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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.
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| Equations of this type commonly satisfy a [[comparison principle]] and have some [[Regularity results for fully nonlinear integro-differential equations|regularity results]].
| | The Hele-Shaw model describes an incompressible flow lying between two nearby horizontal plates<ref name="MR0097227"/>. The following equations are given for a non-negative pressure $u$, supported in in a time dependent domain, |
| | | \begin{align*} |
| 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. | | \Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\ |
| | \frac{\partial_t u}{|Du|} &= |Du| \text{ on } \partial \{u>0\}\cap \Omega |
| | \end{align*} |
| | The first equation expresses the incompressibility of the fluid. The second equation, also known as the free boundary condition, says that the normal speed of the inter-phase (left-hand side) is the velocity of the fluid (right-hand side). |
| | Particular solutions are given for instance by the planar profiles |
| | \[ |
| | P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0 |
| | \] |
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| == Abstract definition <ref name="CS"/><ref name="CS2"/> ==
| | Non-local aspects of the equation can be appreciated by noticing that a given deformation of the domain $\Omega^+$ affects all the values of $|Du|$, at least in the corresponding connected component. To be more precise let us also formally show that the linearization about a planar profile leads to a fractional heat equation of order one. |
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| 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
| | Let $u = P + \varepsilon v$. Then $u$ and $P$ harmonic in their positivity sets imply $v$ harmonic in the intersection, notice that as $\varepsilon\searrow0$, $v$ becomes harmonic in $\{x_n>A(t)\}$. On the other hand, the free boundary relation over $\{x_n=A(t)\}$ gives |
| \[ Iu(x) \leq Iv(x).\] | | \[ |
| | | \frac{a^2+\varepsilon \partial_t v}{|ae_n+\varepsilon Dv|} = |ae_n+\varepsilon Dv| \qquad\Rightarrow\qquad \partial_t v = 2a\partial_n v+\varepsilon |Dv|^2 |
| We now proceed to define the concept of uniform ellipticity. Given the richness of variations of nonlocal equations, we provide a flexible definition of uniform elliticity depending an arbitrary family of linear operators.
| | \] |
| | | By taking the reparametrization $w(x,t) = v(x+Ae_n,t)$ and letting $\varepsilon\searrow0$ we get that $w$ satisfies |
| 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*} | | \begin{align*} |
| M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} \, L u(x) \\
| | \Delta w &= 0 \text{ in } \{x_n>0\}\\ |
| M^-_\mathcal{L} u(x) &= \inf_{L \in \mathcal{L}} \, L u(x)
| | \partial_t w &= a\partial_n w \text{ on } \{x_n=0\} |
| \end{align*} | | \end{align*} |
| | Or in terms of the half-laplacian in $\mathbb R^{n-1} = \{x_n=0\}$, |
| | \[ |
| | \partial_t w = a\Delta_{\mathbb R^{n-1}}^{1/2} w |
| | \] |
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| 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 $v$:
| | == References == |
| \[M^-_\mathcal{L} [u-v](x)\leq Iu(x) - Iv(x) \leq M^+_\mathcal{L} [u-v] (x), \]
| | {{reflist|refs= |
| for any $x \in \Omega$.
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| A fully nonlinear elliptic equation is an equation of the form $Iu=0$ in $\Omega$, for some elliptic operator $I$. | | <ref name="MR0097227">{{Citation | last1=Saffman | first1= P. G. | last2=Taylor | first2= Geoffrey | title=The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid | journal=Proc. Roy. Soc. London. Ser. A | issn=0962-8444 | year=1958 | volume=245 | pages=312--329. (2 plates)}}</ref> |
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| {{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.
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| {{note|text=It is conceivable that any uniformly elliptic integro-differential equation coincides with some [[Isaacs equation]] for some family of linear operators $L_{ab}$. This was proved in the case that the operator $I$ is Frechet differentiable <ref name="Guillen-Schwab"/>.
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| == Another definition==
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| 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.
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| 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
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| \[ F(D^2 u, Du, u, x, \{L_\alpha\}_\alpha) = 0 \qquad \text{in } \Omega.\]
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| 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 $z$.
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| Note that the family of linear operators $\{L_\alpha\}$ can range in an arbitrarily large set $A$ (it could even be uncountable).
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| {{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: $Lu(x) = [F(x,\cdot)^{-1}f(x)]$.
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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 the restriction to one single integro-differential operator instead of a family $\{L_\alpha\}_\alpha$ seems to be taken only for simplicity.
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| {{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 operators are Frechet differentiable}}
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| == Examples ==
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| The two main examples are the following.
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| * The [[Bellman equation]] is the equality
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| \[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \]
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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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| == References ==
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| {{reflist|refs=
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| <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>
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| <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">
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| {{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>
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| <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>
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| <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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| <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>
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| }} | | }} |
This article is a stub. You can help this nonlocal wiki by expanding it.
The Hele-Shaw model describes an incompressible flow lying between two nearby horizontal plates[1]. The following equations are given for a non-negative pressure $u$, supported in in a time dependent domain,
\begin{align*}
\Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\
\frac{\partial_t u}{|Du|} &= |Du| \text{ on } \partial \{u>0\}\cap \Omega
\end{align*}
The first equation expresses the incompressibility of the fluid. The second equation, also known as the free boundary condition, says that the normal speed of the inter-phase (left-hand side) is the velocity of the fluid (right-hand side).
Particular solutions are given for instance by the planar profiles
\[
P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0
\]
Non-local aspects of the equation can be appreciated by noticing that a given deformation of the domain $\Omega^+$ affects all the values of $|Du|$, at least in the corresponding connected component. To be more precise let us also formally show that the linearization about a planar profile leads to a fractional heat equation of order one.
Let $u = P + \varepsilon v$. Then $u$ and $P$ harmonic in their positivity sets imply $v$ harmonic in the intersection, notice that as $\varepsilon\searrow0$, $v$ becomes harmonic in $\{x_n>A(t)\}$. On the other hand, the free boundary relation over $\{x_n=A(t)\}$ gives
\[
\frac{a^2+\varepsilon \partial_t v}{|ae_n+\varepsilon Dv|} = |ae_n+\varepsilon Dv| \qquad\Rightarrow\qquad \partial_t v = 2a\partial_n v+\varepsilon |Dv|^2
\]
By taking the reparametrization $w(x,t) = v(x+Ae_n,t)$ and letting $\varepsilon\searrow0$ we get that $w$ satisfies
\begin{align*}
\Delta w &= 0 \text{ in } \{x_n>0\}\\
\partial_t w &= a\partial_n w \text{ on } \{x_n=0\}
\end{align*}
Or in terms of the half-laplacian in $\mathbb R^{n-1} = \{x_n=0\}$,
\[
\partial_t w = a\Delta_{\mathbb R^{n-1}}^{1/2} w
\]
References
- ↑ Saffman, P. G.; Taylor, Geoffrey (1958), "The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid", Proc. Roy. Soc. London. Ser. A 245: 312--329. (2 plates), ISSN 0962-8444