# Fully nonlinear integro-differential equations

### From Mwiki

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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. | 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. | ||

- | + | The general definition of ellipticity provided below does not require a specific form of the equation. However, the main two applications are the two above. | |

- | == Definition == | + | == Definition <ref name="CS"/><ref name="CS2"/> == |

- | 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) \\ | ||

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The equation is uniformly elliptic with respect to any class $\mathcal{L}$ that contains all the operators $L_{ab}$. | The equation is uniformly elliptic with respect to any class $\mathcal{L}$ that contains all the operators $L_{ab}$. | ||

+ | |||

+ | <div cellspacing="0" style="width:100%;background:transparent;color:inherit;;"> | ||

+ | Hello world | ||

+ | </div> | ||

+ | |||

+ | <div cellspacing="0" style="width:100%;background:#DDEEFF;color:inherit;;"> | ||

+ | <blockquote> | ||

+ | '''Note'''. In several articles <ref name="BI"/><ref name="BIC2"/><ref name="BIC"/>, fully nonlinear integro-differential equations of the form $F(D^2 u, Du, u, x, I)=f(x)$ are analyzed, where $I$ is a [[linear integro-differential operator]]. This is a rigid structure for an equation. For example if an equation is purely integro-differential (it does not depend on $D^2u$, $Du$ or $u$) then it is forced to be linear: $I = [F(x,\cdot)^{-1}f(x)]$. | ||

+ | |||

+ | On the other hand, the results in these papers apply to the more general definitions of fully nonlinear integro-differential equations as well. The reason for that restriction seems to be just to have an equation that is short to write down. | ||

+ | </blockquote> | ||

+ | </div> | ||

+ | |||

+ | == References == | ||

+ | {{reflist|refs= | ||

+ | <ref name="CS2">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity results for nonlocal equations by approximation | publisher=[[Springer-Verlag]] | location=Berlin, New York | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | pages=1–30}}</ref> | ||

+ | <ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref> | ||

+ | <ref name="BIC"> | ||

+ | {{Citation | last1=Barles | first1=Guy | last2=Chasseigne | first2=Emmanuel | last3=Imbert | first3=Cyril | title=Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations | url=http://dx.doi.org/10.4171/JEMS/242 | doi=10.4171/JEMS/242 | year=2011 | journal=Journal of the European Mathematical Society (JEMS) | issn=1435-9855 | volume=13 | issue=1 | pages=1–26}}</ref> | ||

+ | <ref name="BIC2">{{Citation | last1=Barles | first1=G. | last2=Chasseigne | first2=Emmanuel | last3=Imbert | first3=Cyril | title=On the Dirichlet problem for second-order elliptic integro-differential equations | url=http://dx.doi.org/10.1512/iumj.2008.57.3315 | doi=10.1512/iumj.2008.57.3315 | year=2008 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=57 | issue=1 | pages=213–246}}</ref> | ||

+ | <ref name="BI">{{Citation | last1=Barles | first1=Guy | last2=Imbert | first2=Cyril | title=Second-order elliptic integro-differential equations: viscosity solutions' theory revisited | url=http://dx.doi.org/10.1016/j.anihpc.2007.02.007 | doi=10.1016/j.anihpc.2007.02.007 | year=2008 | journal=Annales de l'Institut Henri Poincaré. Analyse Non Linéaire | issn=0294-1449 | volume=25 | issue=3 | pages=567–585}}</ref> | ||

+ | }} |

## Revision as of 02:54, 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.

The general definition of ellipticity provided below does not require a specific form of the equation. However, the main two applications are the two above.

## Definition ^{[1]}^{[2]}

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 $\mathcal{L}$.

## Examples

The two main examples are the following.

- The Bellman equation is the equality

\[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \] where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.

The equation appears naturally in problems of stochastic control with Levy processes.

The equation is uniformly elliptic with respect to any class $\mathcal{L}$ that contains all the operators $L_a$.

- The Isaacs equation is the equality

\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \] where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.

The equation appears naturally in zero sum stochastic games with Levy processes.

The equation is uniformly elliptic with respect to any class $\mathcal{L}$ that contains all the operators $L_{ab}$.

Hello world

Note. In several articles^{[3]}^{[4]}^{[5]}, fully nonlinear integro-differential equations of the form $F(D^2 u, Du, u, x, I)=f(x)$ are analyzed, where $I$ is a linear integro-differential operator. This is a rigid structure for an equation. For example if an equation is purely integro-differential (it does not depend on $D^2u$, $Du$ or $u$) then it is forced to be linear: $I = [F(x,\cdot)^{-1}f(x)]$. On the other hand, the results in these papers apply to the more general definitions of fully nonlinear integro-differential equations as well. The reason for that restriction seems to be just to have an equation that is short to write down.

## References

- ↑ Caffarelli, Luis; Silvestre, Luis (2009), "Regularity theory for fully nonlinear integro-differential equations",
*Communications on Pure and Applied Mathematics***62**(5): 597–638, doi:10.1002/cpa.20274, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20274 - ↑ Caffarelli, Luis; Silvestre, Luis, "Regularity results for nonlocal equations by approximation",
*Archive for Rational Mechanics and Analysis*(Berlin, New York: Springer-Verlag): 1–30, ISSN 0003-9527 - ↑ Barles, Guy; Imbert, Cyril (2008), "Second-order elliptic integro-differential equations: viscosity solutions' theory revisited",
*Annales de l'Institut Henri Poincaré. Analyse Non Linéaire***25**(3): 567–585, doi:10.1016/j.anihpc.2007.02.007, ISSN 0294-1449, http://dx.doi.org/10.1016/j.anihpc.2007.02.007 - ↑ Barles, G.; Chasseigne, Emmanuel; Imbert, Cyril (2008), "On the Dirichlet problem for second-order elliptic integro-differential equations",
*Indiana University Mathematics Journal***57**(1): 213–246, doi:10.1512/iumj.2008.57.3315, ISSN 0022-2518, http://dx.doi.org/10.1512/iumj.2008.57.3315 - ↑
Barles, Guy; Chasseigne, Emmanuel; Imbert, Cyril (2011), "Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations",
*Journal of the European Mathematical Society (JEMS)***13**(1): 1–26, doi:10.4171/JEMS/242, ISSN 1435-9855, http://dx.doi.org/10.4171/JEMS/242