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Grueter, Michael; Widman, Kjell-Ove

The Green function for uniformly elliptic equations. (English) Zbl 0485.35031

Manuscr. Math. 37, 303-342 (1982).
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Cited in 192 Documents

MathOverflow Questions:

Fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix
Integral representation of solution of an elliptic PDE in divergence form
References for Green functions of \(\nabla \cdot a \nabla\) on a domain with \(a \in L^\infty\)

MSC:

35J15 Second-order elliptic equations
35C15 Integral representations of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

Green function; measurable and bounded coefficients; uniqueness; existence
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\textit{M. Grueter} and \textit{K.-O. Widman}, Manuscr. Math. 37, 303--342 (1982; Zbl 0485.35031)
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References:

[1] S. Campanato: Equazioni ellitiche del IIo ordine e spaziL (2,?). Ann. di Mat. Pura e Appl.69, 321-381 (1965) · Zbl 0145.36603
[2] J. Frehse: Capacity Methods in the Theory of Partial Differential Equations. Jber. d. Dt. Math.-Verein84 (1982), 1-44 · Zbl 0486.35002
[3] M. Giaquinta et S. Hildebrandt: Estimation ? priori des solutions faibles de certains syst?mes non lin?aires elliptiques. Seminaire Goulaouic-Meyer-Schwartz 1980-1981, Expos? no XVII. Ecole polytechnique. Centre de math?matiques, Palaiseau
[4] M. Gr?ter: Die Greensche Funktion f?r elliptische Differentialoperatoren mit L?-Koeffizienten. Diplomarbeit, Bonn (1976)
[5] S. Hildebrandt, J. Jost and K.-O. Widman: Harmonic mappings and minimal submanifolds. Inventiones math.62, 269-298 (1980) · Zbl 0446.58006
[6] S. Hildebrandt, H. Kaul and K.-O. Widman: An existence theorem for harmonic mappings of Riemannian manifolds. Acta math.138, 1-16(1977) · Zbl 0356.53015
[7] S. Hildebrandt and K.-O. Widman: Some regularity results for quasilinear elliptic systems of second order, Math.Z.142, 67-86 (1975) · Zbl 0317.35040
[8] S. Hildebrandt and K.-O. Widman: On the H?lder continuity of weak solutions of quasilinear elliptic systems of second order. Ann. Scuola Norm. Sup. Pisa (IV),4, 145-178 (1977) · Zbl 0353.35013
[9] S. Hildebrandt and K.-O. Widman: S?tze vom Liouvilleschen Typ f?r quasilineare elliptische Gleichungen und Systeme. Nachr. Akad. Wiss. G?ttingen, II. Math.- Phys. Klasse, Nr.4, 41-59 (1979) · Zbl 0426.35037
[10] S. Hildebrandt and K.-O. Widman: Variational inequalities for vector-valued functions. J. reine angew. Math.309, 191-220 (1979) · Zbl 0408.49012
[11] P.-A. Ivert: A priori Schranken f?r die Ableitungen der L?sungen gewisser elliptischer Differentialgleichungssysteme, man. math.23, 279-294 (1978) · Zbl 0374.35021
[12] P.-A. Ivert: Regularit?tsuntersuchungen von L?sungen elliptischer Systeme von quasilinearen Differentialgleichungen zweiter Ordnung, man. math.30, 53-88 (1979) · Zbl 0429.35033
[13] D. Kinderlehrer and G. Stampacchia: An Introduction to Variational Inequalities and Their Applications. New York-London-Toronto-Sydney-San Francisco. Academic Press 1980 · Zbl 0457.35001
[14] H. Lewy and G. Stampacchia: On the regularity of the solution of a Variational inequality. Comm. Pure Appl. Math.22, 153-188 (1969) · Zbl 0167.11501
[15] 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) · Zbl 0116.30302
[16] J. Moser: On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math.14, 577-591 (1961) · Zbl 0111.09302
[17] K.-O. Widman: On the boundary behaviour of solutions to a class of elliptic partial differential equations. Ark. f?r Mat. 6.26, 485-533 (1966) · Zbl 0166.37702
[18] 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) · Zbl 0217.12802
[19] K.-O. Widman: Regular points for a class of degenerating elliptic partial differential equations. Uppsala University, Department of Mathematics 29 (1971)
[20] K.-O. Widman: Inequalities for Green functions of second order elliptic operators. Link?ping University, Department of Mathematics 8 (1972)
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