April  2013, 6(4): 1029-1042. doi: 10.3934/dcdss.2013.6.1029

Pointwise estimates for solutions of singular quasi-linear parabolic equations

1. 

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

2. 

Institute of Applied Mathematics and Mechanics, Donetsk 83114, Ukraine

Received  March 2011 Revised  September 2011 Published  December 2012

For a class of singular divergence type quasi-linear parabolicequations with a Radon measure on the right hand side we derivepointwise estimates for solutions via the nonlinear Wolffpotentials.
Citation: Vitali Liskevich, Igor I. Skrypnik. Pointwise estimates for solutions of singular quasi-linear parabolic equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1029-1042. doi: 10.3934/dcdss.2013.6.1029
References:
[1]

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

[2]

E. DiBenedetto, "Degenerate Parabolic Equations,", Springer, (1993). doi: 10.1007/978-1-4612-0895-2. Google Scholar

[3]

E. DiBenedetto, J. M. Urbano and V. Vespri, Current issues on singular and degenerate evolution equations,, in, 1 (2004), 169. Google Scholar

[4]

E. DiBenedetto, U. Gianazza and V. Vespri, A Harnack inequality for a degenerate parabolic equation,, Acta Mathematica, 200 (2008), 181. doi: 10.1007/s11511-008-0026-3. Google Scholar

[5]

F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials,, Amer. J. Math., (). doi: 10.1353/ajm.2011.0023. Google Scholar

[6]

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

[7]

T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137. doi: 10.1007/BF02392793. Google Scholar

[8]

D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1. doi: 10.1215/S0012-7094-02-11111-9. Google Scholar

[9]

O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Uraltceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, 23 (1967). Google Scholar

[10]

V. Liskevich, I. I. Skrypnik and Z. Sobol, Potential estimates for quasi-linear parabolic equations,, preprint 2010., (2010). 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. doi: 10.1016/j.jde.2009.08.018. Google Scholar

[12]

J. Malýand W. Ziemer, "Fine Regularity of Solutions of Elliptic Partial Differential Equations,", Mathematical Surveys and Monographs, 51 (1997). Google Scholar

[13]

N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math. (2), 168 (2008), 859. doi: 10.4007/annals.2008.168.859. Google Scholar

[14]

N. C. Phuc and I. E. Verbitsky, Singular quasilinear and Hessian equations and inequalities,, J. Funct. Anal., 256 (2009), 1875. doi: 10.1016/j.jfa.2009.01.012. Google Scholar

[15]

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

[16]

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

show all references

References:
[1]

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

[2]

E. DiBenedetto, "Degenerate Parabolic Equations,", Springer, (1993). doi: 10.1007/978-1-4612-0895-2. Google Scholar

[3]

E. DiBenedetto, J. M. Urbano and V. Vespri, Current issues on singular and degenerate evolution equations,, in, 1 (2004), 169. Google Scholar

[4]

E. DiBenedetto, U. Gianazza and V. Vespri, A Harnack inequality for a degenerate parabolic equation,, Acta Mathematica, 200 (2008), 181. doi: 10.1007/s11511-008-0026-3. Google Scholar

[5]

F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials,, Amer. J. Math., (). doi: 10.1353/ajm.2011.0023. Google Scholar

[6]

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

[7]

T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137. doi: 10.1007/BF02392793. Google Scholar

[8]

D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1. doi: 10.1215/S0012-7094-02-11111-9. Google Scholar

[9]

O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Uraltceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, 23 (1967). Google Scholar

[10]

V. Liskevich, I. I. Skrypnik and Z. Sobol, Potential estimates for quasi-linear parabolic equations,, preprint 2010., (2010). 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. doi: 10.1016/j.jde.2009.08.018. Google Scholar

[12]

J. Malýand W. Ziemer, "Fine Regularity of Solutions of Elliptic Partial Differential Equations,", Mathematical Surveys and Monographs, 51 (1997). Google Scholar

[13]

N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math. (2), 168 (2008), 859. doi: 10.4007/annals.2008.168.859. Google Scholar

[14]

N. C. Phuc and I. E. Verbitsky, Singular quasilinear and Hessian equations and inequalities,, J. Funct. Anal., 256 (2009), 1875. doi: 10.1016/j.jfa.2009.01.012. Google Scholar

[15]

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

[16]

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

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