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Not be confused with the unconnected Monge equation. 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 derivatives of ''u''. The independent variables (''x'',''y'') vary over a given domain ''D'' of R2. 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. Monge–Ampère equations frequently arise in differential geometry, for example, in the Weyl and 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 : where ''A'', ''B'', ''C'', ''D'', and ''E'' are functions depending on the first order variables ''x'', ''y'', ''u'', ''u''x, and ''u''y only. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Monge–Ampère equation」の詳細全文を読む スポンサード リンク
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