Annali di Matematica Pura ed Applicata

, Volume 104, Issue 1, pp 209–238 | Cite as

Positivity of weak solutions of non-uniformly elliptic equations

  • C. V. Coffman
  • R. J. Duffin
  • V. J. Mizel


Let A be a symmetric N × N real-matrix-valued function on a connected region Ω 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 the boundary value problem a(u, v)=\(\mathop \smallint \limits_\Omega \)fvcdx, for all v ε C 0 (Ω), and the eigenvalue problem a(u, v)=λ\(\mathop \smallint \limits_\Omega \)uvcdx, for all v ε C 0 (Ω). Positivity of the solution operator for the boundary value problem, as well as positivity of the dominant eigenfunction (if there is one) and simplicity of the corresponding eigenvalue are proved to hold in this context.


Weak Solution Eigenvalue Problem Elliptic Equation Solution Operator Connected Region 
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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1974

Authors and Affiliations

  • C. V. Coffman
    • 1
  • R. J. Duffin
    • 1
  • V. J. Mizel
    • 1
  1. 1.Pittsburgh

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