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

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Hölder continuity of the solutions can sometimes be proved only from ellipticity
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.
assumptions on the equation, without depending on smoothness of the
coefficients. This allows great flexibility in terms of applications of the
result. The corresponding result for elliptic equations of second order is the
[[Krylov-Safonov]] theorem in the non-divergence form, or the [[De Giorgi-Nash-Moser theorem]] in the divergence form.


The Hölder estimates are closely related to the [[Harnack inequality]]. In most cases, one can deduce the Hölder estimates from the Harnack inequality. However, there are simple example of integro-differential equations for which the Hölder estimates hold and the Harnack inequality does not <ref name="rang2013h" /> <ref name="bogdan2005harnack" />.
Equations of this type commonly satisfy a [[comparison principle]] and have some [[Regularity results for fully nonlinear integro-differential equations|regularity results]].


There are integro-differential versions of both [[De Giorgi-Nash-Moser theorem]]
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.
and [[Krylov-Safonov theorem]]. The former uses variational techniques and is
stated in terms of Dirichlet forms. The latter is based on comparison
principles.


A Hölder estimate says that a solution to an integro-differential equation with rough coefficients
== Abstract definition <ref name="CS"/><ref name="CS2"/> ==
$L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$
(small). It is very important when an estimate allows for a very rough dependence of
$L_x$ with respect to $x$, since the result then applies to the linearization of
(fully) nonlinear equations without any extra a priori estimate. On the other
hand, the linearization of a [[fully nonlinear integro-differential equation]] (for example the [[Isaacs equation]] or the [[Bellman equation]]) would inherit the initial assumptions regarding for the kernels with
respect to $y$. Therefore, smoothness (or even structural) assumptions for the
kernels with respect to $y$ can be made keeping such result applicable.


In the non variational setting the integro-differential operators $L_x$ are
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
assumed to belong to some family, but no continuity is assumed for its
\[ Iu(x) \leq Iv(x).\]
dependence with respect to $x$. Typically, $L_x u(x)$ has the form
$$ L_x u(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y))
K(x,y) \, dy$$
Since [[linear integro-differential operators]] allow for a great flexibility of
equations, there are several variations on the result: different assumptions on
the kernels, mixed local terms, evolution equations, etc. The linear equation
with rough coefficients is equivalent to the function $u$ satisfying two
inequalities for the [[extremal operators]] corresponding to the family of
operators $L$, which stresses the nonlinear character of the estimates.


As with other estimates in this field too, some Hölder estimates blow up as the
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.
order of the equation converges to two, and others pass to the limit. The
blow-up is a matter of the techniques used in the proof. Only estimates which
are robust are a true generalization of either the [[De Giorgi-Nash-Moser theorem]] or
[[Krylov-Safonov theorem]].  


== The general statement ==
Given a family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:
 
=== Elliptic form ===
The general form of the Hölder estimates for an elliptic problem say that if we have an equation which holds in a domain, and the solution is globally bounded, then the solution is Hölder continuous in the interior of the domain. Typically this is stated in the following form: if $u : \R^d \to \R$ solves
\[
L(u,x) = 0 \ \ \text{in } B_1,
\]
and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$,
\[ \|u\|_{C^\alpha(B_{1/2})} \leq C \|u\|_{L^\infty(\R^d)}.\]
 
There is no lack of generality in assuming that $L$ is a '''linear''' integro-differential operator, provided that there is no regularity assumption on its $x$ dependence.
 
For non variational problems, in order to adapt the situation to the [[viscosity solution]] framework, the equation may be replaced by two inequalities.
\begin{align*}
\begin{align*}
M^+u \geq 0 \ \ \text{in } B_1, \\
M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} \, L u(x) \\
M^-u \leq 0 \ \ \text{in } B_1.
M^-_\mathcal{L} u(x) &= \inf_{L \in \mathcal{L}} \, L u(x)
\end{align*}
\end{align*}
where $M^+$ and $M^-$ are [[extremal operators]] with respect to some class.
=== Parabolic form ===
The general form of the Hölder estimates for a parabolic problem is also an interior regularity statement for solutions of a parabolic equation. Typically this is stated in the following form: if $u : \R^d \times (-1,0] \to \R$ solves
\[
u_t - L(u,x) = 0 \ \ \text{in } (-1,0] \times B_1,
\]
and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$,
\[ \|u\|_{C^\alpha((-1/2,0] \times B_{1/2})} \leq C \|u\|_{L^\infty((-1,0] \times \R^d)}.\]
== Estimates which blow up as the order goes to two ==
=== Non variational case ===
The Hölder estimates were first obtained using probabilistic techniques <ref
name="BL"/> <ref name="BK"/> , and then using purely analytic methods <ref
name="S"/>. The assumptions are that for each $x$ the kernel $K(x,.)$ belongs to
a family satisfying certain set of assumptions. No regularity of any kind is
assumed for $K$ with respect to $x$. The assumption for the family of operators
are
# '''Scaling''': If $L$ belongs to the family, then so does its scaled version
$L_r u(x) = C_{r,L} L [u(x/r)] (x)$ for any $r<1$ and some $C_{r,L}<1$ which
could depend on $L$, but $C_{r,L} \to 0$ as $r \to 0$ uniformly in $L$.
# '''Nondegeneracy''': If $K$ is the kernel associated to $L$,
$\frac{\int_{\R^n} \min(y^2,y^\alpha) K(y) \, dy} {\inf_{B_1} K} \leq C_1$ for
some $C_1$ and $\alpha>0$ independent of $K$.


The right hand side $f$ is assumed to belong to $L^\infty$.
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 particular case in which this result applies is the uniformly elliptic case.
A fully nonlinear elliptic equation is an equation of the form $Iu=0$ in $\Omega$, for some elliptic operator $I$.
$$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$
where $s$ is bounded below and above: $0 < s_0 \leq s(x) \leq s_1 < 2$, but no
continuity of $s$ respect to $x$ is required.
The kernel $K$ is assumed to be symmetric with respect to $y$: $K(x,y)=K(x,-y)$.
However this assumption can be overcome in the following two situations.
* For $s<1$, the symmetry assumption can be removed if the equation does not
contain the drift correction term: $\int_{\R^n} (u(x+y) - u(x)) K(x,y) \, dy =
f(x)$ in $B_1$.
* For $s>1$, the symmetry assumption can be removed if the drift correction term
is global: $\int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) K(x,y) \, dy =
f(x)$ in $B_1$.


The reason for the symmetry assumption, or the modification of the drift
{{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.
correction term, is that in the original formulation the term $y \cdot \nabla
}}
u(x) \, \chi_{B_1}(y)$ is not scale invariant.


=== Variational case ===
{{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"/>.
}}


A typical example of a symmetric nonlocal [[Dirichlet form]] is a bilinear form
== Another definition==
$E(u,v)$ satisfying
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.
$$ E(u,v) = \iint_{\R^n \times \R^n} (v(y)-u(x))(v(y)-v(x)) K(x,y) \, dx
\, dy $$
on the closure of all $L^2$-functions with respect to $J(u)=E(u,u)$. Note
that $K$ can be assumed to be symmetric because the skew-symmetric part
of $K$ would be ignored by the bilinear form.  


Minimizers of the corresponding quadratic forms satisfy the nonlocal Euler
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
equation
\[ F(D^2 u, Du, u, x, \{L_\alpha\}_\alpha) = 0 \qquad \text{in } \Omega.\]
$$ \lim_{\varepsilon \to 0} \int_{|x-y|>\varepsilon} (u(y) - u(x) ) K(x,y) \, dy = 0,$$
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 $z$.
which should be understood in the sense of distributions.


It is known that the gradient flow of a Dirichlet form (parabolic version of the
Note that the family of linear operators $\{L_\alpha\}$ can range in an arbitrarily large set $A$ (it could even be uncountable).
result) becomes instantaneously Hölder continuous <ref name="CCV"/>. The method
of the proof builds an integro-differential version of the parabolic De Giorgi
technique that was developed for the study of critical [[surface
quasi-geostrophic equation]].  


At some point in the original proof of De Giorgi, it is used that the
{{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: $Lu(x) = [F(x,\cdot)^{-1}f(x)]$.
characteristic functions of a set of positive measure do not belong to $H^1$.
Moreover, a quantitative estimate is required about the measure of
''intermediate'' level sets for $H^1$ functions. In the integro-differential
context, the required statement to carry out the proof would be the same with
the $H^{s/2}$ norm. This required statement is not true for $s$ small, and would
even require a non trivial proof for $s$ close to $2$. The difficulty is
bypassed though an argument that takes advantage of the nonlocal character of
the equation, and hence the estimate blows up as the order approaches two.


== Estimates which pass to the second order limit ==
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.
}}


=== Non variational case ===
== Examples ==


An integro-differential generalization of [[Krylov-Safonov]] theorem is
The two main examples are the following.
available both in the elliptic <ref name="CS"/> and parabolic <ref name="lara2011regularity"/> setting. The assumption on the kernels are
# '''Symmetry''': $K(x,y) = K(x,-y)$.
# '''Uniform ellipticity''': $\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq
\frac{(2-s) \Lambda}{|y|^{n+s}}$ for some fixed value $s \in (0,2)$.


The right hand side $f$ is assumed to be in $L^\infty$. The constants in the
* The [[Bellman equation]] is the equality
Hölder estimate do not blow up as $s \to 2$.
\[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \]
where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.


=== Variational case ===
The equation appears naturally in problems of stochastic control with [[Levy processes]].


In the stationary case, it is known that minimizers of Dirichlet forms are
The equation is uniformly elliptic with respect to any class $\mathcal{L}$ that contains all the operators $L_a$.
Hölder continuous by adapting Moser's proof of [[De Giorgi-Nash-Moser theorem]] to the
nonlocal setting <ref name="K"/>.


== Other variants ==
* 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 $b \in \mathcal B$.


* There are Holder estimates for equations in divergence form that are non local in time <ref name="zacher2013" />
The equation appears naturally in zero sum stochastic games with [[Levy processes]].
* If we allow for continuous dependence on the coefficients with respect to $x$, there are Hölder estimates for a very general class of integral equations <ref name="barles2011" />.


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


== References ==
== References ==
{{reflist|refs=
{{reflist|refs=
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder
<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>
estimates for solutions of integro-differential equations like the fractional
<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>
Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 |
<ref name="BIC">
doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University
{{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>
Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 |
<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>
pages=1155–1174}}</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="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre |
<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>
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="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan |
first2=Chi Hin | last3=Vasseur | first3=Alexis | title= |
doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the
American Mathematical Society]] | issn=0894-0347 | issue=24 |
pages=849–869}}</ref>
<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori
estimates for integro-differential operators with measurable kernels |
url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6
| year=2009 | journal=Calculus of Variations and Partial Differential Equations
| issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
<ref name="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann |
first2=Moritz | title=Hölder continuity of harmonic functions with respect to
operators of variable order | url=http://dx.doi.org/10.1080/03605300500257677 |
doi=10.1080/03605300500257677 | year=2005 | journal=Communications in Partial
Differential Equations | issn=0360-5302 | volume=30 | issue=7 |
pages=1249–1259}}</ref>
<ref name="BL">{{Citation | last1=Bass | first1=Richard F. | last2=Levin |
first2=David A. | title=Harnack inequalities for jump processes |
url=http://dx.doi.org/10.1023/A:1016378210944 | doi=10.1023/A:1016378210944 |
year=2002 | journal=Potential Analysis. An International Journal Devoted to the
Interactions between Potential Theory, Probability Theory, Geometry and
Functional Analysis | issn=0926-2601 | volume=17 | issue=4 |
pages=375–388}}</ref>
<ref name="lara2011regularity">{{Citation | last1=Lara | first1= Héctor Chang | last2=Dávila | first2= Gonzalo | title=Regularity for solutions of non local parabolic equations | journal=Calculus of Variations and Partial Differential Equations | year=2011 | pages=1--34}}</ref>
<ref name="zacher2013">{{Citation | last1=Zacher | first1= Rico | title=A De Giorgi--Nash type theorem for time fractional diffusion equations | url=http://dx.doi.org/10.1007/s00208-012-0834-9 | journal=Math. Ann. | issn=0025-5831 | year=2013 | volume=356 | pages=99--146 | doi=10.1007/s00208-012-0834-9}}</ref>
<ref name="barles2011">{{Citation | last1=Barles | first1= Guy | last2=Chasseigne | first2= Emmanuel | last3=Imbert | first3= Cyril | title=H\"older continuity of solutions of second-order non-linear elliptic integro-differential equations | url=http://dx.doi.org/10.4171/JEMS/242 | journal=J. Eur. Math. Soc. (JEMS) | issn=1435-9855 | year=2011 | volume=13 | pages=1--26 | doi=10.4171/JEMS/242}}</ref>
<ref name="rang2013h">{{Citation | last1=Rang | first1= Marcus | last2=Kassmann | first2= Moritz | last3=Schwab | first3= Russell W | title=H$\backslash$" older Regularity For Integro-Differential Equations With Nonlinear Directional Dependence | journal=arXiv preprint arXiv:1306.0082}}</ref>
<ref name="bogdan2005harnack">{{Citation | last1=Bogdan | first1= Krzysztof | last2=Sztonyk | first2= Pawe\l | title=Harnack’s inequality for stable Lévy processes | journal=Potential Analysis | year=2005 | volume=22 | pages=133--150}}</ref>
<ref name="schwab2014regularity">{{Citation | last1=Schwab | first1= Russell W | last2=Silvestre | first2= Luis | title=Regularity for parabolic integro-differential equations with very irregular kernels | journal=arXiv preprint arXiv:1412.3790}}</ref>
<ref name="kassmann2013intrinsic">{{Citation | last1=Kassmann | first1= Moritz | last2=Mimica | first2= Ante | title=Intrinsic scaling properties for nonlocal operators | journal=arXiv preprint arXiv:1310.5371}}</ref>
}}
}}

Revision as of 16:42, 24 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.

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 of ellipticity 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 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. 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 [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.

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. ↑ Guillen, Nestor; Schwab, Russell W, "Neumann Homogenization via Integro-Differential Operators", arXiv preprint arXiv:1403.1980 
  4. ↑ 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 
  5. ↑ 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 
  6. ↑ 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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