|
|
| Line 1: |
Line 1: |
| '''Not be confused with the unconnected [[Monge equation]].'''
| | The writer is known as Irwin. One of the things he enjoys most is ice skating but he is struggling to find time for it. My family members lives in Minnesota and my family members loves it. For many years he's been operating as a meter reader and it's something he truly enjoy.<br><br>Also visit my website [http://dore.gia.ncnu.edu.tw/88ipart/node/1326254 http://dore.gia.ncnu.edu.tw] |
| | |
| In [[mathematics]], a (real) '''Monge–Ampère equation''' is a nonlinear second order [[partial differential equation]] of special kind. A second order equation for the unknown function ''u'' of two variables ''x'',''y'' is of Monge–Ampère type if it is linear in the [[determinant]] of the [[Hessian matrix]] of ''u'' and in the second order [[partial derivative]]s of ''u''. The independent variables (''x'',''y'') vary over a given domain ''D'' of '''R'''<sup>2</sup>. The term also applies to analogous equations with ''n'' independent variables. The most complete results so far have been obtained when the equation is [[elliptic operator|elliptic]].
| |
| | |
| Monge–Ampère equations frequently arise in [[differential geometry]], for example, in the [[Hermann Weyl|Weyl]] and [[Hermann Minkowski|Minkowski]] problems in [[differential geometry of surfaces]]. They were first studied by [[Gaspard Monge]] in 1784 and later by [[André-Marie Ampère]] in 1820. Important results in the theory of Monge–Ampère equations have been obtained by [[Sergei Bernstein]], [[Aleksei Pogorelov]], [[Charles Fefferman]], and [[Louis Nirenberg]].
| |
| | |
| == Description ==
| |
| | |
| Given two independent variables ''x'' and ''y'', and one dependent variable ''u'', the general Monge–Ampère equation is of the form
| |
| | |
| :<math>L[u] = A(u_{xx}u_{yy}-u_{xy}^2)+Bu_{xx}+2Cu_{xy}+Du_{yy}+E = 0\,</math>
| |
| | |
| where ''A'', ''B'', ''C'', ''D'', and ''E'' are functions depending on the first order variables ''x'', ''y'', ''u'', ''u''<sub>x</sub>, and ''u''<sub>y</sub> only.
| |
| | |
| == Rellich's theorem ==
| |
| Let Ω be a bounded domain in '''R'''<sup>3</sup>, and suppose that on Ω ''A'', ''B'', ''C'', ''D'', and ''E'' are continuous functions of ''x'' and ''y'' only. Consider the [[Dirichlet problem]] to find ''u'' so that
| |
| | |
| :<math>L[u]=0,\quad \text{on}\ \Omega</math>
| |
| :<math>u|_{\partial\Omega}=g.</math> | |
| | |
| If
| |
| | |
| :<math>BD-C^2-AE > 0,</math>
| |
| | |
| then the Dirichlet problem has at most two solutions.<ref>{{cite book|last1=Courant |first1=R.|last2=Hilbert |first2=D. |date=1962 |title=Methods of Mathematical Physics |url= |volume= 2|location= |publisher=Interscience Publishers |page=324 |isbn= |accessdate= }}</ref>
| |
| | |
| ==Ellipticity results==
| |
| Suppose now that '''x''' is a variable with values in a domain in '''R'''<sup>n</sup>, and that ''f''('''x''',''u'',''D''<sup>2</sup>''u'') is a positive function. Then the Monge–Ampère equation
| |
| | |
| :<math>L[u] = \det D^2 u - f(\mathbf{x},u,Du)=0\qquad\qquad (1)</math>
| |
| | |
| is a [[nonlinear equation|nonlinear]] [[elliptic partial differential equation]] (in the sense that its [[linearization]] is elliptic), provided one confines attention to [[convex function|convex]] solutions.
| |
| | |
| Accordingly, the operator ''L'' satisfies versions of the [[maximum principle]], and in particular solutions to the Dirichlet problem are unique, provided they exist.
| |
| | |
| ==Applications==
| |
| Monge–Ampère equations arise naturally in several problems in [[Riemannian geometry]], [[conformal geometry]], and [[CR geometry]]. One of the simplest of these applications is to the problem of prescribed [[Gauss curvature]]. Suppose that a real-valued function ''K'' is specified on a domain Ω in '''R'''<sup>n</sup>, the problem of prescribed Gauss curvature seeks to identify a hypersurface of '''R'''<sup>n+1</sup> as a graph ''z''=''u''('''x''') over '''x'''∈Ω so that, at each point of the surface the Gauss curvature is given by ''K''('''x'''). The resulting partial differential equation is
| |
| | |
| :<math>\det D^2 u - K(\mathbf{x})(1+|Du|^2)^{(n+2)/2} = 0.</math>
| |
| | |
| The Monge–Ampère equations are related to the [[Transportation theory (mathematics)#Monge and Kantorovich formulations|Monge-Kantorovich optimal mass transportation problem]], when the "cost functional" therein is given by the Euclidean distance.<ref>{{cite journal|last=Benamou|first=Jean David|coauthors=Yann Brenier|title=A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem|journal=Numerische Mathematik|year=2000|volume=84|issue=3|pages=375–393|doi=10.1007/s002110050002|url=http://dx.doi.org/10.1007/s002110050002}}</ref>
| |
| | |
| ==See also==
| |
| *[[Calabi conjecture|Complex Monge-Ampère equation]]
| |
| | |
| ==References==
| |
| <references />
| |
| | |
| ==Additional References==
| |
| *Gilbarg, D. and [[Neil Trudinger|Trudinger, N. S.]] ''Elliptic Partial Differential Equations of Second Order.'' Berlin: Springer-Verlag, 1983. ISBN 3-540-41160-7 ISBN 978-3540411604
| |
| *{{springer|id=M/m064620|title=Monge–Ampère equation|author=A.V. Pogorelov}}
| |
| | |
| ==External links==
| |
| * {{mathworld|title=Monge-Ampère Differential Equation|urlname=Monge-AmpereDifferentialEquation}}
| |
| | |
| {{DEFAULTSORT:Monge-Ampere equation}}
| |
| [[Category:Partial differential equations]]
| |
The writer is known as Irwin. One of the things he enjoys most is ice skating but he is struggling to find time for it. My family members lives in Minnesota and my family members loves it. For many years he's been operating as a meter reader and it's something he truly enjoy.
Also visit my website http://dore.gia.ncnu.edu.tw