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.

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]

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

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 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.
Another note. 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 has never been proved either.

Another definition

Another definition which gives a more concrete structure to the equation has been suggested [3]. 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 [3][4][5], 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.


Examples

The two main examples are the following.

\[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \] 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 $L_a$.

\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \] 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

  1. 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 
  2. 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 
  3. 3.0 3.1 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 
  4. 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 
  5. 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 
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