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Density estimates for vector minimizers and applications
Variational parabolic capacity
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 |
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. |
[2] |
H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341.
doi: 10.1007/BF01176474. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
T. Kuusi, Lower semicontinuity of weak supersolutions to a nonlinear parabolic equation, Differential Integral Equations, 22 (2009), 1211-1222. |
[20] |
E. Lanconelli, Sul problema di Dirichlet per l'equazione del calore, Ann. Mat. Pura Appl., 97 (1973), 83-114.
doi: 10.1007/BF02414910. |
[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. |
[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. |
[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. |
[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. |
[25] |
M. Pierre, Parabolic capacity and Sobolev spaces, SIAM J. Math. Anal., 14 (1983), 522-533.
doi: 10.1137/0514044. |
[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. |
[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. |
[28] |
T. Ransford, Potential Theory in the Complex Plane, London Mathematical Society Student Texts, 28, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511623776. |
[29] |
N. A. Watson, Thermal capacity, Proc. London Math. Soc., 37 (1978), 342-362.
doi: 10.1112/plms/s3-37.2.342. |
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. |
[2] |
H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341.
doi: 10.1007/BF01176474. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
T. Kuusi, Lower semicontinuity of weak supersolutions to a nonlinear parabolic equation, Differential Integral Equations, 22 (2009), 1211-1222. |
[20] |
E. Lanconelli, Sul problema di Dirichlet per l'equazione del calore, Ann. Mat. Pura Appl., 97 (1973), 83-114.
doi: 10.1007/BF02414910. |
[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. |
[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. |
[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. |
[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. |
[25] |
M. Pierre, Parabolic capacity and Sobolev spaces, SIAM J. Math. Anal., 14 (1983), 522-533.
doi: 10.1137/0514044. |
[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. |
[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. |
[28] |
T. Ransford, Potential Theory in the Complex Plane, London Mathematical Society Student Texts, 28, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511623776. |
[29] |
N. A. Watson, Thermal capacity, Proc. London Math. Soc., 37 (1978), 342-362.
doi: 10.1112/plms/s3-37.2.342. |
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