Monge Ampere equation: Difference between revisions
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imported>Luis (Created page with "The Monge-Ampere equation refers to \[ \det D^2 u = f(x). \] The right hand side $f$ is always nonnegative. We look for a convex function $u$ which solves the equation. The equ...") |
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The Monge-Ampere equation refers to | The Monge-Ampere equation refers to | ||
\[ \det D^2 u = f(x). \] | \[ \det D^2 u = f(x). \] | ||
The right hand side $f$ is always nonnegative. | 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 | The equation is uniformly elliptic when restricted to a subset of functions which are uniformly $C^{1,1}$ and strictly convex. Moreover, the operator | ||
Latest revision as of 15:47, 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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