# Fully nonlinear integro-differential equations

### From Mwiki

Line 7: | Line 7: | ||

Given a family of linear integro-differential operators $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$: | Given a family of linear integro-differential operators $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$: | ||

\begin{align*} | \begin{align*} | ||

- | M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} L u(x) \\ | + | M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} \, L u(x) \\ |

- | M^-_\mathcal{L} u(x) &= \inf_{L \in \mathcal{L}} L u(x) | + | M^-_\mathcal{L} u(x) &= \inf_{L \in \mathcal{L}} \, L u(x) |

\end{align*} | \end{align*} | ||

## Revision as of 01:53, 27 May 2011

Fully nonlinear integro-differential equations are a nonlocal version of fully nonlinear elliptic equations of the form $F(D^2 u, Du, u, x)=0$. The main examples are the integro-differential Bellman equation from optimal control, and the Isaacs equation from stochastic games.

A general definition of ellipticity can be given that does not require a specific form of the equation. However, the main two applications are the two above.

## Definition

Given a family of linear integro-differential operators $\mathcal{L}$, we define the extremal operators $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$: \begin{align*} M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} \, L u(x) \\ M^-_\mathcal{L} u(x) &= \inf_{L \in \mathcal{L}} \, L u(x) \end{align*}

We define a nonlinear operator $I$ to be elliptic in a domain $\Omega$ with respect to the class $\mathcal{L}$ if it assigns a continuous function $Iu$ to every function $u \in L^\infty(\R^n) \cap C^2(\Omega)$, and moreover for any two such functions $u$ and $v$: \[M^-_\mathcal{L} [u-v](x)\leq Iu(x) - Iv(x) \leq M^+_\mathcal{L} [u-v] (x), \] for any $x \in \Omega$.

A fully nonlinear elliptic equation with respect to $\mathcal{L}$ is an equation of the form $Iu=0$ in $\Omega$ with $I$ uniformly elliptic respect to $\mathcal{L}$.