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”.
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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)
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Grüter, M., Widman, KO. The Green function for uniformly elliptic equations. Manuscripta Math 37, 303–342 (1982). https://doi.org/10.1007/BF01166225
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DOI: https://doi.org/10.1007/BF01166225