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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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[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.  Google Scholar

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