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Preface
On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution
1. | Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States |
2. | Dipartimento di Ingegneria Civile, Edile e Ambientale (DICEA), Università di Padova, 35131 Padova, Italy |
References:
[1] |
A. Banerjee and N. Garofalo, Gradient bounds and monotonicity of the energy for some nonlinear singular diffusion equations,, \emph{Indiana Univ. Math. J.}, 62 (2013), 699.
doi: 10.1512/iumj.2013.62.4969. |
[2] |
Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations,, \emph{J. Diff. Geom.}, 33 (1991), 749.
|
[3] |
M. Crandall, M. Kocan and A. Swiech, $L^p$-theory for fully nonlinear uniformly parabolic equations,, \emph{Comm. Partial Differential Equations}, 25 (2000), 1997.
doi: 10.1080/03605300008821576. |
[4] |
K. Does, An evolution equation involving the normalized $p$-Laplacian,, \emph{Comm. Pure Appl. Anal.}, 10 (2011), 361.
doi: 10.3934/cpaa.2011.10.361. |
[5] |
L. Evans and R. Gariepy, Wiener's criterion for the heat equation,, \emph{Arch. Rational Mech. Anal.}, 78 (1982), 293.
doi: 10.1007/BF00249583. |
[6] |
L. C. Evans and J. Spruck, Motions of level sets by mean curvature, Part I,, \emph{J. Diff. Geom.}, 33 (1991), 635.
|
[7] |
E. Fabes, N. Garofalo and E. Lanconelli, Wiener's criterion for divergence form parabolic operators with $C^1$-Dini continuous coefficients,, \emph{Duke Math. J.}, 59 (1989), 191.
doi: 10.1215/S0012-7094-89-05906-1. |
[8] |
A. Friedman, Parabolic equations of the second order,, \emph{Trans. Amer. Math. Soc.}, 93 (1959), 509.
|
[9] |
R. Gariepy and W. P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations,, \emph{Arch. Rat. Mech. Anal.}, 67 (1977), 25.
|
[10] |
S. Granlund, P. Lindqvist and O. Martio, Note on the PWB-method in the nonlinear case,, \emph{Pacific J. Math.}, 125 (1986), 381.
|
[11] |
M. Gruber, Harnack inequalities for solutions of general second order parabolic equations and estimates of their Hölder constants,, \emph{Math. Z.}, 185 (1984), 23.
doi: 10.1007/BF01214972. |
[12] |
J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Oxford Science Publications, (1993).
|
[13] |
C. Imbert and L. Silvestre, Introduction to fully nonlinear parabolic equations,, in \emph{An Introduction to the K\, 2086 (2013), 7.
doi: 10.1007/978-3-319-00819-6_2. |
[14] |
P. Juutinen, Decay estimates in sup norm for the solutions to a nonlinear evolution equation,, \emph{Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, 144 (2014), 557.
doi: 10.1017/S0308210512001163. |
[15] |
P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian,, \emph{Math. Ann.}, 335 (2006), 819.
doi: 10.1007/s00208-006-0766-3. |
[16] |
P. Juutinen, P. Lindqvist and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear equation,, \emph{SIAM J. Math. Anal.}, 33 (2001), 699.
doi: 10.1137/S0036141000372179. |
[17] |
T. Kilpelainen and P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation,, \emph{SIAM J. Math. Anal.}, 27 (1996), 661.
doi: 10.1137/0527036. |
[18] |
T. Kilpelainen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations,, \emph{Acta Math.}, 172 (1994), 137.
doi: 10.1007/BF02392793. |
[19] |
N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients,, \emph{Izv. Akad. Nauk SSSR Ser. Mat.}, 44 (1980), 161.
|
[20] |
O. Ladyzhenskaja and N. Uraltseva, Linear and Quasilinear Elliptic Equations,, Translated from the Russian by Scripta Technica, (1968).
|
[21] |
O. Ladyzhenskaja, V. A. Solonnikov and N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type,, Translations of Mathematical Monographs, (1967). Google Scholar |
[22] |
G. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996).
doi: 10.1142/3302. |
[23] |
G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, \emph{Nonlinear Anal.}, 12 (1988), 1203.
doi: 10.1016/0362-546X(88)90053-3. |
[24] |
J. Maly and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations,, Mathematical Surveys and Monographs, 51 (1997), 0.
doi: 10.1090/surv/051. |
[25] |
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,, \emph{SIAM J. Math Anal.}, 42 (2010), 2058.
doi: 10.1137/100782073. |
[26] |
V. G. Mazya, The continuity at a boundary point of the solutions of quasi-linear elliptic equations,, \emph{Vestnik Leningrad. Univ.}, 25 (1970), 42.
|
[27] |
K. Miller, Barriers on cones for uniformly elliptic operators,, \emph{Ann. Mat. Pura Appl.}, 76 (1967), 93.
|
[28] |
M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications,, \emph{Comm. Partial Differential Equations}, 22 (1997), 381.
doi: 10.1080/03605309708821268. |
[29] |
O. Perron, Die Stabilittsfrage bei Differentialgleichungen,, \emph{Math. Z.}, 32 (1930), 703.
doi: 10.1007/BF01194662. |
[30] |
J. Serrin, Local behavior of solutions of quasi-linear equations,, \emph{Acta Math.}, 111 (1964), 247.
|
[31] |
I. Skrypnik, On the Wiener criterion for quasilinear degenerate parabolic equations,, \emph{Dokl. Akad. Nauk}, 398 (2004), 458.
|
[32] |
W. Sternberg, Über die Gleichung der Wärmeleitung,, \emph{Math. Ann.}, 101 (1929), 394.
doi: 10.1007/BF01454850. |
[33] |
P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, \emph{J. Differential Equations}, 51 (1984), 126.
doi: 10.1016/0022-0396(84)90105-0. |
[34] |
G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains,, \emph{J. Funct. Anal., 59 (1984), 572.
doi: 10.1016/0022-1236(84)90066-1. |
[35] |
L. Wang, On the regularity theory of fully nonlinear parabolic equations: I,, \emph{Comm. Pure Appl. Math.}, 45 (1992), 27.
doi: 10.1002/cpa.3160450103. |
show all references
References:
[1] |
A. Banerjee and N. Garofalo, Gradient bounds and monotonicity of the energy for some nonlinear singular diffusion equations,, \emph{Indiana Univ. Math. J.}, 62 (2013), 699.
doi: 10.1512/iumj.2013.62.4969. |
[2] |
Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations,, \emph{J. Diff. Geom.}, 33 (1991), 749.
|
[3] |
M. Crandall, M. Kocan and A. Swiech, $L^p$-theory for fully nonlinear uniformly parabolic equations,, \emph{Comm. Partial Differential Equations}, 25 (2000), 1997.
doi: 10.1080/03605300008821576. |
[4] |
K. Does, An evolution equation involving the normalized $p$-Laplacian,, \emph{Comm. Pure Appl. Anal.}, 10 (2011), 361.
doi: 10.3934/cpaa.2011.10.361. |
[5] |
L. Evans and R. Gariepy, Wiener's criterion for the heat equation,, \emph{Arch. Rational Mech. Anal.}, 78 (1982), 293.
doi: 10.1007/BF00249583. |
[6] |
L. C. Evans and J. Spruck, Motions of level sets by mean curvature, Part I,, \emph{J. Diff. Geom.}, 33 (1991), 635.
|
[7] |
E. Fabes, N. Garofalo and E. Lanconelli, Wiener's criterion for divergence form parabolic operators with $C^1$-Dini continuous coefficients,, \emph{Duke Math. J.}, 59 (1989), 191.
doi: 10.1215/S0012-7094-89-05906-1. |
[8] |
A. Friedman, Parabolic equations of the second order,, \emph{Trans. Amer. Math. Soc.}, 93 (1959), 509.
|
[9] |
R. Gariepy and W. P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations,, \emph{Arch. Rat. Mech. Anal.}, 67 (1977), 25.
|
[10] |
S. Granlund, P. Lindqvist and O. Martio, Note on the PWB-method in the nonlinear case,, \emph{Pacific J. Math.}, 125 (1986), 381.
|
[11] |
M. Gruber, Harnack inequalities for solutions of general second order parabolic equations and estimates of their Hölder constants,, \emph{Math. Z.}, 185 (1984), 23.
doi: 10.1007/BF01214972. |
[12] |
J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Oxford Science Publications, (1993).
|
[13] |
C. Imbert and L. Silvestre, Introduction to fully nonlinear parabolic equations,, in \emph{An Introduction to the K\, 2086 (2013), 7.
doi: 10.1007/978-3-319-00819-6_2. |
[14] |
P. Juutinen, Decay estimates in sup norm for the solutions to a nonlinear evolution equation,, \emph{Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, 144 (2014), 557.
doi: 10.1017/S0308210512001163. |
[15] |
P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian,, \emph{Math. Ann.}, 335 (2006), 819.
doi: 10.1007/s00208-006-0766-3. |
[16] |
P. Juutinen, P. Lindqvist and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear equation,, \emph{SIAM J. Math. Anal.}, 33 (2001), 699.
doi: 10.1137/S0036141000372179. |
[17] |
T. Kilpelainen and P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation,, \emph{SIAM J. Math. Anal.}, 27 (1996), 661.
doi: 10.1137/0527036. |
[18] |
T. Kilpelainen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations,, \emph{Acta Math.}, 172 (1994), 137.
doi: 10.1007/BF02392793. |
[19] |
N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients,, \emph{Izv. Akad. Nauk SSSR Ser. Mat.}, 44 (1980), 161.
|
[20] |
O. Ladyzhenskaja and N. Uraltseva, Linear and Quasilinear Elliptic Equations,, Translated from the Russian by Scripta Technica, (1968).
|
[21] |
O. Ladyzhenskaja, V. A. Solonnikov and N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type,, Translations of Mathematical Monographs, (1967). Google Scholar |
[22] |
G. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996).
doi: 10.1142/3302. |
[23] |
G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, \emph{Nonlinear Anal.}, 12 (1988), 1203.
doi: 10.1016/0362-546X(88)90053-3. |
[24] |
J. Maly and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations,, Mathematical Surveys and Monographs, 51 (1997), 0.
doi: 10.1090/surv/051. |
[25] |
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,, \emph{SIAM J. Math Anal.}, 42 (2010), 2058.
doi: 10.1137/100782073. |
[26] |
V. G. Mazya, The continuity at a boundary point of the solutions of quasi-linear elliptic equations,, \emph{Vestnik Leningrad. Univ.}, 25 (1970), 42.
|
[27] |
K. Miller, Barriers on cones for uniformly elliptic operators,, \emph{Ann. Mat. Pura Appl.}, 76 (1967), 93.
|
[28] |
M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications,, \emph{Comm. Partial Differential Equations}, 22 (1997), 381.
doi: 10.1080/03605309708821268. |
[29] |
O. Perron, Die Stabilittsfrage bei Differentialgleichungen,, \emph{Math. Z.}, 32 (1930), 703.
doi: 10.1007/BF01194662. |
[30] |
J. Serrin, Local behavior of solutions of quasi-linear equations,, \emph{Acta Math.}, 111 (1964), 247.
|
[31] |
I. Skrypnik, On the Wiener criterion for quasilinear degenerate parabolic equations,, \emph{Dokl. Akad. Nauk}, 398 (2004), 458.
|
[32] |
W. Sternberg, Über die Gleichung der Wärmeleitung,, \emph{Math. Ann.}, 101 (1929), 394.
doi: 10.1007/BF01454850. |
[33] |
P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, \emph{J. Differential Equations}, 51 (1984), 126.
doi: 10.1016/0022-0396(84)90105-0. |
[34] |
G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains,, \emph{J. Funct. Anal., 59 (1984), 572.
doi: 10.1016/0022-1236(84)90066-1. |
[35] |
L. Wang, On the regularity theory of fully nonlinear parabolic equations: I,, \emph{Comm. Pure Appl. Math.}, 45 (1992), 27.
doi: 10.1002/cpa.3160450103. |
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