, Volume 110, Issue 1, pp 223-245

Boundary value problems for nonuniformly elliptic equations with measurable coefficients

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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.

Entrata in Redazione il 21 giugno 1975.
Research was partially supported by the National Science Foundation under Grant GP-28377A2.