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December  2015, 35(12): 5665-5688. doi: 10.3934/dcds.2015.35.5665

Variational parabolic capacity

Benny Avelin 1, , Tuomo Kuusi 2, and  Mikko Parviainen 3,

1. 

Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, 40014 Jyväskylä, Finland
2. 

Department of Mathematics and Systems Analysis, Aalto University School of Science, FI-00076 Aalto, Finland
3. 

Department of Mathematics and Statistics, P.O. Box 35, FI-40014 University of Jyväskylä

Received  April 2014 Published  May 2015

We establish a variational parabolic capacity in a context of degenerate parabolic equations of $p$-Laplace type, and show that this capacity is equivalent to the nonlinear parabolic capacity. As an application, we estimate the capacities of several explicit sets.
Keywords: Capacity, nonlinear potential theory, degenerate parabolic equations, $p$-parabolic equation..
Mathematics Subject Classification: Primary: 35K55; Secondary: 31C4.
Citation: Benny Avelin, Tuomo Kuusi, Mikko Parviainen. Variational parabolic capacity. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5665-5688. doi: 10.3934/dcds.2015.35.5665
References:
[1]

D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften 314, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03282-4.  Google Scholar

[2]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. doi: 10.1007/BF01176474.  Google Scholar

[3]

A. Björn, J. Björn, U. Gianazza and M. Parviainen, Boundary regularity for degenerate and singular parabolic equations, Calc. Var. Partial Differential Equations, 52 (2015), 797-827. doi: 10.1007/s00526-014-0734-9.  Google Scholar

[4]

L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237-258. doi: 10.1006/jfan.1996.3040.  Google Scholar

[5]

V. Bögelein, F. Duzaar and G. Mingione, Degenerate problems with irregular obstacles, J. Reine Angew. Math., 650 (2011), 107-160. doi: 10.1515/CRELLE.2011.006.  Google Scholar

[6]

J. Droniou, A. Porretta and A. Prignet, Parabolic capacity and soft measures for nonlinear equations, Potential Anal., 19 (2003), 99-161. doi: 10.1023/A:1023248531928.  Google Scholar

[7]

L. C. Evans and R. F. Gariepy, Wiener's test for the heat equation, Arch. Rational Mech. Anal., 78 (1982), 293-314. doi: 10.1007/BF00249583.  Google Scholar

[8]

R. Gariepy and W. P. Ziemer, Removable sets for quasilinear parabolic equations, J. London Math. Soc., 21 (1980), 311-318. doi: 10.1112/jlms/s2-21.2.311.  Google Scholar

[9]

R. Gariepy and W. P. Ziemer, Thermal capacity and boundary regularity, J. Differential Equations, 45 (1982), 374-388. doi: 10.1016/0022-0396(82)90034-1.  Google Scholar

[10]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Unabridged republication of the 1993 original, Dover Publications, Inc., Mineola, NY, 2006.  Google Scholar

[11]

T. Kilpeläinen and P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation, SIAM J. Math. Anal., 27 (1996), 661-683. doi: 10.1137/0527036.  Google Scholar

[12]

T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793.  Google Scholar

[13]

J. Kinnunen, R. Korte, T. Kuusi and M. Parviainen, Nonlinear parabolic capacity and polar sets of superparabolic functions, Math. Ann., 355 (2013), 1349-1381. doi: 10.1007/s00208-012-0825-x.  Google Scholar

[14]

K. Kinnunen and P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl., 185 (2006), 411-435. doi: 10.1007/s10231-005-0160-x.  Google Scholar

[15]

J. Kinnunen, T. Lukkari and M. Parviainen, An existence result for superparabolic functions, J. Funct. Anal., 258 (2010), 713-728. doi: 10.1016/j.jfa.2009.08.009.  Google Scholar

[16]

J. Kinnunen, T. Lukkari and M. Parviainen, Local approximation of superharmonic and superparabolic functions in nonlinear potential theory, J. Fixed Point Theory Appl., 13 (2013), 291-307. doi: 10.1007/s11784-013-0108-5.  Google Scholar

[17]

R. Korte, T. Kuusi and M. Parviainen, A connection between a general class of superparabolic functions and supersolutions, J. Evol. Equ., 10 (2010), 1-20. doi: 10.1007/s00028-009-0037-3.  Google Scholar

[18]

R. Korte, T. Kuusi and J. Siljander, Obstacle problem for nonlinear parabolic equations, J. Differential Equations, 246 (2009), 3668-3680. doi: 10.1016/j.jde.2009.02.006.  Google Scholar

[19]

T. Kuusi, Lower semicontinuity of weak supersolutions to a nonlinear parabolic equation, Differential Integral Equations, 22 (2009), 1211-1222.  Google Scholar

[20]

E. Lanconelli, Sul problema di Dirichlet per l'equazione del calore, Ann. Mat. Pura Appl., 97 (1973), 83-114. doi: 10.1007/BF02414910.  Google Scholar

[21]

E. Lanconelli, Sul problema di Dirichlet per equazione paraboliche del secondo ordine a coefficiente discontinui, Ann. Mat. Pura Appl., 106 (1975), 11-38. doi: 10.1007/BF02415021.  Google Scholar

[22]

N. S. Landkof, Foundations of Modern Potential Theory, Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[23]

P. Lindqvist and M. Parviainen, Irregular time dependent obstacles, J. Funct. Anal., 263 (2012), 2458-2482. doi: 10.1016/j.jfa.2012.07.014.  Google Scholar

[24]

V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Second, Revised and Augmented Edition, Grund. der Math. Wiss., 342, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-15564-2.  Google Scholar

[25]

M. Pierre, Parabolic capacity and Sobolev spaces, SIAM J. Math. Anal., 14 (1983), 522-533. doi: 10.1137/0514044.  Google Scholar

[26]

L. M. R. Saraiva, Removable singularities and quasilinear parabolic equations, Proc. London Math. Soc., 48 (1984), 385-400. doi: 10.1112/plms/s3-48.3.385.  Google Scholar

[27]

L. M. R. Saraiva, Removable singularities of solutions of degenerate quasilinear equations, Ann. Mat. Pura Appl., 141 (1985), 187-221. doi: 10.1007/BF01763174.  Google Scholar

[28]

T. Ransford, Potential Theory in the Complex Plane, London Mathematical Society Student Texts, 28, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623776.  Google Scholar

[29]

N. A. Watson, Thermal capacity, Proc. London Math. Soc., 37 (1978), 342-362. doi: 10.1112/plms/s3-37.2.342.  Google Scholar

show all references

References:
[1]

D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften 314, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03282-4.  Google Scholar

[2]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. doi: 10.1007/BF01176474.  Google Scholar

[3]

A. Björn, J. Björn, U. Gianazza and M. Parviainen, Boundary regularity for degenerate and singular parabolic equations, Calc. Var. Partial Differential Equations, 52 (2015), 797-827. doi: 10.1007/s00526-014-0734-9.  Google Scholar

[4]

L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237-258. doi: 10.1006/jfan.1996.3040.  Google Scholar

[5]

V. Bögelein, F. Duzaar and G. Mingione, Degenerate problems with irregular obstacles, J. Reine Angew. Math., 650 (2011), 107-160. doi: 10.1515/CRELLE.2011.006.  Google Scholar

[6]

J. Droniou, A. Porretta and A. Prignet, Parabolic capacity and soft measures for nonlinear equations, Potential Anal., 19 (2003), 99-161. doi: 10.1023/A:1023248531928.  Google Scholar

[7]

L. C. Evans and R. F. Gariepy, Wiener's test for the heat equation, Arch. Rational Mech. Anal., 78 (1982), 293-314. doi: 10.1007/BF00249583.  Google Scholar

[8]

R. Gariepy and W. P. Ziemer, Removable sets for quasilinear parabolic equations, J. London Math. Soc., 21 (1980), 311-318. doi: 10.1112/jlms/s2-21.2.311.  Google Scholar

[9]

R. Gariepy and W. P. Ziemer, Thermal capacity and boundary regularity, J. Differential Equations, 45 (1982), 374-388. doi: 10.1016/0022-0396(82)90034-1.  Google Scholar

[10]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Unabridged republication of the 1993 original, Dover Publications, Inc., Mineola, NY, 2006.  Google Scholar

[11]

T. Kilpeläinen and P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation, SIAM J. Math. Anal., 27 (1996), 661-683. doi: 10.1137/0527036.  Google Scholar

[12]

T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793.  Google Scholar

[13]

J. Kinnunen, R. Korte, T. Kuusi and M. Parviainen, Nonlinear parabolic capacity and polar sets of superparabolic functions, Math. Ann., 355 (2013), 1349-1381. doi: 10.1007/s00208-012-0825-x.  Google Scholar

[14]

K. Kinnunen and P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl., 185 (2006), 411-435. doi: 10.1007/s10231-005-0160-x.  Google Scholar

[15]

J. Kinnunen, T. Lukkari and M. Parviainen, An existence result for superparabolic functions, J. Funct. Anal., 258 (2010), 713-728. doi: 10.1016/j.jfa.2009.08.009.  Google Scholar

[16]

J. Kinnunen, T. Lukkari and M. Parviainen, Local approximation of superharmonic and superparabolic functions in nonlinear potential theory, J. Fixed Point Theory Appl., 13 (2013), 291-307. doi: 10.1007/s11784-013-0108-5.  Google Scholar

[17]

R. Korte, T. Kuusi and M. Parviainen, A connection between a general class of superparabolic functions and supersolutions, J. Evol. Equ., 10 (2010), 1-20. doi: 10.1007/s00028-009-0037-3.  Google Scholar

[18]

R. Korte, T. Kuusi and J. Siljander, Obstacle problem for nonlinear parabolic equations, J. Differential Equations, 246 (2009), 3668-3680. doi: 10.1016/j.jde.2009.02.006.  Google Scholar

[19]

T. Kuusi, Lower semicontinuity of weak supersolutions to a nonlinear parabolic equation, Differential Integral Equations, 22 (2009), 1211-1222.  Google Scholar

[20]

E. Lanconelli, Sul problema di Dirichlet per l'equazione del calore, Ann. Mat. Pura Appl., 97 (1973), 83-114. doi: 10.1007/BF02414910.  Google Scholar

[21]

E. Lanconelli, Sul problema di Dirichlet per equazione paraboliche del secondo ordine a coefficiente discontinui, Ann. Mat. Pura Appl., 106 (1975), 11-38. doi: 10.1007/BF02415021.  Google Scholar

[22]

N. S. Landkof, Foundations of Modern Potential Theory, Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[23]

P. Lindqvist and M. Parviainen, Irregular time dependent obstacles, J. Funct. Anal., 263 (2012), 2458-2482. doi: 10.1016/j.jfa.2012.07.014.  Google Scholar

[24]

V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Second, Revised and Augmented Edition, Grund. der Math. Wiss., 342, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-15564-2.  Google Scholar

[25]

M. Pierre, Parabolic capacity and Sobolev spaces, SIAM J. Math. Anal., 14 (1983), 522-533. doi: 10.1137/0514044.  Google Scholar

[26]

L. M. R. Saraiva, Removable singularities and quasilinear parabolic equations, Proc. London Math. Soc., 48 (1984), 385-400. doi: 10.1112/plms/s3-48.3.385.  Google Scholar

[27]

L. M. R. Saraiva, Removable singularities of solutions of degenerate quasilinear equations, Ann. Mat. Pura Appl., 141 (1985), 187-221. doi: 10.1007/BF01763174.  Google Scholar

[28]

T. Ransford, Potential Theory in the Complex Plane, London Mathematical Society Student Texts, 28, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623776.  Google Scholar

[29]

N. A. Watson, Thermal capacity, Proc. London Math. Soc., 37 (1978), 342-362. doi: 10.1112/plms/s3-37.2.342.  Google Scholar

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