American Institute of Mathematical Sciences

Advanced
July  2013, 12(4): 1731-1744. doi: 10.3934/cpaa.2013.12.1731

Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials

Vitali Liskevich 1, , Igor I. Skrypnik 2, and  Zeev Sobol 1,

1. 

Department of Mathematics, Swansea University, Swansea SA2 8PP, United Kingdom, United Kingdom
2. 

Institute of Applied Mathematics and Mechanics, Donetsk 83114, Ukraine

Received  April 2011 Revised  March 2012 Published  November 2012

For weak solutions to the evolutional $p$-Laplace equation with a time-dependent Radon measure on the right hand side we obtain pointwise estimates via a nonlinear parabolic potential.
Keywords: measure as a source, Parabolic $p$-Laplace equation, pointwise estimates, nonlinear Wolf potential..
Mathematics Subject Classification: Primary: 35K55, 35K65; Secondary: 35B4.
Citation: Vitali Liskevich, Igor I. Skrypnik, Zeev Sobol. Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1731-1744. doi: 10.3934/cpaa.2013.12.1731
References:
[1]

E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (III), 125 (1957), 25-43.

[2]

E. DiBenedetto, "Degenerate Parabolic Equations," Springer, New York, 1993.

[3]

E. DiBenedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci, 13 (1986), 487-535.

[4]

E. DiBenedetto, U. Gianazza and V. Vespri, A Harnack inequality for a degenerate parabolic equation, Acta Mathematica, 200 (2008), 181-209.

[5]

F. Duzaar and G. Mingione, Gradient estimates in non-linear potential theory, Rend. Lincei - Mat. Appl., 20 (2009), 179-190.

[6]

F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials, Amer. J. Math., 133 (2011), 1093-1149.

[7]

M. de Guzmán, A covering lemma with applications to differentiability of measures and singular integral operators, Studia Math., 34 (1970), 299-317.

[8]

M. de Guzmán, "Differentiation of Integrals in $R^n$," Lecture Notes in Math., 481, Springer, 1975.

[9]

T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161.

[10]

D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49.

[11]

V. Liskevich and I. I. Skrypnik, Harnack inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coefficients from Kato-type classes, J. Diff. Eq., 247 (2009), 2740-2777.

[12]

V. Liskevich, I. I. Skrypnik and Z. Sobol, Potential estimates for quasi-linear parabolic equations, Advanced Nonlinear Studies, 11 (2011), 905-915.

[13]

J. Malý and W. Ziemer, "Fine Regularity of Solutions of Elliptic Partial Differential Equations," Mathematical Surveys and Monographs, 51,

[14]

N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math., 168 (2008), 859-914.

[15]

N. C. Phuc and I. E. Verbitsky, Singular quasilinear and Hessian equations and inequalities, J. Funct. Anal., 256 (2009), 1875-1906.

[16]

I. I. Skrypnik, On the Wiener criterion for quasilinear degenerate parabolic equations (Russian), Dokl. Akad. Nauk, 398 (2004), 458-461.

[17]

N. Trudinger and X.-J. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math., 124 (2002), 369-410.

show all references

References:
[1]

E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (III), 125 (1957), 25-43.

[2]

E. DiBenedetto, "Degenerate Parabolic Equations," Springer, New York, 1993.

[3]

E. DiBenedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci, 13 (1986), 487-535.

[4]

E. DiBenedetto, U. Gianazza and V. Vespri, A Harnack inequality for a degenerate parabolic equation, Acta Mathematica, 200 (2008), 181-209.

[5]

F. Duzaar and G. Mingione, Gradient estimates in non-linear potential theory, Rend. Lincei - Mat. Appl., 20 (2009), 179-190.

[6]

F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials, Amer. J. Math., 133 (2011), 1093-1149.

[7]

M. de Guzmán, A covering lemma with applications to differentiability of measures and singular integral operators, Studia Math., 34 (1970), 299-317.

[8]

M. de Guzmán, "Differentiation of Integrals in $R^n$," Lecture Notes in Math., 481, Springer, 1975.

[9]

T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161.

[10]

D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49.

[11]

V. Liskevich and I. I. Skrypnik, Harnack inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coefficients from Kato-type classes, J. Diff. Eq., 247 (2009), 2740-2777.

[12]

V. Liskevich, I. I. Skrypnik and Z. Sobol, Potential estimates for quasi-linear parabolic equations, Advanced Nonlinear Studies, 11 (2011), 905-915.

[13]

J. Malý and W. Ziemer, "Fine Regularity of Solutions of Elliptic Partial Differential Equations," Mathematical Surveys and Monographs, 51,

[14]

N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math., 168 (2008), 859-914.

[15]

N. C. Phuc and I. E. Verbitsky, Singular quasilinear and Hessian equations and inequalities, J. Funct. Anal., 256 (2009), 1875-1906.

[16]

I. I. Skrypnik, On the Wiener criterion for quasilinear degenerate parabolic equations (Russian), Dokl. Akad. Nauk, 398 (2004), 458-461.

[17]

N. Trudinger and X.-J. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math., 124 (2002), 369-410.

[1]

Chao Zhang, Xia Zhang, Shulin Zhou. Gradient estimates for the strong $p(x)$-Laplace equation. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4109-4129. doi: 10.3934/dcds.2017175
[2]

Manas Kar, Jenn-Nan Wang. Size estimates for the weighted p-Laplace equation with one measurement. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2011-2024. doi: 10.3934/dcdsb.2020188
[3]

Margaret Beck. Stability of nonlinear waves: Pointwise estimates. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : 191-211. doi: 10.3934/dcdss.2017010
[4]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427
[5]

Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 51-66. doi: 10.3934/dcdss.2011.4.51
[6]

Jong-Shenq Guo, Bei Hu. Blowup rate estimates for the heat equation with a nonlinear gradient source term. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 927-937. doi: 10.3934/dcds.2008.20.927
[7]

Chao Zhang, Lihe Wang, Shulin Zhou, Yun-Ho Kim. Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2559-2587. doi: 10.3934/cpaa.2014.13.2559
[8]

Vitali Liskevich, Igor I. Skrypnik. Pointwise estimates for solutions of singular quasi-linear parabolic equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1029-1042. doi: 10.3934/dcdss.2013.6.1029
[9]

Shun Uchida. Solvability of doubly nonlinear parabolic equation with p-laplacian. Evolution Equations and Control Theory, 2022, 11 (3) : 975-1000. doi: 10.3934/eect.2021033
[10]

Huafei Di, Yadong Shang, Jiali Yu. Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source. Electronic Research Archive, 2020, 28 (1) : 221-261. doi: 10.3934/era.2020015
[11]

Yongqin Liu, Weike Wang. The pointwise estimates of solutions for dissipative wave equation in multi-dimensions. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1013-1028. doi: 10.3934/dcds.2008.20.1013
[12]

Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016
[13]

J.-P. Raymond. Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 341-370. doi: 10.3934/dcds.1997.3.341
[14]

Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090
[15]

David Gómez-Castro, Juan Luis Vázquez. The fractional Schrödinger equation with singular potential and measure data. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7113-7139. doi: 10.3934/dcds.2019298
[16]

Gang Li, Fen Gu, Feida Jiang. Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1449-1462. doi: 10.3934/cpaa.2020071
[17]

Arrigo Cellina. The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4071-4085. doi: 10.3934/dcds.2018177
[18]

Mikhail D. Surnachev, Vasily V. Zhikov. On existence and uniqueness classes for the Cauchy problem for parabolic equations of the p-Laplace type. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1783-1812. doi: 10.3934/cpaa.2013.12.1783
[19]

N. V. Krylov. Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2073-2090. doi: 10.3934/dcdsb.2012.17.2073
[20]

Yutian Lei. Wolff type potential estimates and application to nonlinear equations with negative exponents. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2067-2078. doi: 10.3934/dcds.2015.35.2067

2021 Impact Factor: 1.273

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy