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October  2020, 40(10): 5955-5972. doi: 10.3934/dcds.2020254

Gradient regularity for a singular parabolic equation in non-divergence form

Amal Attouchi and  Eero Ruosteenoja ,

Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, FI-40014 Jyväskylä, Finland

* Corresponding author: eero.k.ruosteenoja@jyu.fi

Received  February 2020 Revised  April 2020 Published  June 2020

In this paper we consider viscosity solutions of a class of non-homogeneous singular parabolic equations
$ \partial_t u-|Du|^\gamma\Delta_p^N u = f, $
where
$ -1<\gamma<0 $
,
$ 1<p<\infty $
, and
$ f $
 is a given bounded function. We establish interior Hölder regularity of the gradient by studying two alternatives: The first alternative uses an iteration which is based on an approximation lemma. In the second alternative we use a small perturbation argument.
Keywords: Singular parabolic equations, regularity of the gradient, viscosity solutions.
Mathematics Subject Classification: 35K55, 35B65, 35D40, 35K92.
Citation: Amal Attouchi, Eero Ruosteenoja. Gradient regularity for a singular parabolic equation in non-divergence form. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5955-5972. doi: 10.3934/dcds.2020254
References:
[1]

R. Argiolas, F. Charro and I. Peral, On the Aleksandrov-Bakel'man-Pucci estimate for some elliptic and parabolic nonlinear operators, Arch. Ration. Mech. Anal., 202 (2011), 875-917.  doi: 10.1007/s00205-011-0434-y.  Google Scholar

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[3]

A. Attouchi and M. Parviainen, Hölder regularity for the gradient of the inhomogeneous parabolic normalized $p$-Laplacian, Commun. Contemp. Math., 20 (2018), 27 pp. doi: 10.1142/S0219199717500353.  Google Scholar

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A. Attouchi, M. Parviainen and E. Ruosteenoja, $C^{1, \alpha}$ regularity for the normalized $p$-Poisson problem, J. Math. Pures Appl., 108 (2017), 553-591.  doi: 10.1016/j.matpur.2017.05.003.  Google Scholar

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A. Attouchi and E. Ruosteenoja, Remarks on regularity for $p$-Laplacian type equations in non-divergence form, J. Differential Equations, 265 (2018), 1922-1961.  doi: 10.1016/j.jde.2018.04.017.  Google Scholar

[6]

A. Banerjee and N. Garofalo, On the Dirichlet boundary value problem for the normalized $p$-Laplacian evolution, Commun. Pure Appl. Anal., 14 (2015), 1-21.  doi: 10.3934/cpaa.2015.14.1.  Google Scholar

[7]

A. Banerjee and I. H. Munive, Gradient continuity estimates for the normalized $p$-Poisson equation, preprint, arXiv: 1904.13076. doi: 10.1142/S021919971950069X.  Google Scholar

[8]

D. Berti and R. Magnanini, Short-time behavior for game-theoretic $p$-caloric functions, J. Math. Pures Appl., 126 (2019), 249-272.  doi: 10.1016/j.matpur.2018.06.020.  Google Scholar

[9]

T. Bhattacharya and L. Marazzi, On the viscosity solutions to a class of nonlinear degenerate parabolic differential equations, Rev. Mat. Complut., 30 (2017), 621-656.  doi: 10.1007/s13163-017-0229-2.  Google Scholar

[10]

I. Birindelli and F. Demengel, Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators, J. Differential Equations, 249 (2010), 1089-1110.  doi: 10.1016/j.jde.2010.03.015.  Google Scholar

[11]

I. Birindelli and F. Demengel, $C^{1, \beta}$ regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations, ESAIM Control Optim. Calc. Var., 20 (2014), 1009-1024.  doi: 10.1051/cocv/2014005.  Google Scholar

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M. G. Crandall, Viscosity solutions: A primer, in Lecture Notes in Math., 1660, Springer, Berlin, (1997), 1–43. doi: 10.1007/BFb0094294.  Google Scholar

[13]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

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J. V. da Silva and D. dos Prazeres, Schauder type estimates for "flat" viscosity solutions to non-convex fully nonlinear parabolic equations and applications, Potential Anal., 50 (2019), 149-170.  doi: 10.1007/s11118-017-9677-z.  Google Scholar

[15]

F. Demengel, Existence's results for parabolic problems related to fully nonlinear operators degenerate or singular, Potential Anal., 35 (2011), 1-38.  doi: 10.1007/s11118-010-9201-1.  Google Scholar

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

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E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1-22.  doi: 10.1515/crll.1985.357.1.  Google Scholar

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K. Does, An evolution equation involving the normalized $p$-Laplacian, Commun. Pure Appl. Anal., 10 (2011), 361-396.  doi: 10.3934/cpaa.2011.10.361.  Google Scholar

[19]

H. Dong, F. Peng, Y. R.-Y. Zhang and Y. Zhou, Hessian estimates for equations involving $p$-Laplacian via a fundamental inequality, Advances in Mathematics, 370 (2020), 107212. doi: 10.1016/j.aim.2020.107212.  Google Scholar

[20]

J. Han, Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle, Commun. Pure Appl. Anal., 19 (2020), 2617-2640.  doi: 10.3934/cpaa.2020114.  Google Scholar

[21]

F. A. Høeg and P. Lindqvist, Regularity of solutions of the parabolic normalized $p$-Laplace equation, Adv. Nonlinear Anal., 9 (2020), 7-15.  doi: 10.1515/anona-2018-0091.  Google Scholar

[22]

C. Imbert, T. Jin and L. Silvestre, Hölder gradient estimates for a class of singular or degenerate parabolic equations, Adv. Nonlinear Anal., 8 (2019), 845-867.  doi: 10.1515/anona-2016-0197.  Google Scholar

[23]

C. Imbert and L. Silvestre, $C^{1, \alpha}$ regularity of solutions of some degenerate fully non-linear elliptic equations, Adv. Math., 233 (2013), 196-206.  doi: 10.1016/j.aim.2012.07.033.  Google Scholar

[24]

T. Jin and L. Silvestre, Hölder gradient estimates for parabolic homogeneous $p$-Laplacian equations, J. Math. Pures Appl., 108 (2017), 63-87.  doi: 10.1016/j.matpur.2016.10.010.  Google Scholar

[25]

P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal., 33 (2001), 699-717.  doi: 10.1137/S0036141000372179.  Google Scholar

[26]

T. Kuusi and G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl., 98 (2012), 390-427.  doi: 10.1016/j.matpur.2012.02.004.  Google Scholar

[27]

O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[28]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[29]

J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games, SIAM J. Math. Anal., 42 (2010), 2058-2081.  doi: 10.1137/100782073.  Google Scholar

[30]

M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications to the $p$-Laplace diffusion equation, Comm. Partial Differential Equations, 22 (1997), 381-411.  doi: 10.1080/03605309708821268.  Google Scholar

[31]

M. Parviainen and E. Ruosteenoja, Local regularity for time-dependent tug-of-war games with varying probabilities, J. Differential Equations, 261 (2016), 1357-1398.  doi: 10.1016/j.jde.2016.04.001.  Google Scholar

[32]

M. Parviainen and J. L. Vázquez, Equivalence between radial solutions of different parabolic gradient-diffusion equations and applications, preprint, arXiv: 1801.00613. doi: 10.2422/2036-2145.201808_006.  Google Scholar

[33]

O. Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations, 32 (2007), 557-578.  doi: 10.1080/03605300500394405.  Google Scholar

[34]

J. Siltakoski, Equivalence of viscosity and weak solutions for a $p$-parabolic equation, preprint, arXiv: 1901.02507. Google Scholar

[35]

L. Wang, Compactness methods for certain degenerate elliptic equations, J. Differential Equations, 107 (1994), 341-350.  doi: 10.1006/jdeq.1994.1016.  Google Scholar

[36]

Y. Wang, Small perturbation solutions for parabolic equations, Indiana Univ. Math. J., 62 (2013), 671-697.  doi: 10.1512/iumj.2013.62.4961.  Google Scholar

[37]

M. Wiegner, On $C^\alpha$-regularity of the gradient of solutions of degenerate parabolic systems, Ann. Mat. Pura Appl., 145 (1986), 385-405.  doi: 10.1007/BF01790549.  Google Scholar

show all references

References:
[1]

R. Argiolas, F. Charro and I. Peral, On the Aleksandrov-Bakel'man-Pucci estimate for some elliptic and parabolic nonlinear operators, Arch. Ration. Mech. Anal., 202 (2011), 875-917.  doi: 10.1007/s00205-011-0434-y.  Google Scholar

[2]

A. Attouchi, Local regularity for quasi-linear parabolic equations in non-divergence form, preprint, arXiv: 1809.03241. Google Scholar

[3]

A. Attouchi and M. Parviainen, Hölder regularity for the gradient of the inhomogeneous parabolic normalized $p$-Laplacian, Commun. Contemp. Math., 20 (2018), 27 pp. doi: 10.1142/S0219199717500353.  Google Scholar

[4]

A. Attouchi, M. Parviainen and E. Ruosteenoja, $C^{1, \alpha}$ regularity for the normalized $p$-Poisson problem, J. Math. Pures Appl., 108 (2017), 553-591.  doi: 10.1016/j.matpur.2017.05.003.  Google Scholar

[5]

A. Attouchi and E. Ruosteenoja, Remarks on regularity for $p$-Laplacian type equations in non-divergence form, J. Differential Equations, 265 (2018), 1922-1961.  doi: 10.1016/j.jde.2018.04.017.  Google Scholar

[6]

A. Banerjee and N. Garofalo, On the Dirichlet boundary value problem for the normalized $p$-Laplacian evolution, Commun. Pure Appl. Anal., 14 (2015), 1-21.  doi: 10.3934/cpaa.2015.14.1.  Google Scholar

[7]

A. Banerjee and I. H. Munive, Gradient continuity estimates for the normalized $p$-Poisson equation, preprint, arXiv: 1904.13076. doi: 10.1142/S021919971950069X.  Google Scholar

[8]

D. Berti and R. Magnanini, Short-time behavior for game-theoretic $p$-caloric functions, J. Math. Pures Appl., 126 (2019), 249-272.  doi: 10.1016/j.matpur.2018.06.020.  Google Scholar

[9]

T. Bhattacharya and L. Marazzi, On the viscosity solutions to a class of nonlinear degenerate parabolic differential equations, Rev. Mat. Complut., 30 (2017), 621-656.  doi: 10.1007/s13163-017-0229-2.  Google Scholar

[10]

I. Birindelli and F. Demengel, Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators, J. Differential Equations, 249 (2010), 1089-1110.  doi: 10.1016/j.jde.2010.03.015.  Google Scholar

[11]

I. Birindelli and F. Demengel, $C^{1, \beta}$ regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations, ESAIM Control Optim. Calc. Var., 20 (2014), 1009-1024.  doi: 10.1051/cocv/2014005.  Google Scholar

[12]

M. G. Crandall, Viscosity solutions: A primer, in Lecture Notes in Math., 1660, Springer, Berlin, (1997), 1–43. doi: 10.1007/BFb0094294.  Google Scholar

[13]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[14]

J. V. da Silva and D. dos Prazeres, Schauder type estimates for "flat" viscosity solutions to non-convex fully nonlinear parabolic equations and applications, Potential Anal., 50 (2019), 149-170.  doi: 10.1007/s11118-017-9677-z.  Google Scholar

[15]

F. Demengel, Existence's results for parabolic problems related to fully nonlinear operators degenerate or singular, Potential Anal., 35 (2011), 1-38.  doi: 10.1007/s11118-010-9201-1.  Google Scholar

[16]

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

[17]

E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1-22.  doi: 10.1515/crll.1985.357.1.  Google Scholar

[18]

K. Does, An evolution equation involving the normalized $p$-Laplacian, Commun. Pure Appl. Anal., 10 (2011), 361-396.  doi: 10.3934/cpaa.2011.10.361.  Google Scholar

[19]

H. Dong, F. Peng, Y. R.-Y. Zhang and Y. Zhou, Hessian estimates for equations involving $p$-Laplacian via a fundamental inequality, Advances in Mathematics, 370 (2020), 107212. doi: 10.1016/j.aim.2020.107212.  Google Scholar

[20]

J. Han, Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle, Commun. Pure Appl. Anal., 19 (2020), 2617-2640.  doi: 10.3934/cpaa.2020114.  Google Scholar

[21]

F. A. Høeg and P. Lindqvist, Regularity of solutions of the parabolic normalized $p$-Laplace equation, Adv. Nonlinear Anal., 9 (2020), 7-15.  doi: 10.1515/anona-2018-0091.  Google Scholar

[22]

C. Imbert, T. Jin and L. Silvestre, Hölder gradient estimates for a class of singular or degenerate parabolic equations, Adv. Nonlinear Anal., 8 (2019), 845-867.  doi: 10.1515/anona-2016-0197.  Google Scholar

[23]

C. Imbert and L. Silvestre, $C^{1, \alpha}$ regularity of solutions of some degenerate fully non-linear elliptic equations, Adv. Math., 233 (2013), 196-206.  doi: 10.1016/j.aim.2012.07.033.  Google Scholar

[24]

T. Jin and L. Silvestre, Hölder gradient estimates for parabolic homogeneous $p$-Laplacian equations, J. Math. Pures Appl., 108 (2017), 63-87.  doi: 10.1016/j.matpur.2016.10.010.  Google Scholar

[25]

P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal., 33 (2001), 699-717.  doi: 10.1137/S0036141000372179.  Google Scholar

[26]

T. Kuusi and G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl., 98 (2012), 390-427.  doi: 10.1016/j.matpur.2012.02.004.  Google Scholar

[27]

O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[28]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[29]

J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games, SIAM J. Math. Anal., 42 (2010), 2058-2081.  doi: 10.1137/100782073.  Google Scholar

[30]

M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications to the $p$-Laplace diffusion equation, Comm. Partial Differential Equations, 22 (1997), 381-411.  doi: 10.1080/03605309708821268.  Google Scholar

[31]

M. Parviainen and E. Ruosteenoja, Local regularity for time-dependent tug-of-war games with varying probabilities, J. Differential Equations, 261 (2016), 1357-1398.  doi: 10.1016/j.jde.2016.04.001.  Google Scholar

[32]

M. Parviainen and J. L. Vázquez, Equivalence between radial solutions of different parabolic gradient-diffusion equations and applications, preprint, arXiv: 1801.00613. doi: 10.2422/2036-2145.201808_006.  Google Scholar

[33]

O. Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations, 32 (2007), 557-578.  doi: 10.1080/03605300500394405.  Google Scholar

[34]

J. Siltakoski, Equivalence of viscosity and weak solutions for a $p$-parabolic equation, preprint, arXiv: 1901.02507. Google Scholar

[35]

L. Wang, Compactness methods for certain degenerate elliptic equations, J. Differential Equations, 107 (1994), 341-350.  doi: 10.1006/jdeq.1994.1016.  Google Scholar

[36]

Y. Wang, Small perturbation solutions for parabolic equations, Indiana Univ. Math. J., 62 (2013), 671-697.  doi: 10.1512/iumj.2013.62.4961.  Google Scholar

[37]

M. Wiegner, On $C^\alpha$-regularity of the gradient of solutions of degenerate parabolic systems, Ann. Mat. Pura Appl., 145 (1986), 385-405.  doi: 10.1007/BF01790549.  Google Scholar

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