- Hopf maximum principle
The Hopf maximum principle is a
maximum principle in the theory of second orderelliptic partial differential equation s and has been described as the "classic and bedrock result" of that theory. Generalizing the maximum principle forharmonic function s which was already known to Gauss in 1839,Eberhard Hopf proved in 1927 that if a function satisfies a second order partial differential inequality of a certain kind in a domain of R"n" and attains amaximum in the domain then the function is constant. The simple idea behind Hopf's proof, the comparison technique he introduced for this purpose, has led to an enormous range of important applications and generalizations.Mathematical formulation
Let "u" = "u"("x"), "x" = ("x"1, …, "x""n") be a "C"2 function which satisfies the differential inequality
:
in an open domain Ω, where the
symmetric matrix "a""ij" = "a""ij"("x") is locally uniformly positive definite in Ω and the coefficients "a""ij", "b""i" = "b""i"("x") are locallybounded . If "u" takes a maximum value "M" in Ω then "u" ≡ "M".It is usually thought that the Hopf maximum principle applies only to
linear differential operator s "L". In particular, this is the point of view taken by Courant and Hilbert's "Methods of Mathematical Physics ". However, in the later sections of his original paper Hopf considered a more general situation, which permits certain nonlinear operators "L" and leads to uniqueness statements in theDirichlet problem for the operator of themean curvature and theMonge–Ampère equation in some cases.References
* "Selected works of Eberhard Hopf with commentaries". Edited by Cathleen S. Morawetz, James B. Serrin and Yakov G. Sinai.
American Mathematical Society , Providence, RI, 2002. xxiv+386 pp ISBN 0-8218-2077-X MathSciNet|id=1985954
* Patrizia Pucci and James Serrin, doi-inline|id=10.1016/j.jde.2003.05.001|"The strong maximum principle revisited", J. Differential Equations 196 (2004), no. 1, 1–66, MathSciNet|id=2025185
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