May  2021, 20(5): 1783-1820. doi: 10.3934/cpaa.2021041

Partial regularity for parabolic systems with VMO-coefficients

Department Mathematik, Universität Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany

Received  October 2020 Revised  January 2021 Published  April 2021

Fund Project: The author has been supported by the FWF-Project P31956-N32 "Doubly nonlinear evolution equations"

In this article we establish a partial Hölder continuity result for weak solutions of parabolic systems, where the nonlinear vector field $ A(\cdot) $ satisfies a standard $ p $-growth condition and a non-degenerate ellipticity condition with respect to the gradient variable, while in the space-time variable $ z = (x,t) $ it verifies a VMO-type condition. Thus, no continuity in the space-time variable is assumed. The proof is based on the method of $ \mathcal{A} $-caloric approximation, applied on suitably chosen intrinsic cylinders.

Citation: Leon Mons. Partial regularity for parabolic systems with VMO-coefficients. Communications on Pure & Applied Analysis, 2021, 20 (5) : 1783-1820. doi: 10.3934/cpaa.2021041
References:
[1]

P. Baroni, Regularity in parabolic Dini continuous systems, Forum Math., 23 (2011), 1281-1322.  doi: 10.1515/FORM.2011.049.  Google Scholar

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V. B$\ddot{o}$geleinF. DuzaarJ. Habermann and C. Scheven, Partial H$\ddot{o}$lder continuity for discontinuous elliptic problems with VMO-coefficients, Proc. Lond. Math. Soc., 103 (2011), 371-404.  doi: 10.1112/plms/pdr009.  Google Scholar

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V. B$\ddot{o}$gelein, F. Duzaar and G. Mingione, The regularity of general parabolic systems with degenerate diffusion, Mem. Amer. Math. Soc., 221 (2013), no. 1041. doi: 10.1090/S0065-9266-2012-00664-2.  Google Scholar

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V. B$\ddot{o}$geleinM. Foss and G. Mingione, Regularity in parabolic systems with continuous coefficients, Math. Z., 270 (2012), 903-938.  doi: 10.1007/s00209-010-0832-0.  Google Scholar

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S. Campanato, Differentiability of the solutions of nonlinear elliptic systems with natural growth, Ann. Mat. Pura Appl., 131 (1982), 75-106.  doi: 10.1007/BF01765147.  Google Scholar

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M. CarozzaN. Fusco and G. Mingione, Partial regularity of minimizers of quasiconvex integrals with subquadratic growth, Ann. Mat. Pura Appl., 175 (1998), 141-164.  doi: 10.1007/BF01783679.  Google Scholar

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E. De Giorgi, Frontiere orientate di misura minima, Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61.  Google Scholar

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E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital. (4), 1 (1968), 135-137.   Google Scholar

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E. DiBenedetto and A. Friedman, Regularity of solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math., 349 (1984), 83-128.   Google Scholar

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E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

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F. Duzaar and J. F. Grotowski, Optimal interior partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation, Manuscripta Math., 103 (2000), 267-298.  doi: 10.1007/s002290070007.  Google Scholar

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F. Duzaar and G. Mingione, Second order parabolic systems, optimal regularity, and singular sets of solutions, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 22 (2005), 705–751. doi: 10.1016/j.anihpc.2004.10.011.  Google Scholar

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F. Duzaar and G. Mingione, Harmonic type approximation lemmas, J. Math. Anal. Appl., 352 (2009), 301-335.  doi: 10.1016/j.jmaa.2008.09.076.  Google Scholar

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F. Duzaar, G. Mingione and K. Steffen, Parabolic systems with polynomial growth and regularity, Mem. Amer. Math. Soc., 214 (2011), no. 1005. doi: 10.1090/S0065-9266-2011-00614-3.  Google Scholar

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F. Duzaar and K. Steffen, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math., 546 (2002), 73-138.  doi: 10.1515/crll.2002.046.  Google Scholar

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M. Foss and J. Geisbauer, Partial regularity for subquadratic parabolic systems with continuous coefficients, Manuscripta Math., 139 (2012), 1-47.  doi: 10.1007/s00229-011-0502-5.  Google Scholar

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M. Foss and G. Mingione, Partial continuity for elliptic problems, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 25 (2008), 471–503. doi: 10.1016/j.anihpc.2007.02.003.  Google Scholar

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E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003. doi: 10.1142/9789812795557.  Google Scholar

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M. Kronz, Partial regularity results for minimizers of quasiconvex functionals of higher order, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 19 (2002), 81–112. doi: 10.1016/S0294-1449(01)00072-5.  Google Scholar

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O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs Vol. 23, American Mathematical Society, Providence R.I., 1968.  Google Scholar

[23]

G. Mingione, Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. Math., 51 (2006), 355-425.  doi: 10.1007/s10778-006-0110-3.  Google Scholar

[24]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.  doi: 10.2307/2372841.  Google Scholar

[25]

C. Scheven, Partial regularity for subquadratic parabolic systems by $\mathcal{A}$-caloric approximation, Rev. Mat. Iberoam., 27 (2011), 751-801.  doi: 10.4171/RMI/652.  Google Scholar

[26]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[27]

J. Star$\acute{a}$ and O. John, Some (new) counterexamples of parabolic systems, Comment. Math. Univ. Carolin., 36 (1995), 503-510.   Google Scholar

show all references

References:
[1]

P. Baroni, Regularity in parabolic Dini continuous systems, Forum Math., 23 (2011), 1281-1322.  doi: 10.1515/FORM.2011.049.  Google Scholar

[2]

V. B$\ddot{o}$geleinF. DuzaarJ. Habermann and C. Scheven, Partial H$\ddot{o}$lder continuity for discontinuous elliptic problems with VMO-coefficients, Proc. Lond. Math. Soc., 103 (2011), 371-404.  doi: 10.1112/plms/pdr009.  Google Scholar

[3]

V. B$\ddot{o}$gelein, F. Duzaar and G. Mingione, The regularity of general parabolic systems with degenerate diffusion, Mem. Amer. Math. Soc., 221 (2013), no. 1041. doi: 10.1090/S0065-9266-2012-00664-2.  Google Scholar

[4]

V. B$\ddot{o}$geleinM. Foss and G. Mingione, Regularity in parabolic systems with continuous coefficients, Math. Z., 270 (2012), 903-938.  doi: 10.1007/s00209-010-0832-0.  Google Scholar

[5]

S. Campanato, Equazioni paraboliche del secondo ordine e spazi $\mathcal{L}^{2, \theta}(\Omega, \delta)$, Ann. Mat. Pura Appl., 73 (1966), 55-102.  doi: 10.1007/BF02415082.  Google Scholar

[6]

S. Campanato, Differentiability of the solutions of nonlinear elliptic systems with natural growth, Ann. Mat. Pura Appl., 131 (1982), 75-106.  doi: 10.1007/BF01765147.  Google Scholar

[7]

M. CarozzaN. Fusco and G. Mingione, Partial regularity of minimizers of quasiconvex integrals with subquadratic growth, Ann. Mat. Pura Appl., 175 (1998), 141-164.  doi: 10.1007/BF01783679.  Google Scholar

[8]

G. Da Prato, Spazi $\mathfrak{L}^{(p, \theta)}(\Omega, \delta)$ e loro propriet$\grave{a}$, Ann. Mat. Pura Appl., 69 (1965), 383-392.  doi: 10.1007/BF02414378.  Google Scholar

[9]

E. De Giorgi, Frontiere orientate di misura minima, Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61.  Google Scholar

[10]

E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital. (4), 1 (1968), 135-137.   Google Scholar

[11]

E. DiBenedetto and A. Friedman, Regularity of solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math., 349 (1984), 83-128.   Google Scholar

[12]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[13]

F. Duzaar and J. F. Grotowski, Optimal interior partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation, Manuscripta Math., 103 (2000), 267-298.  doi: 10.1007/s002290070007.  Google Scholar

[14]

F. Duzaar and G. Mingione, Second order parabolic systems, optimal regularity, and singular sets of solutions, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 22 (2005), 705–751. doi: 10.1016/j.anihpc.2004.10.011.  Google Scholar

[15]

F. Duzaar and G. Mingione, Harmonic type approximation lemmas, J. Math. Anal. Appl., 352 (2009), 301-335.  doi: 10.1016/j.jmaa.2008.09.076.  Google Scholar

[16]

F. Duzaar, G. Mingione and K. Steffen, Parabolic systems with polynomial growth and regularity, Mem. Amer. Math. Soc., 214 (2011), no. 1005. doi: 10.1090/S0065-9266-2011-00614-3.  Google Scholar

[17]

F. Duzaar and K. Steffen, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math., 546 (2002), 73-138.  doi: 10.1515/crll.2002.046.  Google Scholar

[18]

M. Foss and J. Geisbauer, Partial regularity for subquadratic parabolic systems with continuous coefficients, Manuscripta Math., 139 (2012), 1-47.  doi: 10.1007/s00229-011-0502-5.  Google Scholar

[19]

M. Foss and G. Mingione, Partial continuity for elliptic problems, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 25 (2008), 471–503. doi: 10.1016/j.anihpc.2007.02.003.  Google Scholar

[20]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003. doi: 10.1142/9789812795557.  Google Scholar

[21]

M. Kronz, Partial regularity results for minimizers of quasiconvex functionals of higher order, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 19 (2002), 81–112. doi: 10.1016/S0294-1449(01)00072-5.  Google Scholar

[22]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs Vol. 23, American Mathematical Society, Providence R.I., 1968.  Google Scholar

[23]

G. Mingione, Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. Math., 51 (2006), 355-425.  doi: 10.1007/s10778-006-0110-3.  Google Scholar

[24]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.  doi: 10.2307/2372841.  Google Scholar

[25]

C. Scheven, Partial regularity for subquadratic parabolic systems by $\mathcal{A}$-caloric approximation, Rev. Mat. Iberoam., 27 (2011), 751-801.  doi: 10.4171/RMI/652.  Google Scholar

[26]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[27]

J. Star$\acute{a}$ and O. John, Some (new) counterexamples of parabolic systems, Comment. Math. Univ. Carolin., 36 (1995), 503-510.   Google Scholar

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