solutions to quasilinear systems, Manuscripta math., 24, 217-220, (1978) · Zbl 0373.35027  Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, (1983), Princeton Univ. Press Princeton, NJ · Zbl 0516.49003  Giaquinta, M.; Modica, G., Local existence for quasilinear parabolic systems under nonlinear boundary conditions, Ann. mat. pura appl. (4), 149, 41-59, (1987) · Zbl 0655.35049  J. Grotowski, Boundary regularity results for non-linear elliptic systems in divergence form, Habilitationsschrift, 2000  Grotowski, J., Boundary regularity for nonlinear elliptic systems, Calc. var. partial differential equations, 15, 353-388, (2002) · Zbl 1148.35315  Kristensen, J.; Mingione, G., The singular set of minima of integral functionals, Arch. ration. mech. anal., 180, 331-398, (2006) · Zbl 1116.49010  J. Kristensen, G. Mingione, Boundary regularity in variational problems, in press · Zbl 1228.49043  Kristensen, J.; Mingione, G., Boundary regularity of minima, Rend. lincei mat. appl., 19, 265-277, (2008) · Zbl 1194.49048  Kronz, M., Quasimonotone systems of higher order, Boll. unione mat. ital. sez. B artic. ric. mat. (8), 6, 459-480, (2003) · Zbl 1150.35385  Mingione, G., The singular set of solutions to non-differentiable elliptic systems, Arch. ration. mech. anal., 166, 287-301, (2003) · Zbl 1142.35391  Mingione, G., Bounds for the singular set of solutions to non linear elliptic systems, Calc. var. partial differential equations, 18, 373-400, (2003) · Zbl 1045.35024  Mingione, G., Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. math., 51, 355-425, (2006) · Zbl 1164.49324  Simon, J., Compact sets in the space $$L^p(0, T; B)$$, Ann. mat. pura appl. (4), 146, 65-96, (1987) · Zbl 0629.46031  Simon, L., Theorems on regularity and singularity of energy minimizing maps, Lectures math. ETH Zürich, (1996), Birkhäuser Basel  Stará, J.; John, O.; Malý, J., Counterexamples to the regularity of weak solutions of the quasilinear parabolic system, Comment. math. univ. carolin., 27, 123-136, (1986) · Zbl 0625.35047
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