Boundary value problems for nonuniformly elliptic equations with measurable coefficients
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.
KeywordsHilbert Space Elliptic Equation Integral Representation Connected Region Measurable Coefficient
Unable to display preview. Download preview PDF.
- N. Dunford -J. Schwartz,Linear Operators, Part I, Interscience, New York, 1958.Google Scholar
- R. E. Edwards,Functional Analysis, Holt, Rinehart and Winston, Inc., New York, 1965.Google Scholar
- A. Friedman,Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York, 1969.Google Scholar
- A. Ionescu-Tulcea -C. Ionescu-Tulcea,Topics in the Theory of Lifting, Springer-Verlag, New York, 1969.Google Scholar
- W. Rudin,Functional Analysis, McGraw-Hill, New York, 1973.Google Scholar
- 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