# Fully nonlinear integro-differential equations

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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. | ||

- | + | Equations of this type commonly satisfy a [[comparison principle]] and have some [[Regularity results for fully nonlinear integro-differential equations|regularity results]]. | |

- | + | 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. | |

- | Given a family of linear integro-differential operators $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$: | + | == Abstract definition <ref name="CS"/><ref name="CS2"/> == |

+ | |||

+ | A nonlocal operator is any rule that assigns a value to $Iu(x)$ whenever $u$ is a bounded function in $\mathbb R^n$ that is $C^2$ around the point $x$. The most basic requirement to call $I$ '''elliptic''' is that whenever $u-v$ achieves a global nonnegative maximum at the point $x$, then | ||

+ | \[ Iu(x) \leq Iv(x).\] | ||

+ | |||

+ | We now proceed to define the concept of uniform ellipticity. Given the richness of variations of nonlocal equations, we provide a flexible definition of uniform elliticity depending an arbitrary family of linear operators. | ||

+ | |||

+ | 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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\end{align*} | \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 $v$: | + | We define a nonlinear operator $I$ to be '''uniformly 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 $v$: |

\[M^-_\mathcal{L} [u-v](x)\leq Iu(x) - Iv(x) \leq M^+_\mathcal{L} [u-v] (x), \] | \[M^-_\mathcal{L} [u-v](x)\leq Iu(x) - Iv(x) \leq M^+_\mathcal{L} [u-v] (x), \] | ||

for any $x \in \Omega$. | for any $x \in \Omega$. | ||

- | A fully nonlinear elliptic equation with | + | A fully nonlinear elliptic equation is an equation of the form $Iu=0$ in $\Omega$, for some elliptic operator $I$. |

+ | |||

+ | {{note|text= If $\mathcal L$ consists of purely second order operators of the form $\mathrm{tr} \, A \cdot D^2 u$ with $\lambda I \leq A \leq \Lambda I$, then $M^+_{\mathcal L}$ and $M^-_{\mathcal L}$ denote the usual extremal Pucci operators. It is a ''folklore'' statement that then nonlinear operator $I$ elliptic respect to $\mathcal L$ in the sense described above must coincide with a fully nonlinear elliptic operator of the form $Iu = F(D^2u,x)$. However, this proof may have never been written anywhere. | ||

+ | }} | ||

+ | |||

+ | {{note|text=Any uniformly elliptic integro-differential equation with respect to some class $\mathcal L$ satisfies the following $\inf-\sup$ representation | ||

+ | \[ Iu(x) = \inf_{v \in C^2 \cap L^\infty} \sup_{L \in \mathcal L} \bigg\{ (Iv(x) - Lv(v)) + Lu(x) \bigg\}.\] | ||

+ | Therefore, $I$ coincides with some [[Isaacs equation]] for some family of linear operators $L_{ab}$ (the index $a$ ranges over all functions $v$ and the index $b$ over all linear operators $L$). | ||

+ | |||

+ | In the case that $I$ is not uniformly elliptic with respect to any given class, an $\inf-\sup$ formula was also obtained when the operator $I$ is Frechet differentiable <ref name="Guillen-Schwab"/>. | ||

+ | }} | ||

+ | |||

+ | == Another definition== | ||

+ | Another definition which gives a more concrete structure to the equation has been suggested <ref name="BI"/>. It is not clear if both definitions are equivalent, but both include the most important examples and are amenable of approximately the same methods. | ||

+ | |||

+ | Given a family of [[linear integro-differential operators]] $L_\alpha$ indexed by a parameter $\alpha$ which ranges in an arbitrary set $A$, a fully nonlinear elliptic equation is an equation of the form | ||

+ | \[ F(D^2 u, Du, u, x, \{L_\alpha\}_\alpha) = 0 \qquad \text{in } \Omega.\] | ||

+ | Where the function $F(X,p,z,x,\{i_\alpha\}_\alpha)$ is monotone increasing with respect to $X$ and $\{i_\alpha\}$ and monotone decreasing with respect to $z$. | ||

+ | |||

+ | Note that the family of linear operators $\{L_\alpha\}$ can range in an arbitrarily large set $A$ (it could even be uncountable). | ||

+ | |||

+ | {{note|text= In several articles <ref name="BI"/><ref name="BIC2"/><ref name="BIC"/>, fully nonlinear integro-differential equations of the form $F(D^2 u, Du, u, x, Lu)=f(x)$ are analyzed, where $L$ is one fixed [[linear integro-differential operator]]. This is a rigid structure for purely integro-differential equations because such equation (which would not depend on $D^2u$, $Du$ or $u$) would be forced to be linear: $Lu(x) = [F(x,\cdot)^{-1}f(x)]$. | ||

+ | |||

+ | 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 the restriction to one single integro-differential operator instead of a family $\{L_\alpha\}_\alpha$ seems to be taken only for simplicity. | ||

+ | }} | ||

+ | |||

+ | {{note|text= Whenever a general nonlocal operator corresponding to the first definition has an $\inf-\sup$ representation, it also fits the second definition. Any operator in the form of the second definition obviously fits the first definition. Therefore, for all practical purposes, the two definitions are practically equivalent. | ||

+ | }} | ||

== Examples == | == Examples == | ||

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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 $L_{ab}$. | ||

+ | |||

+ | == 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> | ||

+ | <ref name="Guillen-Schwab">{{Citation | last1=Guillen | first1= Nestor | last2=Schwab | first2= Russell W | title=Neumann Homogenization via Integro-Differential Operators | journal=arXiv preprint arXiv:1403.1980}}</ref> | ||

+ | }} |

## Latest revision as of 14:12, 26 June 2015

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.

Equations of this type commonly satisfy a comparison principle and have some regularity results.

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.

## Contents |

## Abstract definition ^{[1]}^{[2]}

A nonlocal operator is any rule that assigns a value to $Iu(x)$ whenever $u$ is a bounded function in $\mathbb R^n$ that is $C^2$ around the point $x$. The most basic requirement to call $I$ **elliptic** is that whenever $u-v$ achieves a global nonnegative maximum at the point $x$, then
\[ Iu(x) \leq Iv(x).\]

We now proceed to define the concept of uniform ellipticity. Given the richness of variations of nonlocal equations, we provide a flexible definition of uniform elliticity depending an arbitrary family of linear operators.

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 **uniformly 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 is an equation of the form $Iu=0$ in $\Omega$, for some elliptic operator $I$.

Note. If $\mathcal L$ consists of purely second order operators of the form $\mathrm{tr} \, A \cdot D^2 u$ with $\lambda I \leq A \leq \Lambda I$, then $M^+_{\mathcal L}$ and $M^-_{\mathcal L}$ denote the usual extremal Pucci operators. It is afolklorestatement that then nonlinear operator $I$ elliptic respect to $\mathcal L$ in the sense described above must coincide with a fully nonlinear elliptic operator of the form $Iu = F(D^2u,x)$. However, this proof may have never been written anywhere.

Note. Any uniformly elliptic integro-differential equation with respect to some class $\mathcal L$ satisfies the following $\inf-\sup$ representation \[ Iu(x) = \inf_{v \in C^2 \cap L^\infty} \sup_{L \in \mathcal L} \bigg\{ (Iv(x) - Lv(v)) + Lu(x) \bigg\}.\] Therefore, $I$ coincides with some Isaacs equation for some family of linear operators $L_{ab}$ (the index $a$ ranges over all functions $v$ and the index $b$ over all linear operators $L$). In the case that $I$ is not uniformly elliptic with respect to any given class, an $\inf-\sup$ formula was also obtained when the operator $I$ is Frechet differentiable^{[3]}.

## Another definition

Another definition which gives a more concrete structure to the equation has been suggested ^{[4]}. It is not clear if both definitions are equivalent, but both include the most important examples and are amenable of approximately the same methods.

Given a family of linear integro-differential operators $L_\alpha$ indexed by a parameter $\alpha$ which ranges in an arbitrary set $A$, a fully nonlinear elliptic equation is an equation of the form \[ F(D^2 u, Du, u, x, \{L_\alpha\}_\alpha) = 0 \qquad \text{in } \Omega.\] Where the function $F(X,p,z,x,\{i_\alpha\}_\alpha)$ is monotone increasing with respect to $X$ and $\{i_\alpha\}$ and monotone decreasing with respect to $z$.

Note that the family of linear operators $\{L_\alpha\}$ can range in an arbitrarily large set $A$ (it could even be uncountable).

Note. In several articles^{[4]}^{[5]}^{[6]}, fully nonlinear integro-differential equations of the form $F(D^2 u, Du, u, x, Lu)=f(x)$ are analyzed, where $L$ is one fixed linear integro-differential operator. This is a rigid structure for purely integro-differential equations because such equation (which would not depend on $D^2u$, $Du$ or $u$) would be forced to be linear: $Lu(x) = [F(x,\cdot)^{-1}f(x)]$. 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 the restriction to one single integro-differential operator instead of a family $\{L_\alpha\}_\alpha$ seems to be taken only for simplicity.

Note. Whenever a general nonlocal operator corresponding to the first definition has an $\inf-\sup$ representation, it also fits the second definition. Any operator in the form of the second definition obviously fits the first definition. Therefore, for all practical purposes, the two definitions are practically equivalent.

## 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 $L_{ab}$.

## 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 - ↑ Guillen, Nestor; Schwab, Russell W, "Neumann Homogenization via Integro-Differential Operators",
*arXiv preprint arXiv:1403.1980* - ↑
^{4.0}^{4.1}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