Annali di Matematica Pura ed Applicata

, Volume 110, Issue 1, pp 223–245 | Cite as

Boundary value problems for nonuniformly elliptic equations with measurable coefficients

  • C. V. Coffman
  • M. M. Marcus
  • V. J. Mizel


Let A be a symmetric N × N real-matrix-valued function on a connected region in Rn, with A positive definite a.e. and A, A−1 locally integrable. Let b and c be locally integrable, non-negative, real-valued functions on Ω, with c positive a.e. Put a(u, v) = =\(\mathop \smallint \limits_\Omega \)((A∇u, ∇v) + buv) dx. We consider in X the weak boundary value problem a(u, v) = =\(\mathop \smallint \limits_\Omega \)fvcdx, all v ε X; where X is a suitable Hilbert space contained in H loc 1,1 (Ω). Criteria are given in order that the Green's operator for this problem have an integral representation and bounded eigenfunctions; in addition, criteria for compactness are given.


Hilbert Space Elliptic Equation Integral Representation Connected Region Measurable Coefficient 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [1]
    C. V. Coffman -R. Duffin -V. J. Mizel,Positivity of weak solutions of non-uniformly elliptic equations, Ann. di Mat. pura e appl.,104 (1975), pp. 209–238.CrossRefMathSciNetGoogle Scholar
  2. [2]
    N. Dunford -J. Schwartz,Linear Operators, Part I, Interscience, New York, 1958.Google Scholar
  3. [3]
    R. E. Edwards,Functional Analysis, Holt, Rinehart and Winston, Inc., New York, 1965.Google Scholar
  4. [4]
    A. Friedman,Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York, 1969.Google Scholar
  5. [4a]
    E. Gagliardo,Proprietà di alcune classi di funzioni in più variabili, Ricerche di Mat.,7 (1958), pp. 102–137.zbMATHMathSciNetGoogle Scholar
  6. [5]
    A. Grothendieck, Sur certains sous-espaces vectoriels de Lp, Canad. J. Math.,6 (1954), pp. 158–160.zbMATHMathSciNetGoogle Scholar
  7. [6]
    A. Ionescu-Tulcea -C. Ionescu-Tulcea,Topics in the Theory of Lifting, Springer-Verlag, New York, 1969.Google Scholar
  8. [7]
    M. Marcus -V. J. Mizel,Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rat. Mech. Anal.,45 (1972), pp. 294–320.CrossRefMathSciNetGoogle Scholar
  9. [8]
    M. K. V. Murthy -G. Stampacchia,Boundary value problems for some degenerate elliptic operators, Ann. di Mat. pura e appl.,80 (1968), pp. 1–122.CrossRefMathSciNetGoogle Scholar
  10. [9]
    W. Rudin,Functional Analysis, McGraw-Hill, New York, 1973.Google Scholar
  11. [10]
    G. Stampacchia,Equations elliptiques du second ordere a coefficients discontinuus, Seminaires de Math. Superieures, Ete 1965, Les Presses de l'Universite de Montreal.Google Scholar
  12. [11]
    M. Trudinger,Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa,27 (1973), pp. 265–308.zbMATHMathSciNetGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1975

Authors and Affiliations

  • C. V. Coffman
    • 1
  • M. M. Marcus
    • 1
  • V. J. Mizel
    • 1
  1. 1.Pittsburgh

Personalised recommendations