×

Random relaxed Dirichlet problems. (English) Zbl 0673.60067

Summary: We investigate sequences of relaxed Dirichlet problems of the form: \[ - \Delta u_ h+\mu_ hu_ h=0 \] where \(\mu_ h\) are random Borel measures belonging to a suitable class \({\mathcal M}_ 0\). By means of a variational approach, necessary and sufficient conditions for the convergence in probability of the sequence \(u_ h\) toward the solution of a deterministic relaxed Dirichlet problem are given. Some applications to Dirichlet problems in random perturbated domains and to a Schrödinger equation with random singular potentials are considered.

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
60J45 Probabilistic potential theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ash, R., Real analysis and probability (1972), New York: Academic Press, New York · Zbl 1381.28001
[2] Attouch, H., Variational convergence for functions and operators (1984), London: Pitman, London · Zbl 0561.49012
[3] H.Attouch - F.Murat,Potentiels fortement oscillant, to appear.
[4] Baxter, J. R.; Chacon, R. V.; Jain, N. C., Weak limits of stopped diffusions, Trans. Amer. Math. Soc., 293, 767-792 (1986) · Zbl 0591.60040
[5] Baxter, J. R.; Maso, G. Dal; Mosco, U., Stopping times and Γ-convergence, Trans. Amer. Math. Soc., 303, 1-38 (1987) · Zbl 0627.60071
[6] Baxter, J. R.; Jain, N. C., Asymptotic capacities for finely divided bodies and stopped diffusions, Illinois J. Math., 31, 469-495 (1987) · Zbl 0618.60072
[7] Brillard, A., Quelques questions de convergence... and calcul des variations, These (1983), Orsay: Université de Paris Sud, Orsay
[8] Buttazzo, G.; Maso, G. Dal; Mosco, U., A derivation theorem for capacities with respect to a Radon measure, J. Funct. Anal., 71, 263-278 (1987) · Zbl 0622.28006
[9] Chavel, I., Eigenvalues in Riemannian Geometry (1984), New York: Academic Press, New York
[10] I.Chavel - E. A.Feldman,The Lenz shift and Wiener sausage in Riemannian manifolds, to appear. · Zbl 0607.60066
[11] I.Chavel - E. A.Feldman,The Wiener sausage, and a theorem of Spitzer,in Riemannian manifolds, to appear. · Zbl 0597.58047
[12] Choquet, G., Forme abstraite du théoreme de capacitabilité, Ann. Inst. Fourier (Grenoble), 9, 83-89 (1959) · Zbl 0093.29701
[13] Cioranescu, D.; Murat, F., Un terme étrange venu d’ailleurs, I, Nonlinear partial differential equations and their applications, 98-138 (1982), London: Pitman, London · Zbl 0496.35030
[14] Cioranescu, D.; Murat, F., Un terme étrange venu d’ailleurs, II, Nonlinear partial differential equations and their applications, 154-178 (1983), London: Pitman, London · Zbl 0496.35030
[15] Ciobanescu, D.; Paulin, J. Saint Jean, Homogénéisation dans des ouverts à cavités, C. R. Acad. Sci. Paris Sér. A, 284, 857-860 (1977) · Zbl 0361.35009
[16] Cioranescu, D.; Paulin, J. Saint Jean, Homogenization in open sets with holes, J. Math. Anal. Appl., 71, 590-607 (1979) · Zbl 0427.35073
[17] Dal Maso, G., Γ-convergence and μ-capacities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14, 423-464 (1987) · Zbl 0657.49005
[18] G.Dal Maso - E.De Giorgi - L.Modica,Weak convergence of measures on spaces of lower semicontinuous functions, in «Integral functionals in calculus of variations» (Trieste, 1985), pp. 59-100, Supplemento ai Rend. Circ. Mat. Palermo,15 (1987). · Zbl 0638.46019
[19] Dal Maso, G.; Modica, L., Nonlinear stochastic homogenization, Ann. Mat. Pura Appl., (4), 144, 347-389 (1986) · Zbl 0607.49010
[20] Dal Maso, G.; Mosco, U., Wiener criteria and energy decay for relaxed Dirichlet problems, Arch. Rational Mech. Anal., 95, 345-387 (1986) · Zbl 0634.35033
[21] Dal Maso, G.; Mosco, U., Wiener’s criterion and Γ-convergence, Appl. Math. Optim., 15, 15-63 (1987) · Zbl 0644.35033
[22] G.Dal Maso - U.Mosco,The Wiener modulus of a radial measure, Houston J. Math., to appear. · Zbl 0696.31009
[23] E.De Giorgi,G-operators and Γ-convergence, Proceedings of the «International Congress of Mathematicians», pp. 1175-1191, Warszaw, 1983.
[24] De Giorgi, E.; Franzoni, T., Su un tipo di convergenza variasionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (8), 58, 842-850 (1975) · Zbl 0339.49005
[25] De Giorgi, E.; Franzoni, T., Su un tipo di convergenza variazionale, Rend. Sem. Mat. Brescia, 3, 63-101 (1979)
[26] De Giorgi, E.; Letta, G., Une notion générale de convergence faible pour des fonctions croissantes d’ensemble, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4), 4, 61-99 (1977) · Zbl 0405.28008
[27] Dellacherie, C., Ensembles analytiques, capacités, mesures de Hausdorff, Lecture Notes in Math.,295 (1972), Berlin: Springer Berlin Heidelberg, Berlin · Zbl 0259.31001
[28] Federer, H.; Ziemer, W., The Lebesgue set of a function whose distribution derivatives are p-th power summable, Indiana Univ. Math. J., 22, 139-158 (1972) · Zbl 0238.28015
[29] Figari, R.; Orlandi, E.; Teta, S., The Laplacian in regions with many small obstacles: fluctuation around the limit operator, J. Statist. Phys., 41, 465-487 (1985) · Zbl 0642.60053
[30] Hruslov, E. Ya., The method of orthogonal projections and the Dirichlet problems in domains with a fine-grained boundary, Math. U.S.S.R. Sb., 17, 37-59 (1972) · Zbl 0251.35032
[31] Hruslov, E. Ya., The first boundary value problem in domains with a complicated boundary for higher order equations, Math. U.S.S.R. Sb., 32, 535-549 (1977) · Zbl 0396.35039
[32] Kac, M., Probabilistic methods in some problems of scattering theory, Rocky Mountain J. Math., 4, 511-538 (1974) · Zbl 0314.47006
[33] Kinderlehrer, D.; Stampacchia, G., Introduction to variational inequalities and their applications (1980), New Tork: Academic Press, New Tork · Zbl 0457.35001
[34] Marchenko, A. V.; Hruslov, E. Ya., Boundary value problems in domains with closedgrained boundaries (Russian) (1974), Kiev: Naukova Dumka, Kiev
[35] Marchenko, A. V.; Hruslov, E. Ya., New results in the theory of boundary value problems for regions with closed-grained boundaries, Uspekhi Mat. Nauk, 33, 127-137 (1978)
[36] Ozawa, S., On an elaboration of M. Kac’s theorem concerning eigenvalues of the Laplacian in a region with randomly distributed small obstacles, Comm. Math. Phys., 91, 473-487 (1983) · Zbl 0541.35019
[37] Ozawa, S., Random media and the eigenvalues of the Laplacian, Comm. Math. Phys., 94, 421-437 (1984) · Zbl 0555.35101
[38] G. C.Papanicolaou - S. R. S.Varadhan,Diffusion in regions with many small holes, in «Stochastic differential systems, filtering and control», Proceedings of the IFIP-WG 7/1 Working Conference (Vilnius, Lithuania, 1978), pp. 190-206, Lecture Notes in Control and Information Sci., Springer-Verlag,25 (1980).
[39] Parthasarathy, K. R., Probability measures on metric spaces (1967), New York: Academic Press, New York · Zbl 0153.19101
[40] Rauch, J.; Taylor, M., Potential and scattering theory on wildly perturbed domains, J. Funct. Anal., 18, 27-59 (1975) · Zbl 0293.35056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.