Skip to main content Accessibility help
×
Home

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

  • Gianni Dal Maso (a1) and Annalisa Malusa (a1)

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.

Copyright

References

Hide All
1Aizenman, M. and Simon, B.. Brownian motion and Harnack inequality for Schrödinger operators. Comm. Pure Appl. Math. 35 (1982), 209273.
2Baxter, J. R., Dal, G. Maso and Mosco, U.. Stopping times and Γ-convergence. Trans. Amer. Math. Soc. 303 (1987), 138.
3Buttazzo, G., Dal, G. Maso and Mosco, U.. A derivation theorem for capacities with respect to a Radon measure. J. Funct. Anal. 71 (1987), 263278.
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), 98138; 70 (1983), 154–178 (London: Pitman).
5Maso, G. Dal. Γ-convergence and μ-capacities. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 423464.
6Maso, G. Dal and Mosco, U.. Wiener criteria and energy decay for relaxed Dirichlet problems. Arch. Rational Mech. Anal. 95 (1986), 345387.
7Maso, G. Dal and Mosco, U.. Wiener criterion and Γ-convergence. Appl. Math. Optim. 15 (1987), 1563.
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), 517535.
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, 661696 (Boston: Birkhaüser, 1989).
10Kato, T.. Schrödinger operators with singular potentials. Israel J. Math. 22 (1972), 139158.
11Kinderlehrer, D. and Stampacchia, G.. An Introduction to Variational Inequalities and Their Applications (New York: Academic Press, 1980).
12Littman, W., Stampacchia, G. and Weinberger, H. F.. Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa 17 (1963), 4579.
13Schechter, M.. Spectra of Partial Differential Opertors (Amsterdam: North-Holland, 1986).
14Stampacchia, G.. Le probléme de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965), 189258.
15Zamboni, P.. Some function spaces and elliptic partial differential equations. Matematiche 42 (1987), 171178.
16Ziemer, W. P.. Weakly Differentiate Functions (Berlin: Springer, 1989).

Related content

Powered by UNSILO

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

  • Gianni Dal Maso (a1) and Annalisa Malusa (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.