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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 nonlocal porous medium equation of order $\sigma$ is the name currently given to two very different equations, namely |
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| 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.
| | \[ u_t = \nabla \cdot \left ( u \nabla \mathcal{K_\alpha} (u) \right )\] |
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| == Definition <ref name="CS"/><ref name="CS2"/> == | | \[\mbox{ where } \mathcal{K}_\alpha(u) := C_{n,\alpha}\; u * |x|^{-n+\alpha},\;\; \alpha+2=\sigma \] |
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| Given a family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:
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| \begin{align*}
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| M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} \, L u(x) \\
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| M^-_\mathcal{L} u(x) &= \inf_{L \in \mathcal{L}} \, L u(x)
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| \end{align*}
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| 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 $v$:
| | \[ u_t +(-\Delta)^{s}(u^m) = 0 \] |
| \[M^-_\mathcal{L} [u-v](x)\leq Iu(x) - Iv(x) \leq M^+_\mathcal{L} [u-v] (x), \]
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| for any $x \in \Omega$.
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| 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 $\mathcal{L}$.
| | These equations agree when $s=1$ and $m=2$, otherwise they are not only different superficially, they also exhibit extremely different behaviors. They are both fractional order [[Quasilinear equations]]. |
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| == Examples ==
| | The first of the two has the remarkable property (for nonlocal equations at least) that any initial data with compact support remains with compact support for all later times, the opposite is true of the second equation, for which [[instantaneous speed of propagation]] holds. |
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| The two main examples are the following.
| | This means that the first model presents us with a [[free boundary problem]]. For this model global existence and Hölder continuity of weak solutions have been recently obtained <ref name="CV1"/>, there is almost nothing known about the properties of its free boundary, making it a rich source of open questions. |
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| * The [[Bellman equation]] is the equality
| | For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years <ref name="PQRV" /ref>. |
| \[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \]
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| where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.
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| The equation appears naturally in problems of stochastic control with [[Levy processes]].
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| The equation is uniformly elliptic with respect to any class $\mathcal{L}$ that contains all the operators $L_a$.
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| * The [[Isaacs equation]] is the equality
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| \[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \]
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| where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.
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| The equation appears naturally in zero sum stochastic games with [[Levy processes]].
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| The equation is uniformly elliptic with respect to any class $\mathcal{L}$ that contains all the operators $L_{ab}$.
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| <div cellspacing="0" style="width:100%;background:#DDEEFF;color:inherit;;">
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| <blockquote>
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| '''Note'''. In several articles <ref name="BI"/><ref name="BIC2"/><ref name="BIC"/>, fully nonlinear integro-differential equations of the form $F(D^2 u, Du, u, x, I)=f(x)$ are analyzed, where $I$ 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: $I = [F(x,\cdot)^{-1}f(x)]$.
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| 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.
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| </blockquote>
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| </div>
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| == 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="CV1"> {{Citation | last1=Caffarelli | first1=Luis | last2=Vazquez | first2=Juan | title=Nonlinear Porous Medium Flow with Fractional Potential Pressure | url=http://dx.doi.org/10.1007/s00205-011-0420-4 | publisher=Springer Berlin / Heidelberg | year=2011 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | pages=1–29}} </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>
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| <ref name="BIC">
| | <ref name="PQRV">{{Citation | last1=Pablo | first1=Arturo de | last2=Quirós | first2=Fernando | last3=Rodríguez | first3=Ana | last4=Vazquez | first4=Juan Luis | title=A fractional porous medium equation | url=http://www.sciencedirect.com/science/article/pii/S0001870810003130 | doi=DOI: 10.1016/j.aim.2010.07.017 | year=2011 | journal=Advances in Mathematics | issn=0001-8708 | volume=226 | issue=2 | pages=1378–1409}} |
| {{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> |
| <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>
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| }} | | }} |
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| | [[Category:Quasilinear equations]] [[Category:Evolution equations]] [[Category:Free boundary problems]] |
The nonlocal porous medium equation of order $\sigma$ is the name currently given to two very different equations, namely
\[ u_t = \nabla \cdot \left ( u \nabla \mathcal{K_\alpha} (u) \right )\]
\[\mbox{ where } \mathcal{K}_\alpha(u) := C_{n,\alpha}\; u * |x|^{-n+\alpha},\;\; \alpha+2=\sigma \]
and
\[ u_t +(-\Delta)^{s}(u^m) = 0 \]
These equations agree when $s=1$ and $m=2$, otherwise they are not only different superficially, they also exhibit extremely different behaviors. They are both fractional order Quasilinear equations.
The first of the two has the remarkable property (for nonlocal equations at least) that any initial data with compact support remains with compact support for all later times, the opposite is true of the second equation, for which instantaneous speed of propagation holds.
This means that the first model presents us with a free boundary problem. For this model global existence and Hölder continuity of weak solutions have been recently obtained [1], there is almost nothing known about the properties of its free boundary, making it a rich source of open questions.
For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years Cite error: Invalid <ref> tag; invalid names, e.g. too many
[2]
}}