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<p><b>New page</b></p><div>The '''Hopf maximum principle''' is a [[maximum principle]] in the theory of second order [[elliptic partial differential equation]]s and has been described as the "classic and bedrock result" of that theory. Generalizing the maximum principle for [[harmonic function]]s which was already known to [[Carl Friedrich Gauss|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'''<sup>''n''</sup> and attains a [[maximum]] 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.<br />
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== Mathematical formulation ==<br />
<br />
Let ''u'' = ''u''(''x''), ''x'' = (''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>) be a ''C''<sup>2</sup> function which satisfies the differential inequality<br />
<br />
: <math> Lu = \sum_{ij} a_{ij}(x)\frac{\partial^2 u}{\partial x_i\partial x_j} + <br />
\sum_i b_i\frac{\partial u}{\partial x_i} \geq 0 </math><br />
<br />
in an [[open set|open domain]] &Omega;, where the [[symmetric matrix]] ''a''<sub>''ij''</sub> = ''a''<sub>''ij''</sub>(''x'') is locally uniformly [[positive definite matrix|positive definite]] in &Omega; and the coefficients ''a''<sub>''ij''</sub>, ''b''<sub>''i''</sub> = ''b''<sub>''i''</sub>(''x'') are locally [[bounded]]. If ''u'' takes a maximum value ''M'' in &Omega; then ''u'' &equiv; ''M''.<br />
<br />
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 [[Richard Courant|Courant]] and [[David Hilbert|Hilbert's]] ''[[Methods of Mathematical Physics]]''. In the later sections of his original paper, however, Hopf considered a more general situation which permits certain nonlinear operators ''L'' and, in some cases, leads to uniqueness statements in the [[Dirichlet problem]] for the [[mean curvature]] operator and the [[Monge–Ampère equation]].<br />
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== References ==<br />
<br />
*{{citation<br />
| last = Hopf | first = Eberhard<br />
| editor1-last = Morawetz | editor1-first = Cathleen S.<br />
| editor2-last = Serrin | editor2-first = James B.<br />
| editor3-last = Sinai | editor3-first = Yakov G.<br />
| isbn = 0-8218-2077-X<br />
| location = Providence, RI<br />
| mr = 1985954<br />
| publisher = American Mathematical Society<br />
| title = Selected works of Eberhard Hopf with commentaries<br />
| year = 2002}}.<br />
*{{citation<br />
| last1 = Pucci | first1 = Patrizia<br />
| last2 = Serrin | first2 = James<br />
| doi = 10.1016/j.jde.2003.05.001<br />
| issue = 1<br />
| journal = Journal of Differential Equations<br />
| mr = 2025185<br />
| pages = 1–66<br />
| title = The strong maximum principle revisited<br />
| volume = 196<br />
| year = 2004}}.<br />
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[[Category:Elliptic partial differential equations]]<br />
[[Category:Mathematical principles]]</div>en>D.Lazard