Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-16T21:15:35.269Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation of relaxed Dirichlet problems by boundary value problems in perforated domains

Published online by Cambridge University Press:  14 November 2011

Gianni Dal Maso  and
Annalisa Malusa
Gianni Dal Maso
Affiliation:
SISSA, Via Beirut 2/4, 34014 Trieste, Italy
Annalisa Malusa
Affiliation:
SISSA, Via Beirut 2/4, 34014 Trieste, Italy
Get access Rights & Permissions [Opens in a new window]

Abstract

Given an elliptic operator L on a bounded domain Ω ⊆ Rn, and a positive Radon measure μ on Ω, not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of domains Ωh ⊇ Ω with the following property: for every f ∈ H−1(Ω) the sequence uh of the solutions of the Dirichlet problems Luh = f in Ωh, uh = 0 on ∂Ωh, extended to 0 in Ω\Ωh, converges to the solution of the “relaxed Dirichlet problem” Lu + μu = f in Ω, u = 0 on ∂Ω.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 1 , 1995 , pp. 99 - 114
Copyright
Copyright © Royal Society of Edinburgh 1995

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

1Aizenman, M. and Simon, B.. Brownian motion and Harnack inequality for Schrödinger operators. Comm. Pure Appl. Math. 35 (1982), 209–273.CrossRefGoogle Scholar
2Baxter, J. R., Dal, G. Maso and Mosco, U.. Stopping times and Γ-convergence. Trans. Amer. Math. Soc. 303 (1987), 1–38.Google Scholar
3Buttazzo, G., Dal, G. Maso and Mosco, U.. A derivation theorem for capacities with respect to a Radon measure. J. Funct. Anal. 71 (1987), 263–278.CrossRefGoogle Scholar
4Cioranescu, D. and Murat, F.. Un terme étrange venu d'ailleurs I & II. In Nonlinear Partial Differential Equations and their applications. Collège de France Seminar, Vols II and III, Research Notes in Mathematics 60 (1982), 98–138; 70 (1983), 154–178 (London: Pitman).Google Scholar
5Maso, G. Dal. Γ-convergence and μ-capacities. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 423–464.Google Scholar
6Maso, G. Dal and Mosco, U.. Wiener criteria and energy decay for relaxed Dirichlet problems. Arch. Rational Mech. Anal. 95 (1986), 345–387.CrossRefGoogle Scholar
7Maso, G. Dal and Mosco, U.. Wiener criterion and Γ-convergence. Appl. Math. Optim. 15 (1987), 15–63.CrossRefGoogle Scholar
8Fukushima, M., Sato, K. and Taniguchi, S.. On the closable part of pre-Dirichlet forms and the fine supports of underlying measures. Osaka J. Math. 28 (1991), 517–535.Google Scholar
9Kacimi, H. and Murat, F.. Estimation de l'erreur dans des problèmes de Dirichlet où apparaît un term étrange. In Partial Differential Equations and the Calculus of Variations, Essays in Honor of Ennio De Giorgi, 661–696 (Boston: Birkhaüser, 1989).Google Scholar
10Kato, T.. Schrödinger operators with singular potentials. Israel J. Math. 22 (1972), 139–158.Google Scholar
11Kinderlehrer, D. and Stampacchia, G.. An Introduction to Variational Inequalities and Their Applications (New York: Academic Press, 1980).Google Scholar
12Littman, W., Stampacchia, G. and Weinberger, H. F.. Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa 17 (1963), 45–79.Google Scholar
13Schechter, M.. Spectra of Partial Differential Opertors (Amsterdam: North-Holland, 1986).Google Scholar
14Stampacchia, G.. Le probléme de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965), 189–258.CrossRefGoogle Scholar
15Zamboni, P.. Some function spaces and elliptic partial differential equations. Matematiche 42 (1987), 171–178.Google Scholar
16Ziemer, W. P.. Weakly Differentiate Functions (Berlin: Springer, 1989).Google Scholar