Intro to nonlocal equations and Monge Ampere equation: Difference between pages
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The Monge-Ampere equation refers to | |||
\[ \det D^2 u = f(x). \] | |||
The right hand side $f$ is always nonnegative. The unknown function $u$ should be convex. | |||
The equation is uniformly elliptic when restricted to a subset of functions which are uniformly $C^{1,1}$ and strictly convex. Moreover, the operator | |||
\[ MA(D^2 u) := \left( \det (D^2 u) \right)^{1/n} = \inf \{ a_{ij} \partial_{ij} u : \det \{a_{ij}\} = 1 \},\] | |||
is concave. | |||
{{stub}} | |||
Latest revision as of 16:47, 8 April 2015
The Monge-Ampere equation refers to \[ \det D^2 u = f(x). \] The right hand side $f$ is always nonnegative. The unknown function $u$ should be convex.
The equation is uniformly elliptic when restricted to a subset of functions which are uniformly $C^{1,1}$ and strictly convex. Moreover, the operator \[ MA(D^2 u) := \left( \det (D^2 u) \right)^{1/n} = \inf \{ a_{ij} \partial_{ij} u : \det \{a_{ij}\} = 1 \},\] is concave.
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