Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T16:07:40.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Very weak solutions of boundary value problems for the Laplace operator and the Lamé system on polyhedral domains in ℝ3

Published online by Cambridge University Press:  14 November 2011

Ding Hua
Ding Hua
Affiliation:
Institute of Mechanics, CAS, 100080 Beijing, China
Get access Rights & Permissions [Opens in a new window]

Abstract

The notion of very weak solutions is introduced in this paper in order to solve the boundary value problems for the Laplace operator and for the Lamé system with nonsmooth data in polyhedral domains. A continuity theorem is given for variational solutions of the above problems. This result may be used to solve problems with concentrated loads.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 4 , 1994 , pp. 645 - 672
Copyright
Copyright © Royal Society of Edinburgh 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Agmon, S., Douglis, A. and Nirenberg, L.. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Comm. Pure Appl. Math. XII (1959), 623–727.CrossRefGoogle Scholar
2Courant, R. and Hilbert, D.. Methods of Mathematical Physics, Vol. II, Partial Differential Equations (New York: Interscience, 1962).Google Scholar
3Dauge, M.. Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics 1341 (Berlin: Springer, 1988).CrossRefGoogle Scholar
4Ding, H.. Etude qualitative du comportement d'un corps elastique sous l'effet de charges concentrées (Thèse de Doctorat, Université de Nice, France, 1986).Google Scholar
5Ding, H.. Traces of nonsmooth functions on polyhedral domains. Science in China (Ser. A) 35 (1992), 452–462.Google Scholar
6Ding, H.. A regularity result for boundary value problems on Lipschitz domains. Ann. Fac. Sci. Toulouse 10 (1989), 325–333.Google Scholar
7Grisvard, P.. Elliptic Problems in Nonsmooth Domains (London: Pitman, 1985).Google Scholar
8Grisvard, P.. Problèmes aux limites dans les polygones mode d'emploi (Preprint, Université de Nice, 45, 1984).Google Scholar
9Grisvard, P.. Singularités en élasticité. Arch. Rational Mech. Anal. 107 (1989), 157–180.CrossRefGoogle Scholar
10Kato, T.. Perturbation Theory for Linear Operators (New York: Springer, 1966).Google Scholar
11Khoan, Vo-Khac. Distribution, Analyse de Fourier, Opérateurs aux dérivées partielles (Paris: Librairie Viubert, 1972).Google Scholar
12Kozlov, V. A. and Maz'ja, V. G.. Spectral properties of the operator bundles generated by elliptic value problems in a cone. Funktsional Anal. i. Prilozhen 22 (1988), 38–46; translated in Functional Anal. Appl. 22 (1988), 114–121.CrossRefGoogle Scholar
13Landau, L. D. and Lifshitz, E. M.. Theory of Elasticity (Oxford: Pergamon, 1986).Google Scholar
14Lions, J. L. and Magenes, E.. Problème aux limites non homogrnos (Problemi ai limiti non omogenei). (I) Ann. Scuola Norm. Sup. Pisa 14(3) (1960), 269–308; (II) Ann. Inst. Fourier 11 (1961), 137–178; (III) Ann. Scuola Norm. Sup. Pisa 15 (1961), 39–101.Google Scholar
15Lions, J. L. and Peetre, J.. Sur une classe d'espaces d'interpolation. Publ. I.H.E.S. 19 (1964), 5–68.Google Scholar
16Mazja, V. G. and Plamenevskii, B. A.. Estimates in L p and Hölder classes and the Miranda–Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary. Amer. Math. Soc. Transl. (2) 123 (1964), 1–56.Google Scholar
17Maz'ja, V. G. and Rossmann, J.. On the asymptotics of solutions of elliptic boundary value problems near edges. Math. Nachr. 138 (1988), 27–53.CrossRefGoogle Scholar
18Nečas, J.. Les methodes directes en théorie des equations elliptiques (Paris: Masson, 1976).Google Scholar
19Riesz, F. and Sz. Nagy, B.. Leçons d'Analyse Fonctionnelle. (Budapest: Akadçmiai Kado and Paris: Gauthier-Villars, 1982).Google Scholar
20Stampacchia, G.. Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier 15 (1965), 189–258.CrossRefGoogle Scholar
21Wildenhain, G.. Darstellung von Lösungen Linearer Elliptischer Differentialgleichungen, Mathematical Research 8 (Berlin: Akademie, 1981).Google Scholar