Periodic behavior of solutions to parabolic-elliptic free boundary problems

Nobuyuki KENMOCHI; Masahiro KUBO

doi:10.2969/jmsj/04140625

Nobuyuki KENMOCHI, Masahiro KUBO

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JOURNAL FREE ACCESS

1989 Volume 41 Issue 4 Pages 625-640

DOI https://doi.org/10.2969/jmsj/04140625

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Published: 1989 Received: January 29, 1988 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : Dedicated to Professor Niro Yanagihara on his 60th birthday

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341.
2) H. W. Alt, S. Luckhaus and A. Visintin, On nonstationary flow through porous media, Ann. Mat. Pura Appl., 136 (1984), 303-316.
3) M. Bertsch and J. Hulshof, Regularity results for an elliptic-parabolic free boundary problem, Trans. Amer. Math. Soc., 297 (1986), 337-350.
4) H. Brézis, Operateurs Maximaux Monotones et Semi-grouper de Contractions Bans les Espaces de Hilbert, North-Holland, Amsterdam-London, 1973.
5) E. DiBenedetto and A. Friedman, Periodic behavior for the evolutionary dam problem and related free boundary problems, Comm. Partial Duff. Equations, 11 (1986), 1297-1377.
6) C. J. van Duyn and L. A. Peletier, Nonstationary filtration in partially saturated porous media, Arch. Rational Mech. Anal., 78 (1982), 173-198.
7) A. Fasano and M. Primicerio, Liquid flow in partially saturated porous media, J. Inst. Math. Applics., 23 (1979), 503-517.
8) U. Hornung, A parabolic-elliptic variational inequality, Manuscripta Math., 39 (1982), 155-172.
9) J. Hulshof, An elliptic-parabolic free boundary problem; continuity of the interface, Proc. Roy. Soc. Edinburgh, 106 (1987), 327-339.
10) J. Hulshof, Bounded weak solutions of an elliptic-parabolic Neumann problem, Trans. Amer. Math. Soc., 303 (1987), 211-227.
11) J. Hulshof and L. A. Peletier, An elliptic-parabolic free boundary problem, Nonlinear Anal., 10 (1986), 1327-1346.
12) N. Kenmochi and M. Kubo, Periodic solutions to a class of nonlinear variational inequalities with time-dependent constraints, Funkcial. Ekvac., 30 (1987), 333-349.
13) N. Kenmochi and Y. Mizuta, Potential theoretic properties of the gradient of a convex function on a functional space, Nagoya Math. J., 59 (1975), 199-215.
14) N. Kenmochi and I. Pawlow, A class of nonlinear elliptic-parabolic equations with time-dependent constraints, Nonlinear Anal., 10 (1986), 1181-1202.
15) N. Kenmochi and I. Pawlow, Parabolic-elliptic free boundary problems with time-dependent obstacles, Japan J. Appl. Math., 5 (1988), 87-121.
16) N. Kenmochi and I. Pawlow, Asymptotic behavior of solutions to parabolic-elliptic variational inequalities, to appear in Nonlinear Anal., 1989.
17) D. Kröner, Parabolic regularization and behaviour of the free boundary for unsaturated flow in a porous medium, J. Reine Angew. Math., 348 (1984), 180-196.
18) D. Kröner and J. F. Rodrigues, Global behaviour for bounded solutions of a porous media equation of elliptic-parabolic type, J. Math. Pures Appl., 64 (1985), 105-120.
19) U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math., 3 (1969), 510-585.
20) J. Watanabe, Approximation of nonlinear problems of a certain type, Numer. Appl. Anal., 1 (1979), 147-163.

Right : [1] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341.
[2] H. W. Alt, S. Luckhaus and A. Visintin, On nonstationary flow through porous media, Ann. Mat. Pura Appl., 136 (1984), 303-316.
[3] M. Bertsch and J. Hulshof, Regularity results for an elliptic-parabolic free boundary problem, Trans. Amer. Math. Soc., 297 (1986), 337-350.
[4] H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam-London, 1973.
[5] E. DiBenedetto and A. Friedman, Periodic behavior for the evolutionary dam problem and related free boundary problems, Comm. Partial Duff. Equations, 11 (1986), 1297-1377.
[6] C. J. van Duyn and L. A. Peletier, Nonstationary filtration in partially saturated porous media, Arch. Rational Mech. Anal., 78 (1982), 173-198.
[7] A. Fasano and M. Primicerio, Liquid flow in partially saturated porous media, J. Inst. Math. Applics., 23 (1979), 503-517.
[8] U. Hornung, A parabolic-elliptic variational inequality, Manuscripta Math., 39 (1982), 155-172.
[9] J. Hulshof, An elliptic-parabolic free boundary problem: continuity of the interface, Proc. Roy. Soc. Edinburgh, 106 (1987), 327-339.
[10] J. Hulshof, Bounded weak solutions of an elliptic-parabolic Neumann problem, Trans. Amer. Math. Soc., 303 (1987), 211-227.
[11] J. Hulshof and L. A. Peletier, An elliptic-parabolic free boundary problem, Nonlinear Anal., 10 (1986), 1327-1346.
[12] N. Kenmochi and M. Kubo, Periodic solutions to a class of nonlinear variational inequalities with time-dependent constraints, Funkcial. Ekvac., 30 (1987), 333-349.
[13] N. Kenmochi and Y. Mizuta, Potential theoretic properties of the gradient of a convex function on a functional space, Nagoya Math. J., 59 (1975), 199-215.
[14] N. Kenmochi and I. Pawlow, A class of nonlinear elliptic-parabolic equations with time-dependent constraints, Nonlinear Anal., 10 (1986), 1181-1202.
[15] N. Kenmochi and I. Pawlow, Parabolic-elliptic free boundary problems with time-dependent obstacles, Japan J. Appl. Math., 5 (1988), 87-121.
[16] N. Kenmochi and I. Pawlow, Asymptotic behavior of solutions to parabolic-elliptic variational inequalities, to appear in Nonlinear Anal., 1989.
[17] D. Kröner, Parabolic regularization and behaviour of the free boundary for unsaturated flow in a porous medium, J. Reine Angew. Math., 348 (1984), 180-196.
[18] D. Kröner and J. F. Rodrigues, Global behaviour for bounded solutions of a porous media equation of elliptic-parabolic type, J. Math. Pures Appl., 64 (1985), 105-120.
[19] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math., 3 (1969), 510-585.
[20] J. Watanabe, Approximation of nonlinear problems of a certain type, Numer. Appl. Anal., 1 (1979), 147-163.

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