Applied Mathematics and Optimization

, Volume 15, Issue 1, pp 15–63 | Cite as

Wiener's criterion and Γ-convergence

  • Gianni Dal Maso
  • Umberto Mosco


Dirichlet problems with homogeneous boundary conditions in (possibly irregular) domains and stationary Schrödinger equations with (possibly singular) nonnegative potentials are considered as special cases of more general equations of the form −Δu + µu = 0, whereµ is an arbitrary given nonnegative Borel measure in ℝn. The stability and compactness of weak solutions under suitable variational perturbations ofµ is investigated and stable pointwise estimates for the modulus of continuity and the “energy” of local solutions are obtained.


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  1. 1.
    Aizenman M, Simon B (1982) Brownian motion and Harnack inequality for Schrödinger operators. Comm Pure Appl Math 35:209–273Google Scholar
  2. 2.
    Attouch H (1984) Variational Convergence for Functions and Operators. Pitman, LondonGoogle Scholar
  3. 3.
    Attouch H, Picard C (1983) Variational inequalities with varying obstacles: the general form of the limit problem. J Funct Anal 50:329–389Google Scholar
  4. 4.
    Baxter JR, Chacon RV, Jain NC (1985) Weak Limits of Stopped Diffusions. University of Minnesota Mathematics Report 84-165, 1-44Google Scholar
  5. 5.
    Carbone L, Colombini F (1980) On convergence of functionals with unilateral constraints. J Math Pures Appl (9) 59:465–500Google Scholar
  6. 6.
    Cioranescu D (1980) Calcul des variations sur des sous-espaces variables. C R Acad Sci Paris Sér A 291:19–22, 87–90Google Scholar
  7. 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 154–178Google Scholar
  8. 8.
    Cioranescu D, Saint Jean Paulin J (1979) Homogenization in open sets with holes. J Math Pures Appl 71:590–607Google Scholar
  9. 9.
    Dal Maso G (1981) Asymptotic behavior of minimum problems with bilateral obstacles. Ann Mat Pura Appl (4) 129:327–366Google Scholar
  10. 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 79–100Google Scholar
  11. 11.
    Dal Maso G, Longo P (1980) Γ-limits of obstacles. Ann Mat Pura Appl (4) 128:1–50Google Scholar
  12. 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)Google Scholar
  13. 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)Google Scholar
  14. 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 3:63–101Google Scholar
  15. 15.
    De Giorgi E, Dal Maso G, Longo P (1980) Γ-limiti di ostacoli. Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur (8) 68:481–487Google Scholar
  16. 16.
    Federer H, Ziemer W (1972) The Lebesgue set of a function whose distribution derivatives arep-th power summable. Indiana Univ Math J 22:139–158Google Scholar
  17. 17.
    Frehse J (1982) Capacity methods in the theory of partial differential equations. Jahresber Deutsch Math-Verein 84:1–44Google Scholar
  18. 18.
    Frehse J, Mosco U (1982) Irregular obstacles and quasivariational inequalities of the stochastic impulse control. Ann Scuola Norm Sup Pisa Cl Sci (4), 105–157Google Scholar
  19. 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 225–257Google Scholar
  20. 20.
    Gariepy R, Ziemer W (1977) A regularity condition at the boundary for solutions of quasilinear elliptic equations. Arch Rational Mech Anal 67:25–39Google Scholar
  21. 21.
    Hruslov EYa (1972) The method of orthogonal projections and the Dirichlet problem in domains with a fine grained boundary. Math USSR-Sb 17:37–59Google Scholar
  22. 22.
    Hruslov EYa (1977) The first boundary value problem in domains with a complicated boundary for higher-order equations. Math USSR-Sb 32:535–549Google Scholar
  23. 23.
    Kac M (1974) Probabilistic methods in some problems of scattering theory. Rocky Mountain J Math 4:511–538Google Scholar
  24. 24.
    Kato T (1973) Schrödinger operators with singular potentials. Israel J Math 13:135–148Google Scholar
  25. 25.
    Marchenko AV, Hruslov EYa (1974) Boundary Value Problems in Domains with Close-Grained Boundaries. Naukova Dumka, Kiev (in Russian)Google Scholar
  26. 26.
    Marchenko AV, Hruslov EYa (1978) New results in the theory of boundary value problems for regions with close-grained boundaries. Uspeki Mat Nauk 33:127Google Scholar
  27. 27.
    Maz'ja VG (1976) On the continuity at a boundary point of solutions of quasi-linear elliptic equations. Vestnik Leningrad Univ Math 3:225–242Google Scholar
  28. 28.
    Mosco U (1969) Convergence of convex sets and solutions of variational inequalities. Adv in Math 3:510–585Google Scholar
  29. 29.
    Mosco U (1985) Wiener Criterion and Potential Estimates for the Obstacle Problems. IMA Preprint Series No 135, Minneapolis, pp 1–56Google Scholar
  30. 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 190–206Google Scholar
  31. 31.
    Rauch J, Taylor M (1975) Potential and scattering theory on wildly perturbed domains. J Funct Anal 18:27–59Google Scholar
  32. 32.
    Wiener N (1924) The Dirichlet problem. J Math Phys 3:127–146Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Gianni Dal Maso
    • 1
  • Umberto Mosco
    • 2
  1. 1.Istituto di MatematicaUniversita di UdineUdineItaly
  2. 2.Dipartimento di MatematicaUniversita di RomaRomaItaly

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