Abstract
The authors discuss a generalization of the usual Green function to equations with only measurable and bounded coefficients. The existence and uniqueness as well as several other important properties are shown. Such a Green function proves useful in connection with quasilinear elliptic systems of “diagonal type”.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Campanato: Equazioni ellitiche del IIo ordine e spaziL (2,λ). Ann. di Mat. Pura e Appl.69, 321–381 (1965)
J. Frehse: Capacity Methods in the Theory of Partial Differential Equations. Jber. d. Dt. Math.-Verein84 (1982), 1–44
M. Giaquinta et S. Hildebrandt: Estimation à priori des solutions faibles de certains systèmes non linéaires elliptiques. Seminaire Goulaouic-Meyer-Schwartz 1980–1981, Exposé no XVII. Ecole polytechnique. Centre de mathématiques, Palaiseau
M. Grüter: Die Greensche Funktion für elliptische Differentialoperatoren mit L∞-Koeffizienten. Diplomarbeit, Bonn (1976)
S. Hildebrandt, J. Jost and K.-O. Widman: Harmonic mappings and minimal submanifolds. Inventiones math.62, 269–298 (1980)
S. Hildebrandt, H. Kaul and K.-O. Widman: An existence theorem for harmonic mappings of Riemannian manifolds. Acta math.138, 1–16(1977)
S. Hildebrandt and K.-O. Widman: Some regularity results for quasilinear elliptic systems of second order, Math.Z.142, 67–86 (1975)
S. Hildebrandt and K.-O. Widman: On the Hölder continuity of weak solutions of quasilinear elliptic systems of second order. Ann. Scuola Norm. Sup. Pisa (IV),4, 145–178 (1977)
S. Hildebrandt and K.-O. Widman: Sätze vom Liouvilleschen Typ für quasilineare elliptische Gleichungen und Systeme. Nachr. Akad. Wiss. Göttingen, II. Math.- Phys. Klasse, Nr.4, 41–59 (1979)
S. Hildebrandt and K.-O. Widman: Variational inequalities for vector-valued functions. J. reine angew. Math.309, 191–220 (1979)
P.-A. Ivert: A priori Schranken für die Ableitungen der Lösungen gewisser elliptischer Differentialgleichungssysteme, man. math.23, 279–294 (1978)
P.-A. Ivert: Regularitätsuntersuchungen von Lösungen elliptischer Systeme von quasilinearen Differentialgleichungen zweiter Ordnung, man. math.30, 53–88 (1979)
D. Kinderlehrer and G. Stampacchia: An Introduction to Variational Inequalities and Their Applications. New York-London-Toronto-Sydney-San Francisco. Academic Press 1980
H. Lewy and G. Stampacchia: On the regularity of the solution of a Variational inequality. Comm. Pure Appl. Math.22, 153–188 (1969)
W. Littman, G. Stampacchia and H.F. Weinberger: Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa (III),17, 43–77 (1963)
J. Moser: On Harnack's theorem for elliptic differential equations. Comm. Pure Appl. Math.14, 577–591 (1961)
K.-O. Widman: On the boundary behaviour of solutions to a class of elliptic partial differential equations. Ark. för Mat. 6.26, 485–533 (1966)
K.-O. Widman: The singularity of the Green function for non-uniformly elliptic partial differential equations with discontinuous coefficients. Uppsala University, Department of Mathematics 12 (1970)
K.-O. Widman: Regular points for a class of degenerating elliptic partial differential equations. Uppsala University, Department of Mathematics 29 (1971)
K.-O. Widman: Inequalities for Green functions of second order elliptic operators. Linköping University, Department of Mathematics 8 (1972)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Grüter, M., Widman, KO. The Green function for uniformly elliptic equations. Manuscripta Math 37, 303–342 (1982). https://doi.org/10.1007/BF01166225
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01166225