imported>Luis |
imported>Luis |
| Line 1: |
Line 1: |
| 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.
| | ''Viscosity solutions'' are a type of weak solutions for elliptic or parabolic fully nonlinear integro-differential equations that is particularly suitable for the [[comparison principle]] to hold and for stability under uniform limits. |
|
| |
|
| Equations of this type commonly satisfy a [[comparison principle]] and have some [[Regularity results for fully nonlinear integro-differential equations|regularity results]].
| | == Definition == |
|
| |
|
| 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. | | The following is a definition of viscosity solutions for [[fully nonlinear integro-differential equations]] of the form $Iu=0$, assuming that the functions involved are defined in the whole space. Other situations can be considered with straight forward modifications in the definition. The nonlocal operator $I$ is not necessarily assumed to be translation invariant. All we assume about $I$ is that it is ''black box'' operator such that |
| | # $Iv(x_0)$ is well defined every time $v \in C^2(x_0)$. |
| | # $Iv(x_0) \geq Iw(x_0)$ if $v(x_0)=w(x_0)$ and $v\geq w$ in $\R^n$. |
|
| |
|
| == Abstract definition <ref name="CS"/><ref name="CS2"/> ==
| | An function $u : \R^n \to \R$, which is upper semicontinuous in $\Omega$, is said to be a '''viscosity subsolution''' in $\Omega$ if the following statement holds |
| | <blockquote style="background:#DDDDDD;"> |
| | For any open $U \subset \Omega$, $x_0 \in U$ and $\varphi \in C^2(U)$ such that $\varphi(x_0)=u(x_0)$ and $\varphi \geq u$ in $U$, we construct the auxiliary function $v$ as |
| | \[ v(x) = \begin{cases} |
| | \varphi(x) & \text{if } x \in U \\ |
| | u(x) & \text{if } x \notin U |
| | \end{cases}\] |
| | Then, we have the inequality \[ Iv(x_0) \geq 0. \] |
| | </blockquote> |
|
| |
|
| 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 of ellipticity is that whenever $u-v$ achieves a global nonnegative maximum at the point $x$, then
| | An function $u : \R^n \to \R$, which is lower semicontinuous in $\Omega$, is said to be a '''viscosity supersolution''' in $\Omega$ if the following statement holds |
| \[ Iu(x) \leq Iv(x).\] | | <blockquote style="background:#DDDDDD;"> |
| | For any open $U \subset \Omega$, $x_0 \in U$ and $\varphi \in C^2(U)$ such that $\varphi(x_0)=u(x_0)$ and $\varphi \leq u$ in $U$, , we construct the auxiliary function $v$ as |
| | \[ v(x) = \begin{cases} |
| | \varphi(x) & \text{if } x \in U \\ |
| | u(x) & \text{if } x \notin U |
| | \end{cases}\] |
| | Then, we have the inequality \[ Iv(x_0) \leq 0. \] |
| | </blockquote> |
|
| |
|
| 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.
| | A continuous function in $\Omega$ which is at the same time a subsolution and a supersolution is said to be a '''viscosity solution'''. |
|
| |
|
| Given a family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:
| | Note that if the operator $I$ happens to be local, the construction of the function $v$ is unnecessary since $Iv(x_0) = I\varphi(x_0)$. Thus for local equations the definition is given evaluating $I\varphi(x_0)$ instead. |
| \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|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=It is conceivable that any uniformly elliptic integro-differential equation coincides with some [[Isaacs equation]] for some family of linear operators $L_{ab}$, at least in the translation invariant case. This was proved in the case that 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= In view of the nonlinear version of Courrege theorem given by Guillen and Schwab <ref name="Guillen-Schwab" />, both definitions of nonlocal operators coincide, at least when the operator are Frechet differentiable}}
| |
| | |
| == 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 ==
| |
| {{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>
| |
| }}
| |
Viscosity solutions are a type of weak solutions for elliptic or parabolic fully nonlinear integro-differential equations that is particularly suitable for the comparison principle to hold and for stability under uniform limits.
Definition
The following is a definition of viscosity solutions for fully nonlinear integro-differential equations of the form $Iu=0$, assuming that the functions involved are defined in the whole space. Other situations can be considered with straight forward modifications in the definition. The nonlocal operator $I$ is not necessarily assumed to be translation invariant. All we assume about $I$ is that it is black box operator such that
- $Iv(x_0)$ is well defined every time $v \in C^2(x_0)$.
- $Iv(x_0) \geq Iw(x_0)$ if $v(x_0)=w(x_0)$ and $v\geq w$ in $\R^n$.
An function $u : \R^n \to \R$, which is upper semicontinuous in $\Omega$, is said to be a viscosity subsolution in $\Omega$ if the following statement holds
For any open $U \subset \Omega$, $x_0 \in U$ and $\varphi \in C^2(U)$ such that $\varphi(x_0)=u(x_0)$ and $\varphi \geq u$ in $U$, we construct the auxiliary function $v$ as
\[ v(x) = \begin{cases}
\varphi(x) & \text{if } x \in U \\
u(x) & \text{if } x \notin U
\end{cases}\]
Then, we have the inequality \[ Iv(x_0) \geq 0. \]
An function $u : \R^n \to \R$, which is lower semicontinuous in $\Omega$, is said to be a viscosity supersolution in $\Omega$ if the following statement holds
For any open $U \subset \Omega$, $x_0 \in U$ and $\varphi \in C^2(U)$ such that $\varphi(x_0)=u(x_0)$ and $\varphi \leq u$ in $U$, , we construct the auxiliary function $v$ as
\[ v(x) = \begin{cases}
\varphi(x) & \text{if } x \in U \\
u(x) & \text{if } x \notin U
\end{cases}\]
Then, we have the inequality \[ Iv(x_0) \leq 0. \]
A continuous function in $\Omega$ which is at the same time a subsolution and a supersolution is said to be a viscosity solution.
Note that if the operator $I$ happens to be local, the construction of the function $v$ is unnecessary since $Iv(x_0) = I\varphi(x_0)$. Thus for local equations the definition is given evaluating $I\varphi(x_0)$ instead.