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Wiener’s criterion and \(\Gamma\)-convergence. (English) Zbl 0644.35033
The authors describe a class of equations, the relaxed Dirichlet problems, and a variational convergence for their perturbations, the gamma-convergence. The relaxed Dirichlet problem is a general formulation with a potential given by a nonnegative Borel measure in \({\mathbb{R}}^ n.\) This formulation includes both the Dirichlet problems with homogeneous boundary conditions in (possibly irregular) domains and the Schrödinger equations with (possibly singular) nonnegative potentials. The stability and compactness of weak solutions under suitable variational perturbations of the potential is investigated and stable pointwise estimates for the modulus of continuity and the energy of local solutions are obtained.
Reviewer: M.Codegone

35J25 Boundary value problems for second-order elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
49J20 Existence theories for optimal control problems involving partial differential equations
35J10 Schrödinger operator, Schrödinger equation
Full Text: DOI
[1] Aizenman M, Simon B (1982) Brownian motion and Harnack inequality for Schrödinger operators. Comm Pure Appl Math 35:209-273 · Zbl 0475.60063 · doi:10.1002/cpa.3160350206
[2] Attouch H (1984) Variational Convergence for Functions and Operators. Pitman, London · Zbl 0561.49012
[3] Attouch H, Picard C (1983) Variational inequalities with varying obstacles: the general form of the limit problem. J Funct Anal 50:329-389 · doi:10.1016/0022-1236(83)90009-5
[4] Baxter JR, Chacon RV, Jain NC (1985) Weak Limits of Stopped Diffusions. University of Minnesota Mathematics Report 84-165, 1-44
[5] Carbone L, Colombini F (1980) On convergence of functionals with unilateral constraints. J Math Pures Appl (9) 59:465-500 · Zbl 0415.49010
[6] Cioranescu D (1980) Calcul des variations sur des sous-espaces variables. C R Acad Sci Paris Sér A 291:19-22, 87-90 · Zbl 0447.49003
[7] Cioranescu D, Murat F (1982, 1983) Un terme étrange venu d’ailleurs, I and II. In Brezis H, Lions JL (eds) Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, vols II and III. Research Notes in Mathematics, Pitman, London, vol 60, pp 98-138, vol 70 pp 154-178
[8] Cioranescu D, Saint Jean Paulin J (1979) Homogenization in open sets with holes. J Math Pures Appl 71:590-607 · Zbl 0427.35073
[9] Dal Maso G (1981) Asymptotic behavior of minimum problems with bilateral obstacles. Ann Mat Pura Appl (4) 129:327-366 · doi:10.1007/BF01762149
[10] Dal Maso G (1982) Limiti di problemi di minimo con ostacoli. In Atti del Convegno su Studio dei Problemi-limite in Analisi Funzionale, Bressanone, 7-9 Settembre 1981, Pitagora, Bologna, pp 79-100
[11] Dal Maso G, Longo P (1980) ?-limits of obstacles. Ann Mat Pura Appl (4) 128:1-50 · Zbl 0467.49004
[12] Dal Maso G, Mosco U (1985) Wiener Criteria and Energy Decay for Relaxed Dirichlet Problems. IMA Preprint Series No 197, Minneapolis, pp 1-64 (to appear in Arch Rational Mech Anal)
[13] Dal Maso G, Mosco U (1985) The Wiener Modulus of a Radial Measure. IMA Preprint Series No 194, Minneapolis, pp 1-26 (to appear in Houston J Math) · Zbl 0696.31009
[14] De Giorgi E, Franzoni T (1975) Su un tipo di convergenza variazionale. Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur (8) 58:842-850; (1979) Rend Sem Mat Brescia 3:63-101 · Zbl 0339.49005
[15] De Giorgi E, Dal Maso G, Longo P (1980) ?-limiti di ostacoli. Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur (8) 68:481-487
[16] Federer H, Ziemer W (1972) The Lebesgue set of a function whose distribution derivatives arep-th power summable. Indiana Univ Math J 22:139-158 · Zbl 0238.28015 · doi:10.1512/iumj.1972.22.22013
[17] Frehse J (1982) Capacity methods in the theory of partial differential equations. Jahresber Deutsch Math-Verein 84:1-44 · Zbl 0486.35002
[18] Frehse J, Mosco U (1982) Irregular obstacles and quasivariational inequalities of the stochastic impulse control. Ann Scuola Norm Sup Pisa Cl Sci (4), 105-157 · Zbl 0503.49008
[19] Frehse J, Mosco U (1984) Wiener obstacles. In Brezis H, Lions JL (eds) Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, vol 6. Research Notes in Mathematics. Pitman, London, pp 225-257 · Zbl 0583.35038
[20] Gariepy R, Ziemer W (1977) A regularity condition at the boundary for solutions of quasilinear elliptic equations. Arch Rational Mech Anal 67:25-39 · Zbl 0389.35023 · doi:10.1007/BF00280825
[21] Hruslov EYa (1972) The method of orthogonal projections and the Dirichlet problem in domains with a fine grained boundary. Math USSR-Sb 17:37-59 · Zbl 0251.35032 · doi:10.1070/SM1972v017n01ABEH001490
[22] Hruslov EYa (1977) The first boundary value problem in domains with a complicated boundary for higher-order equations. Math USSR-Sb 32:535-549 · Zbl 0396.35039 · doi:10.1070/SM1977v032n04ABEH002405
[23] Kac M (1974) Probabilistic methods in some problems of scattering theory. Rocky Mountain J Math 4:511-538 · Zbl 0314.47006 · doi:10.1216/RMJ-1974-4-3-511
[24] Kato T (1973) Schrödinger operators with singular potentials. Israel J Math 13:135-148 · Zbl 0246.35025 · doi:10.1007/BF02760233
[25] Marchenko AV, Hruslov EYa (1974) Boundary Value Problems in Domains with Close-Grained Boundaries. Naukova Dumka, Kiev (in Russian)
[26] Marchenko AV, Hruslov EYa (1978) New results in the theory of boundary value problems for regions with close-grained boundaries. Uspeki Mat Nauk 33:127
[27] Maz’ja VG (1976) On the continuity at a boundary point of solutions of quasi-linear elliptic equations. Vestnik Leningrad Univ Math 3:225-242
[28] Mosco U (1969) Convergence of convex sets and solutions of variational inequalities. Adv in Math 3:510-585 · Zbl 0192.49101 · doi:10.1016/0001-8708(69)90009-7
[29] Mosco U (1985) Wiener Criterion and Potential Estimates for the Obstacle Problems. IMA Preprint Series No 135, Minneapolis, pp 1-56
[30] Papanicolaou GC, Varadhan SRS (1980) Diffusion in regions with many small holes. In Grigelionis B (ed) Stochastic Differential Systems, Filtering and Control. Proceedings of the IFIP-WG 7/1 Working Conference, Vilnius, Lithuania, 27 August?2 September 1978. Lecture Notes in Control and Information Sciences 25, Springer-Verlag, New York, pp 190-206
[31] Rauch J, Taylor M (1975) Potential and scattering theory on wildly perturbed domains. J Funct Anal 18:27-59 · Zbl 0293.35056 · doi:10.1016/0022-1236(75)90028-2
[32] Wiener N (1924) The Dirichlet problem. J Math Phys 3:127-146 · JFM 51.0361.01
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