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Capacity Theory for Monotone Operators

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Abstract

If Au = - div(a(x, Du)) is a monotone operator defined on the Sobolev space W1,P(R n, 1 < p < + ∞, with a(x,0) = 0 for a.e. x ∈ Rn, the capacity C_A(E,F) relative to A can be defined for every pair (E,F) of bounded sets in Rn with E ⊂ F. The main properties of the set function CA(E,F) are investigated. In particular it is proved that CA(E,F) is increasing and countably subadditive with respect to E, decreasing with respect to F, and continuous, in a suitable sense, with respect to E and F.

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References

  1. Casado Diaz, J. and Garroni, A.: Asymptotic behaviour of nonlinear elliptic systems on varying domains. In preparation.

  2. Choquet, G.: ‘Forme abstraite du théorème de capacitabilité’, Ann. Inst. Fourier(Grenoble) 9(1959), 83-89.

    Google Scholar 

  3. Dal Maso, G.: ‘On the integral representation of certain local functionals’, Ricerche Mat. 32(1983), 85-113.

    Google Scholar 

  4. Dal Maso, G. and Defranceschi, A.: ‘Limits of nonlinear Dirichlet problems in varying domains’, Manuscripta Math. 61(1988), 251-278.

    Google Scholar 

  5. Dal Maso, G. and Garroni, A.: Capacity theory for non-symmetric elliptic operators.Preprint SISSA, Trieste, 1993.

    Google Scholar 

  6. Dal Maso, G. and Garroni, A.: ‘New results on the asymptotic behaviour of Dirichlet problems in perforated domains’, Math. Models Methods Appl. Sci. 3(1994), 373-407.

    Google Scholar 

  7. Dal Maso, G. and Garroni, A.: ‘The capacity method for asymptotic Dirichlet problems’, Asymptotic Anal., to appear.

  8. Dal Maso, G. and Murat, F.: ‘Dirichlet problems in preforated domains for homogeneous monotone operators on H10’. Calculus of Variations, Homogenization and Continuum Mechanics (CIRM-Luminy, Marseille, 1993), 177-202, World Scientific, Singapore, 1994.

    Google Scholar 

  9. Dal Maso, G. and Murat, F.: Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators.Preprint SISSA, Trieste, 1996.

    Google Scholar 

  10. Dal Maso, G. and Skrypnik, I.V.: ‘Asymptotic behaviour of nonlinear Dirichlet problems in perforated domains’, Ann. Mat. Pura Appl., to appear.

  11. Evans, L. C. and Gariepy, R. F.: Measure Theory and Fine Properties of Functions.CRC Press, Boca Raton, 1992.

    Google Scholar 

  12. Federer, H. and Ziemer, W. P.: ‘The Lebesgue set of a function whose distribution derivatives are pth power summable’, Indiana Univ. Math. J. 22(1972), 139-158.

    Google Scholar 

  13. Feyel, D. and de La Pradelle, A.: ‘Le rôle des espaces de Sobolev en topologie fine.’ Séminaire de Theorie du Potentiel, Paris, No. 2, 43-61, Lecture Notes in Math. 563, Springer-Verlag, Berlin, 1976.

    Google Scholar 

  14. Heinonen, J. Kilpeläinen, T. and Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Clarendon Press, Oxford, 1993.

    Google Scholar 

  15. Kinderlehrer, D. and Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications.Academic Press, New York, 1980.

    Google Scholar 

  16. Ma, Z.-M and Röckner, M.: Introduction to the Theory of(Non-Symmetric) Dirichlet Forms. Springer-Verlag, Berlin, 1992.

    Google Scholar 

  17. Maz'ya, V. G.: Sobolev Spaces.Springer-Verlag, Berlin, 1985.

    Google Scholar 

  18. Skrypnik, I. V.: Nonlinear Elliptic Boundary Value Problems.Teubner-Verlag, Leipzig, 1986.

    Google Scholar 

  19. Skrypnik, I.V.: Methods for Analysis of Nonlinear Elliptic Boundary Value Problems (inRussian). Nauka, Moscow, 1990. English translation in: Translations of Mathematical Monographs 139, American Mathematical Society, Providence, 1994.

    Google Scholar 

  20. Skrypnik, I. V.: ‘Averaging nonlinear Dirichlet problems in domains with channels’, Soviet Math. Dokl. 42(1991), 853-857.

    Google Scholar 

  21. Skrypnik, I. V.: ‘Asymptotic behaviour of solutions of nonlinear elliptic problems in perforated domains’, Math. Sb.(N.S.) 184(1993), 67-90.

    Google Scholar 

  22. Skrypnik, I. V.: ‘Homogenization of nonlinear Dirichlet problems in perforated domains of general structure’, Mat. Sb.(N.S.), to appear.

  23. Stampacchia, G.: ‘Le problème de Dirichlet pour équations elliptiques du second ordre `a coeffi-cients discontinus’, Ann. Inst. Fourier(Grenoble) 15(1965), 189-258.

    Google Scholar 

  24. Ziemer, W. P.: Weakly Differentiable Functions. Springer-Verlag, Berlin, 1989.

    Google Scholar 

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Authors
  1. Gianni Dal Maso
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  2. Igor V. Skrypnik
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Dal Maso, G., Skrypnik, I.V. Capacity Theory for Monotone Operators. Potential Analysis 7, 765–803 (1997). https://doi.org/10.1023/A:1017987405983

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