Fully nonlinear integro-differential equations and Differentiability estimates: Difference between pages

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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.
Given a [[fully nonlinear integro-differential equation]] $Iu=0$, [[uniformly elliptic]] with respect to certain [[class of operators]], sometimes an interior $C^{1,\alpha}$ estimate holds for some $\alpha>0$ (typically very small). Assume $I0=0$. The $C^{1,\alpha}$ estimate is a result like the following.


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.
'''Theorem'''. Let $u \in L^\infty(R^n) \cap C(\overline B_1)$ solve the equation \[Iu = 0 \ \ \text{in } B_1.\]
Then $u \in C^{1,\alpha}(B_{1/2})$ and the following estimate holds
\[ ||u||_{C^{1,\alpha}(B_{1/2})} \leq C ||u||_{L^\infty}. \]


== Definition <ref name="CS"/><ref name="CS2"/> ==
A theorem as above is known to hold under some assumptions on the [[nonlocal operator]] $I$. A list of valid assumptions is provided below.


Given a family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:
Note that the result is stated for general [[fully nonlinear integro-differential equations]], but the most important cases to apply it are the [[Isaacs equation]] and [[Bellman equation]].
\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 $v$:
== Idea of the proof ==
\[M^-_\mathcal{L} [u-v](x)\leq Iu(x) - Iv(x) \leq M^+_\mathcal{L} [u-v] (x), \]
The idea to prove a $C^{1,\alpha}$ estimate is to apply [[Holder estimates]] to the derivatives of the solutions $u$. The directional derivatives $u_e$ satisfy the two inequalities
for any $x \in \Omega$.
\[ M^+_{\mathcal L} u_e \geq 0 \text{ and } M^-_{\mathcal L} u_e \geq 0 \]
where $M^\pm_{\mathcal L}$ are the [[extremal operatos]] with respect to the corresponding class of operators $\mathcal L$. If the [[Holder estimates]] apply to this class of operators, one would expect that $u_e \in C^\alpha$ for any vector $e$, and therefore $u \in C^{1,\alpha}$.


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 $\mathcal{L}$.
There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.


== Examples ==
The only known solution to this difficulty is to add an extra smoothness assumption to the family of kernels that allows to integrate by parts the tails in the integral representation of each linear operator $L u_e$ in the class $\mathcal L$ and write its tail as an integral in terms of $u$. It is an interesting [[open problem]] whether a better solution exist.


The two main examples are the following.
==Classes of kernels for which the estimate holds ==


* The [[Bellman equation]] is the equality
=== Translation invariant, uniformly elliptic of order $s$, and some smoothness in the tails of the kernels ===
\[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \]
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 first situation in which the interior $C^{1,\alpha}$ estimate was proved for a nonlocal equation was if $I$ is translation invariant and [[uniformly elliptic]] with respect to the class of kernels satisfying the following hypothesis for some $\rho_0$ small enough<ref name="CS"/>.
\begin{align*}
\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{(standard unif. ellipticity of order $s$)}\\
\int_{\R^n \setminus B_{\rho_0}} \frac{|K(y)-K(y-h)|}{|h|} \mathrm d y &\leq C \qquad \text{every time $|h|<\frac {\rho_0} 2$} && \text{(kernel tails in $W^{1,1}$)}
\end{align*}
 
=== Variant if the kernel tails are $C^1$ ===
 
A small variation of the previous result is to assume the class of kernels satisfying the slightly stronger assumptions. A scale invariant class for which interior $C^{1,\alpha}$ regularity holds is <ref name="CS2"/>
\begin{align*}
\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{(standard unif. ellipticity of order $s$)}\\
\nabla K(y) &\leq \frac{\Lambda}{|y|^{n+s+1}} && \text{appropriate decay of the kernel in $C^1$.}
\end{align*}


The equation is uniformly elliptic with respect to any class $\mathcal{L}$ that contains all the operators $L_a$.
Then, any solution of $Iu=0$ in $B_r$ satisfies the estimate
\[ [u]_{C^{1,\alpha}(B_{r/2})} \leq C \left(\frac 1 {r^{1+\alpha}} ||u||_{L^\infty(B_r)} + \frac 1 {r^{1+\alpha-s}} \int_{\R^n \setminus B_r} \frac{|u(y)|}{|y|^{n+s}} \mathrm d y \right). \]
Other $C^{1,\alpha}$ estimates are obtained from this one using [[perturbation methods]] <ref name="CS2"/>.


* The [[Isaacs equation]] is the equality
=== A class of non-differentiable kernels ===
\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \]
A scale invariant class for which interior $C^{1,\alpha}$ regularity holds and the kernels can be very irregular is given by the following hypothesis<ref name="CS2"/>
where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.
\begin{align*}
K(y) &= (2-s) \frac{a_1(y) + a_2(y)}{|y|^{n+s}} \\
\lambda &\leq a_1(y) \leq \Lambda \\
|a_2| &\leq \eta \\
|\nabla a_1(y)| &\leq \frac{C_1}{|y|} \qquad \text{in } \R^n \setminus \{0\}
\end{align*}
for $s>1$ and $\eta$ small enough (depending on $\lambda$, $\Lambda$, $C_1$ and dimension)


The equation appears naturally in zero sum stochastic games with [[Levy processes]].
=== Isaacs equation with variable coefficients but close to constant ===
If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates <ref name="CS2"/>. The family of integro-differential operators has kernels which are the sum of a fixed term $a_0$ (the same for all kernels in the class) and a small term which can depend on $x$.
\[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)(a_0(y) + a_{\alpha \beta}(x,y))}{|y|^{n+s}} \mathrm d y =0\]
such that we have for $\eta$ small enough and any $\alpha$, $\beta$,
\begin{align*}
|a_{\alpha \beta}(x,y)| &< \eta \qquad \text{ for every } \alpha, \beta \\
\lambda &\leq a_0(y) \leq \Lambda \\
|\nabla a_0(y)| &\leq C |y|^{-1}
\end{align*}
(note that this $C^{1,\alpha}$ estimate is nontrivial in the linear case as well)


The equation is uniformly elliptic with respect to any class $\mathcal{L}$ that contains all the operators $L_{ab}$.
=== Isaacs equation with continuous coefficients ===
If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates <ref name="CS2"/>.
\[ \inf_\alpha \sup_\beta \int_{\R^n} \delta u(x,y) \frac{(2-s)a_{\alpha \beta}(x,y)}{|y|^{n+s}} \mathrm d y = 0\]
such that for every $\alpha$, $\beta$ we have
\begin{align*}
\lambda \leq a_{\alpha \beta}(x,y) &\leq \Lambda \\
\nabla_y a_{\alpha \beta}(x,y) &\leq C_1/((2-s)|y|)\\
|a_{\alpha \beta}(x_1,y) - a_{\alpha \beta}(x_2,y)| &\leq c(|x_1-x_2|) && \text{for some uniform modulus of continuity $c$}.
\end{align*}


<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 $F(D^2 u, Du, u, x, Lu)=f(x)$ are analyzed, where $L$ is a [[linear integro-differential operator]]. This is a rigid structure for an equation because if, for example, an equation is purely integro-differential (it does not depend on $D^2u$, $Du$ or $u$) then it is 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 that restriction seems to be just to have an equation that is short to write down.
</blockquote>
</div>


== References ==
== References ==
{{reflist|refs=
{{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="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">
<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 | year=2009 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | pages=1–30}}</ref>
{{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 12:45, 29 May 2011

Given a fully nonlinear integro-differential equation $Iu=0$, uniformly elliptic with respect to certain class of operators, sometimes an interior $C^{1,\alpha}$ estimate holds for some $\alpha>0$ (typically very small). Assume $I0=0$. The $C^{1,\alpha}$ estimate is a result like the following.

Theorem. Let $u \in L^\infty(R^n) \cap C(\overline B_1)$ solve the equation \[Iu = 0 \ \ \text{in } B_1.\] Then $u \in C^{1,\alpha}(B_{1/2})$ and the following estimate holds \[ ||u||_{C^{1,\alpha}(B_{1/2})} \leq C ||u||_{L^\infty}. \]

A theorem as above is known to hold under some assumptions on the nonlocal operator $I$. A list of valid assumptions is provided below.

Note that the result is stated for general fully nonlinear integro-differential equations, but the most important cases to apply it are the Isaacs equation and Bellman equation.

Idea of the proof

The idea to prove a $C^{1,\alpha}$ estimate is to apply Holder estimates to the derivatives of the solutions $u$. The directional derivatives $u_e$ satisfy the two inequalities \[ M^+_{\mathcal L} u_e \geq 0 \text{ and } M^-_{\mathcal L} u_e \geq 0 \] where $M^\pm_{\mathcal L}$ are the extremal operatos with respect to the corresponding class of operators $\mathcal L$. If the Holder estimates apply to this class of operators, one would expect that $u_e \in C^\alpha$ for any vector $e$, and therefore $u \in C^{1,\alpha}$.

There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.

The only known solution to this difficulty is to add an extra smoothness assumption to the family of kernels that allows to integrate by parts the tails in the integral representation of each linear operator $L u_e$ in the class $\mathcal L$ and write its tail as an integral in terms of $u$. It is an interesting open problem whether a better solution exist.

Classes of kernels for which the estimate holds

Translation invariant, uniformly elliptic of order $s$, and some smoothness in the tails of the kernels

The first situation in which the interior $C^{1,\alpha}$ estimate was proved for a nonlocal equation was if $I$ is translation invariant and uniformly elliptic with respect to the class of kernels satisfying the following hypothesis for some $\rho_0$ small enough[1]. \begin{align*} \frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{(standard unif. ellipticity of order $s$)}\\ \int_{\R^n \setminus B_{\rho_0}} \frac{|K(y)-K(y-h)|}{|h|} \mathrm d y &\leq C \qquad \text{every time $|h|<\frac {\rho_0} 2$} && \text{(kernel tails in $W^{1,1}$)} \end{align*}

Variant if the kernel tails are $C^1$

A small variation of the previous result is to assume the class of kernels satisfying the slightly stronger assumptions. A scale invariant class for which interior $C^{1,\alpha}$ regularity holds is [2] \begin{align*} \frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{(standard unif. ellipticity of order $s$)}\\ \nabla K(y) &\leq \frac{\Lambda}{|y|^{n+s+1}} && \text{appropriate decay of the kernel in $C^1$.} \end{align*}

Then, any solution of $Iu=0$ in $B_r$ satisfies the estimate \[ [u]_{C^{1,\alpha}(B_{r/2})} \leq C \left(\frac 1 {r^{1+\alpha}} ||u||_{L^\infty(B_r)} + \frac 1 {r^{1+\alpha-s}} \int_{\R^n \setminus B_r} \frac{|u(y)|}{|y|^{n+s}} \mathrm d y \right). \] Other $C^{1,\alpha}$ estimates are obtained from this one using perturbation methods [2].

A class of non-differentiable kernels

A scale invariant class for which interior $C^{1,\alpha}$ regularity holds and the kernels can be very irregular is given by the following hypothesis[2] \begin{align*} K(y) &= (2-s) \frac{a_1(y) + a_2(y)}{|y|^{n+s}} \\ \lambda &\leq a_1(y) \leq \Lambda \\ |a_2| &\leq \eta \\ |\nabla a_1(y)| &\leq \frac{C_1}{|y|} \qquad \text{in } \R^n \setminus \{0\} \end{align*} for $s>1$ and $\eta$ small enough (depending on $\lambda$, $\Lambda$, $C_1$ and dimension)

Isaacs equation with variable coefficients but close to constant

If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates [2]. The family of integro-differential operators has kernels which are the sum of a fixed term $a_0$ (the same for all kernels in the class) and a small term which can depend on $x$. \[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)(a_0(y) + a_{\alpha \beta}(x,y))}{|y|^{n+s}} \mathrm d y =0\] such that we have for $\eta$ small enough and any $\alpha$, $\beta$, \begin{align*} |a_{\alpha \beta}(x,y)| &< \eta \qquad \text{ for every } \alpha, \beta \\ \lambda &\leq a_0(y) \leq \Lambda \\ |\nabla a_0(y)| &\leq C |y|^{-1} \end{align*} (note that this $C^{1,\alpha}$ estimate is nontrivial in the linear case as well)

Isaacs equation with continuous coefficients

If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates [2]. \[ \inf_\alpha \sup_\beta \int_{\R^n} \delta u(x,y) \frac{(2-s)a_{\alpha \beta}(x,y)}{|y|^{n+s}} \mathrm d y = 0\] such that for every $\alpha$, $\beta$ we have \begin{align*} \lambda \leq a_{\alpha \beta}(x,y) &\leq \Lambda \\ \nabla_y a_{\alpha \beta}(x,y) &\leq C_1/((2-s)|y|)\\ |a_{\alpha \beta}(x_1,y) - a_{\alpha \beta}(x_2,y)| &\leq c(|x_1-x_2|) && \text{for some uniform modulus of continuity $c$}. \end{align*}


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. 2.0 2.1 2.2 2.3 2.4 Caffarelli, Luis; Silvestre, Luis (2009), "Regularity results for nonlocal equations by approximation", Archive for Rational Mechanics and Analysis (Berlin, New York: Springer-Verlag): 1–30, ISSN 0003-9527