Abstract
In the present paper the concept and properties of the residual functional in Sobolev space are investigated. The weak compactness, force condition, lower semi-continuity and convex of the residual functional are proved. In Sobolev space, the minimum principle of the residual functional is proposed. The minimum existence theoreom for J(u)=0 is given by the modern critical point theory. And the equivalence theorem or five equivalence forms for the residual functional equation are also proved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Proceedings of the Third National Conference on MWR, Emei, Sichuan, China (1989). (in Chinese)
Xu Ci-da,The Method of Weighted Residuals on Solid Mechanics, Tong-ji University Press, Shanghai (1987), (in Chinese)
Qiu Ji-bao, On The search for the theoretical foundation of MWR,Acta Mechanica Sinica,19, Sup, July (1987). (in Chinese)
Ling yong-yong, A residual inequality of one class of an nonlinear differential equation,Proceedings of the 3rd National Conference on MWR, Emei, Sichuan, China (1989), 38. (in Chinese)
Ling Yong-yong and Ling Bei-bei, Periodic solution and error bounds for a second order nonlinear differential equation,Proceedings of the first National Conference on CMANS, Changsha, Hunan, China, (1990) 853. (in Chinese)
Adams, R.A.,Sobolev Spaces, British Columbia Uni. (1977).
Lee Li-kun, Guo yi-dao,Introduction to the Sobolev Spaces, Shanghai Science-Technics Publisher (1981), (in Chinese)
Chang Gong-qing,Critical Point Theory and its Applications, Shanghai-Technics Publisher 91986). (in Chinese)
Lipgschutz, S.,General topology, Schaum Publishing Company (1965).
Ciesielski, Z. et al., Construction of an orthonormal basis inC m(Id) andW mp (Id),Studia Math.,41 (1972), 211.
Mitrinovic, D.S.,Analytic Inequalities, Springer-Verlag (1970).
Author information
Authors and Affiliations
Additional information
Cummunicated by Hsu Tzu-ta
Rights and permissions
About this article
Cite this article
Yong-yong, L. An extremum theory of the residual functional in Sobolev spacesW m,p(ω). Appl Math Mech 13, 273–279 (1992). https://doi.org/10.1007/BF02457373
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02457373