Dirichlet form and Fully nonlinear integro-differential equations: 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.
\newcommand{\dd}{\mathrm{d}}
\newcommand{\R}{\mathbb{R}}
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A Dirichlet form in $\mathbb{R}^n$ is a bilinear function
A general definition of ellipticity can be given that does not require a specific form of the equation. However, the main two applications are the two above.


\begin{equation*}
== Definition ==
\mathcal{E}: D\times D \to \mathbb{R}
\end{equation*}


with the following properties
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*}


1) The domain $D$ is a dense subset of $\mathbb{R}^n$
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 $v$:
\[M^-_\mathcal{L} u(x)\leq Iu(x) - Iv(x) \leq M^+_\mathcal{L} u(x), \]
for any $x \in \Omega$.


2) $\mathcal{E}$ is symmetric, that is $\mathcal{E}(u,v)=\mathcal{E}(v,u)$ for any $u,v \in D$.
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 $\mathcal{L}$.
 
3) $\mathcal{E}(u,u) \geq 0$ for any $u \in D$.
 
4) The set $D$ equipped with the inner product defined by $(u,v)_{\mathcal{E}} := (u,v)_{L^2(\mathbb{R}^n)} + \mathcal{E}(u,v)$ is a real Hilbert space.
 
5) For any $u \in D$ we have that $u_* = (u\vee 0) \wedge 1 \in D$ and $\mathcal{E}(u_*,u_*)\leq \mathcal{E}(u,u)$
 
 
An example of a Dirichlet form is given by  any integral of the form
\mathcal{E}(u,v) = \[ \iint_{\R^n \times \R^n} (u(y)-u(x))(v(y)-v(x))k(x,y)\, \dd x \dd y, \]
where $K$ is some non-negative symmetric kernel.
 
If the kernel $K$ satisfies the bound $K(x,y) \leq \Lambda |x-y|^{-n-s}$, then the quadratic form is bounded in $\dot H^{s/2}$ . If moreover, $\lambda |x-y|^{-n-s} \leq K(x,y)$, then the form is comparable to the norm in $\dot H^{s/2}$ squared and in that case the set $D \subset L^2(\mathbb{R}^n)$ defined above is given by  $H^{s/2}(\mathbb{R}^n)$
 
Dirichlet forms are natural generalizations of the Dirichlet integrals
\[ \int a_{ij}(x) \partial_i u \partial_j u \dd x, \]
where $a_{ij}$ is elliptic.
 
The Euler-Lagrange equation of a Dirichlet form is a fractional order version of elliptic equations in divergence form. They are studied using variational methods and they are expected to satisfy similar properties <ref name="BBCK"/><ref name="K"/><ref name="CCV"/>.
 
== References ==
(There should be a lot more references here)
{{reflist|refs=
<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="BBCK">{{Citation | last1=Barlow | first1=Martin T. | last2=Bass | first2=Richard F. | last3=Chen | first3=Zhen-Qing | last4=Kassmann | first4=Moritz | title=Non-local Dirichlet forms and symmetric jump processes | url=http://dx.doi.org/10.1090/S0002-9947-08-04544-3 | doi=10.1090/S0002-9947-08-04544-3 | year=2009 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=361 | issue=4 | pages=1963–1999}}</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>
}}
 
 
{{stub}}

Revision as of 20:46, 26 May 2011

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.

A general definition of ellipticity can be given that does not require a specific form of the equation. However, the main two applications are the two above.

Definition

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 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 $v$: \[M^-_\mathcal{L} u(x)\leq Iu(x) - Iv(x) \leq M^+_\mathcal{L} u(x), \] 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 $\mathcal{L}$.