# American Institute of Mathematical Sciences

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

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

 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 , and $ f $is a given bounded function. We establish interior Hölder regularity of the gradient by studying two alternatives: The first alternative uses an iteration which is based on an approximation lemma. In the second alternative we use a small perturbation argument. Citation: Amal Attouchi, Eero Ruosteenoja. Gradient regularity for a singular parabolic equation in non-divergence form. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5955-5972. doi: 10.3934/dcds.2020254 ##### References:  [1] R. Argiolas, F. Charro and I. Peral, On the Aleksandrov-Bakel'man-Pucci estimate for some elliptic and parabolic nonlinear operators, Arch. Ration. Mech. Anal., 202 (2011), 875-917. doi: 10.1007/s00205-011-0434-y. Google Scholar [2] A. Attouchi, Local regularity for quasi-linear parabolic equations in non-divergence form, preprint, arXiv: 1809.03241. Google Scholar [3] A. Attouchi and M. Parviainen, Hölder regularity for the gradient of the inhomogeneous parabolic normalized$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$-regularity of the gradient of solutions of degenerate parabolic systems, Ann. Mat. Pura Appl., 145 (1986), 385-405. doi: 10.1007/BF01790549. Google Scholar show all references ##### References:  [1] R. Argiolas, F. Charro and I. Peral, On the Aleksandrov-Bakel'man-Pucci estimate for some elliptic and parabolic nonlinear operators, Arch. Ration. Mech. Anal., 202 (2011), 875-917. doi: 10.1007/s00205-011-0434-y. Google Scholar [2] A. Attouchi, Local regularity for quasi-linear parabolic equations in non-divergence form, preprint, arXiv: 1809.03241. Google Scholar [3] A. Attouchi and M. Parviainen, Hölder regularity for the gradient of the inhomogeneous parabolic normalized$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. 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