Fully nonlinear elliptic equations: Difference between revisions
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imported>Luis (Created page with "A fully nonlinear elliptic equation is an expression of the form \[ F(D^2u, \nabla u, u, x) = 0 \text{ in } \Omega.\] The function $F$ is supposed to satisfy the following two b...") |
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* If $A \geq B$, $F(A,p,u,x) \geq F(B,p,u,x)$ for any values of $p \in \R^n$, $u\in \R$ and $x \in \Omega$. | * If $A \geq B$, $F(A,p,u,x) \geq F(B,p,u,x)$ for any values of $p \in \R^n$, $u\in \R$ and $x \in \Omega$. | ||
* If $u \geq v$, $F(A,p,u,x) \leq F(A,p,v,x)$ for any values of $A \in \R^{n \times n}$, $p \in \R^n$ and $x \in \Omega$. | * If $u \geq v$, $F(A,p,u,x) \leq F(A,p,v,x)$ for any values of $A \in \R^{n \times n}$, $p \in \R^n$ and $x \in \Omega$. | ||
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Latest revision as of 15:51, 8 April 2015
A fully nonlinear elliptic equation is an expression of the form \[ F(D^2u, \nabla u, u, x) = 0 \text{ in } \Omega.\]
The function $F$ is supposed to satisfy the following two basic assumptions
- If $A \geq B$, $F(A,p,u,x) \geq F(B,p,u,x)$ for any values of $p \in \R^n$, $u\in \R$ and $x \in \Omega$.
- If $u \geq v$, $F(A,p,u,x) \leq F(A,p,v,x)$ for any values of $A \in \R^{n \times n}$, $p \in \R^n$ and $x \in \Omega$.
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