Boundary values of the solutions of the parabolic equation
Abstract
The paper deals with the problem of the behaviour of a given solution of a quasi-linear parabolic equation near the parabolic boundary. Necessary and sufficient conditions for weak and strong convergence in the Sobolev space Wp1,1, p ≥ 2, are given.
References
1. J. Chabrowski, B. Thompson, On traces of solutions of a quasilinear partial differential equation of elliptic type, Department of Mathematics, University of Queensland, Brisbane 1979.
2. E. Gagliardo, Proprieta di alcuni classi di funzioni in piuvariabli, Ricerche Mat. 7 (1958), 102-137.
3. D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-Heidelberg-New York 1977.
4. A. Kufner, O. John, S. Fučik, Function spaces, Noordhoff International Publishing Leyden and Academia Publishing House of the Czechoslovak Academy of Sciences, Prague 1977.
5. M. Marcus, V.J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rational Mech. Anal. 45 (1972), 294-320.
6. M. Marcus, V.J. Mizel, Nemytsky operators on Sobolev spaces, Arch. Rational Mech. Anal. 51 (1973), 347-370.
7. W.P. Mihailov, Boundary values of solutions of elliptic equations in the ball, Mat. SC. 100 142 (1976), 1-13.
8. W.P. Mihailov, Boundary values of solutions of elliptic equations in a domain with smooth boundary, Mat. SC. 101 143 (1976), 163-188.
9. W.P. Mihailov, Partial Differential Equations, Moskwa 1976.
10. M.M. Vainberg, Variational methods for the study of non-linear operators, Holden-Day, Inc. San Francisco, London, Amsterdam (1964).
2. E. Gagliardo, Proprieta di alcuni classi di funzioni in piuvariabli, Ricerche Mat. 7 (1958), 102-137.
3. D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-Heidelberg-New York 1977.
4. A. Kufner, O. John, S. Fučik, Function spaces, Noordhoff International Publishing Leyden and Academia Publishing House of the Czechoslovak Academy of Sciences, Prague 1977.
5. M. Marcus, V.J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rational Mech. Anal. 45 (1972), 294-320.
6. M. Marcus, V.J. Mizel, Nemytsky operators on Sobolev spaces, Arch. Rational Mech. Anal. 51 (1973), 347-370.
7. W.P. Mihailov, Boundary values of solutions of elliptic equations in the ball, Mat. SC. 100 142 (1976), 1-13.
8. W.P. Mihailov, Boundary values of solutions of elliptic equations in a domain with smooth boundary, Mat. SC. 101 143 (1976), 163-188.
9. W.P. Mihailov, Partial Differential Equations, Moskwa 1976.
10. M.M. Vainberg, Variational methods for the study of non-linear operators, Holden-Day, Inc. San Francisco, London, Amsterdam (1964).
LichawskiK. (1985). Boundary values of the solutions of the parabolic equation. Annales Mathematicae Silesianae, 1, 66-88. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14337
Kazimierz Lichawski
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
