Fully nonlinear integro-differential equations: Difference between revisions
imported>Luis (Created page with "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-...") |
imported>Luis No edit summary |
||
| Line 5: | Line 5: | ||
== Definition == | == Definition == | ||
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) \\ | ||
| Line 12: | Line 12: | ||
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 $v$: | 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 $v$: | ||
\[M^-_\mathcal{L} u(x)\leq Iu(x) - Iv(x) \leq M^+_\mathcal{L} u(x), \] | \[M^-_\mathcal{L} [u-v](x)\leq Iu(x) - Iv(x) \leq M^+_\mathcal{L} [u-v] (x), \] | ||
for any $x \in \Omega$. | 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 $\mathcal{L}$. | 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 $\mathcal{L}$. | ||
Revision as of 19:53, 26 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 $\mathcal{L}$.