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The two-component $ \mu $-Camassa–Holm system with peaked solutions
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 |
$ \partial_t u-|Du|^\gamma\Delta_p^N u = f, $ |
$ -1<\gamma<0 $ |
$ 1<p<\infty $ |
$ f $ |
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. |
[2] |
A. Attouchi, Local regularity for quasi-linear parabolic equations in non-divergence form, preprint, arXiv: 1809.03241. |
[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. |
[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. |
[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. |
[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. |
[7] |
A. Banerjee and I. H. Munive, Gradient continuity estimates for the normalized $p$-Poisson equation, preprint, arXiv: 1904.13076.
doi: 10.1142/S021919971950069X. |
[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. |
[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. |
[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. |
[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. |
[12] |
M. G. Crandall, Viscosity solutions: A primer, in Lecture Notes in Math., 1660, Springer, Berlin, (1997), 1–43.
doi: 10.1007/BFb0094294. |
[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. |
[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. |
[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. |
[16] |
E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[28] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
doi: 10.1142/3302. |
[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. |
[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. |
[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. |
[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. |
[33] |
O. Savin,
Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations, 32 (2007), 557-578.
doi: 10.1080/03605300500394405. |
[34] |
J. Siltakoski, Equivalence of viscosity and weak solutions for a $p$-parabolic equation, preprint, arXiv: 1901.02507. |
[35] |
L. Wang,
Compactness methods for certain degenerate elliptic equations, J. Differential Equations, 107 (1994), 341-350.
doi: 10.1006/jdeq.1994.1016. |
[36] |
Y. Wang,
Small perturbation solutions for parabolic equations, Indiana Univ. Math. J., 62 (2013), 671-697.
doi: 10.1512/iumj.2013.62.4961. |
[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. |
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. |
[2] |
A. Attouchi, Local regularity for quasi-linear parabolic equations in non-divergence form, preprint, arXiv: 1809.03241. |
[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. |
[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. |
[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. |
[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. |
[7] |
A. Banerjee and I. H. Munive, Gradient continuity estimates for the normalized $p$-Poisson equation, preprint, arXiv: 1904.13076.
doi: 10.1142/S021919971950069X. |
[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. |
[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. |
[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. |
[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. |
[12] |
M. G. Crandall, Viscosity solutions: A primer, in Lecture Notes in Math., 1660, Springer, Berlin, (1997), 1–43.
doi: 10.1007/BFb0094294. |
[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. |
[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. |
[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. |
[16] |
E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[28] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
doi: 10.1142/3302. |
[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. |
[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. |
[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. |
[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. |
[33] |
O. Savin,
Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations, 32 (2007), 557-578.
doi: 10.1080/03605300500394405. |
[34] |
J. Siltakoski, Equivalence of viscosity and weak solutions for a $p$-parabolic equation, preprint, arXiv: 1901.02507. |
[35] |
L. Wang,
Compactness methods for certain degenerate elliptic equations, J. Differential Equations, 107 (1994), 341-350.
doi: 10.1006/jdeq.1994.1016. |
[36] |
Y. Wang,
Small perturbation solutions for parabolic equations, Indiana Univ. Math. J., 62 (2013), 671-697.
doi: 10.1512/iumj.2013.62.4961. |
[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. |
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