Summary
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.
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Entrata in Redazione il 13 giugno 1973.
Research partially supported by the National Science Foundation under Grant GP-21512.
Research Grant-DA-AROD-31-124-71-G17 Army ResearchOdce (Durham).
Partially supported by the National Science Foundation under Grant GP-28377.
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Coffman, C.V., Duffin, R.J. & Mizel, V.J. Positivity of weak solutions of non-uniformly elliptic equations. Annali di Matematica 104, 209–238 (1975). https://doi.org/10.1007/BF02417017
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DOI: https://doi.org/10.1007/BF02417017