American Institute of Mathematical Sciences

Advanced
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 - A, 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, (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. 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. 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. 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. 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. 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. 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. 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. 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, (1993). 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar

[20]

E. Lanconelli, Sul problema di Dirichlet per l'equazione del calore,, Ann. Mat. Pura Appl., 97 (1973), 83. 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. doi: 10.1007/BF02415021. Google Scholar

[22]

N. S. Landkof, Foundations of Modern Potential Theory,, Translated from the Russian by A. P. Doohovskoy, (1972). Google Scholar

[23]

P. Lindqvist and M. Parviainen, Irregular time dependent obstacles,, J. Funct. Anal., 263 (2012), 2458. 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., (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. doi: 10.1137/0514044. Google Scholar

[26]

L. M. R. Saraiva, Removable singularities and quasilinear parabolic equations,, Proc. London Math. Soc., 48 (1984), 385. 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. doi: 10.1007/BF01763174. Google Scholar

[28]

T. Ransford, Potential Theory in the Complex Plane,, London Mathematical Society Student Texts, (1995). doi: 10.1017/CBO9780511623776. Google Scholar

[29]

N. A. Watson, Thermal capacity,, Proc. London Math. Soc., 37 (1978), 342. 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, (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. 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. 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. 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. 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. 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. 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. 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. 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, (1993). 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar

[20]

E. Lanconelli, Sul problema di Dirichlet per l'equazione del calore,, Ann. Mat. Pura Appl., 97 (1973), 83. 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. doi: 10.1007/BF02415021. Google Scholar

[22]

N. S. Landkof, Foundations of Modern Potential Theory,, Translated from the Russian by A. P. Doohovskoy, (1972). Google Scholar

[23]

P. Lindqvist and M. Parviainen, Irregular time dependent obstacles,, J. Funct. Anal., 263 (2012), 2458. 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., (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. doi: 10.1137/0514044. Google Scholar

[26]

L. M. R. Saraiva, Removable singularities and quasilinear parabolic equations,, Proc. London Math. Soc., 48 (1984), 385. 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. doi: 10.1007/BF01763174. Google Scholar

[28]

T. Ransford, Potential Theory in the Complex Plane,, London Mathematical Society Student Texts, (1995). doi: 10.1017/CBO9780511623776. Google Scholar

[29]

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

[1]

Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 51-66. doi: 10.3934/dcdss.2011.4.51
[2]

Changchun Liu. A fourth order nonlinear degenerate parabolic equation. Communications on Pure & Applied Analysis, 2008, 7 (3) : 617-630. doi: 10.3934/cpaa.2008.7.617
[3]

Chi-Cheung Poon. Blowup rate of solutions of a degenerate nonlinear parabolic equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5317-5336. doi: 10.3934/dcdsb.2019060
[4]

Takesi Fukao, Masahiro Kubo. Nonlinear degenerate parabolic equations for a thermohydraulic model. Conference Publications, 2007, 2007 (Special) : 399-408. doi: 10.3934/proc.2007.2007.399
[5]

Alexandre Montaru. Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 231-256. doi: 10.3934/dcdsb.2014.19.231
[6]

Giuseppe Floridia. Well-posedness for a class of nonlinear degenerate parabolic equations. Conference Publications, 2015, 2015 (special) : 455-463. doi: 10.3934/proc.2015.0455
[7]

Chunlai Mu, Zhaoyin Xiang. Blowup behaviors for degenerate parabolic equations coupled via nonlinear boundary flux. Communications on Pure & Applied Analysis, 2007, 6 (2) : 487-503. doi: 10.3934/cpaa.2007.6.487
[8]

Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081
[9]

Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-26. doi: 10.3934/dcds.2019226
[10]

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
[11]

Francisco Ortegón Gallego, María Teresa González Montesinos. Existence of a capacity solution to a coupled nonlinear parabolic--elliptic system. Communications on Pure & Applied Analysis, 2007, 6 (1) : 23-42. doi: 10.3934/cpaa.2007.6.23
[12]

Young-Sam Kwon. Strong traces for degenerate parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1275-1286. doi: 10.3934/dcds.2009.25.1275
[13]

Jiebao Sun, Boying Wu, Jing Li, Dazhi Zhang. A class of doubly degenerate parabolic equations with periodic sources. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1199-1210. doi: 10.3934/dcdsb.2010.14.1199
[14]

H. Gajewski, I. V. Skrypnik. To the uniqueness problem for nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 315-336. doi: 10.3934/dcds.2004.10.315
[15]

Jan Prüss, Gieri Simonett, Rico Zacher. On normal stability for nonlinear parabolic equations. Conference Publications, 2009, 2009 (Special) : 612-621. doi: 10.3934/proc.2009.2009.612
[16]

Wolfgang Walter. Nonlinear parabolic differential equations and inequalities. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 451-468. doi: 10.3934/dcds.2002.8.451
[17]

Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130
[18]

Patrick Martinez, Judith Vancostenoble. The cost of boundary controllability for a parabolic equation with inverse square potential. Evolution Equations & Control Theory, 2019, 8 (2) : 397-422. doi: 10.3934/eect.2019020
[19]

M. Sango. Weak solutions for a doubly degenerate quasilinear parabolic equation with random forcing. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 885-905. doi: 10.3934/dcdsb.2007.7.885
[20]

Michael Winkler. Nontrivial ordered ω-limit sets in a linear degenerate parabolic equation. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 739-750. doi: 10.3934/dcds.2007.17.739

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences