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A Harnack inequality for degenerate elliptic equations on minimal surfaces

Published online by Cambridge University Press:  20 January 2009

Frances Cooper
Frances Cooper
Affiliation:
University of Glasgow, Glasgow G12 8QW
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Harnack inequalities are known to be of great importance in the theory of quasilinear elliptic partial differential equations. In the case of such equations defined over a domain Ω in Rn, inequalities of this type have been proved for solutions of second-order equations in divergence form which are of either elliptic or degenerate elliptic structure. More recently Bombieri and Giusti (2) have proved a Harnack inequality for solutions of linear elliptic equations on a minimal surface in Rn+1. The equations are of the form

where summation over i, j = 1, …, n+l is understood, and δ = (δ1, …, δn+1) is the tangential derivative on S. In (2), the inequality is used to give much simplified proofs of some classical results on minimal surfaces, and to generalise some more recent ones.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 20 , Issue 1 , March 1976 , pp. 29 - 39
Copyright
Copyright © Edinburgh Mathematical Society 1976

References

REFERENCES

(1) Bombeeri, E., Theory of minimal surfaces and a counterexample to the Bernstein conjecture in high dimensions, lecture notes (Courant Institute of Mathematical Sciences, New York, 1970).Google Scholar
(2) Bombieri, E. and Giusti, E., Harnack's inequality for elliptic differential equations on minimal surfaces, Invent. Math. 15 (1972), 24–46.Google Scholar
(3) Edmunds, D. E. and Peletier, L. A., A Harnack inequality for weak solutions of degenerate elliptic equations, J. London Math. Soc. 5 (1972), 21–31.CrossRefGoogle Scholar
(4) Federer, H., Geometric measure theory (Springer, Berlin-Heidelberg-New York 1969).Google Scholar
(5) Miranda, M., Disuguaglianze di Sobolev sulle ipersuprefici minimali, Rend. Sem. Mat. Univ. Padova 38 (1967), 69–79.Google Scholar
(6) Moser, J., On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591.CrossRefGoogle Scholar
(7) Murthy, M. K. V. and Stampacchia, G., Boundary value problems for some degenerate elliptic equations, Ann. Mat. Pura Appl. 80 (1968), 1–122.Google Scholar