Extremal operators - Revision history

imported>Luis at 02:43, 16 April 2015

2015-04-16T02:43:23Z

← Older revision		Revision as of 21:43, 15 April 2015
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	M^-\, u &= \int_{\R^n} \left( \lambda \delta u(x,y)^+ - \Lambda \delta u(x,y)^- \right) \frac{(2-s)}{\|y\|^{n+s}} \mathrm d y		M^-\, u &= \int_{\R^n} \left( \lambda \delta u(x,y)^+ - \Lambda \delta u(x,y)^- \right) \frac{(2-s)}{\|y\|^{n+s}} \mathrm d y
	\end{align*}		\end{align*}
	where $\delta u(x,y) = (u(x+y) + u(x-y) - 2u(x))$. These two extremal operator are sometimes called "the ''monster'' Pucci operators".		where $\delta u(x,y) = (u(x+y) + u(x-y) - 2u(x))$. These two extremal operator are sometimes called "the ''monster'' Pucci operators" (even though there are other operators that are easily more "monstruous")

	== References ==		== References ==

imported>Nestor at 06:23, 2 June 2011

2011-06-02T06:23:59Z

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	<ref name="S">{{Citation \| last1=Silvestre \| first1=Luis \| title=Hölder estimates for solutions of integro-differential equations like the fractional Laplace \| url=http://dx.doi.org/10.1512/iumj.2006.55.2706 \| doi=10.1512/iumj.2006.55.2706 \| year=2006 \| journal=Indiana University Mathematics Journal \| issn=0022-2518 \| volume=55 \| issue=3 \| pages=1155–1174}}</ref>		<ref name="S">{{Citation \| last1=Silvestre \| first1=Luis \| title=Hölder estimates for solutions of integro-differential equations like the fractional Laplace \| url=http://dx.doi.org/10.1512/iumj.2006.55.2706 \| doi=10.1512/iumj.2006.55.2706 \| year=2006 \| journal=Indiana University Mathematics Journal \| issn=0022-2518 \| volume=55 \| issue=3 \| pages=1155–1174}}</ref>
	}}		}}

			[[Category:Fully nonlinear equations]]

imported>Luis at 18:06, 29 May 2011

2011-05-29T18:06:04Z

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			The extremal operator associated to some class of linear operators $\mathcal{L}$ represent the maximal and minimal value that $Lu(x)$ can take from all possible choices of $L \in \mathcal L$.

			The extremal operators are used to define [[uniformly elliptic\|uniform ellipticity]] for nonlocal operators. In fact, the extremal operators are also the maximal and minimal nonlinear uniformly elliptic operators with respect to $\mathcal L$ that vanish at zero.

	Given any family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:		Given any family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:
	\begin{align*}		\begin{align*}
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	If $\mathcal L$ consists of purely second order operators of the form $Lu = \mathrm{tr} \, A \cdot D^2 u$ with $\lambda I \leq A \leq \Lambda I$, then $M^+_{\mathcal L}$ and $M^-_{\mathcal L}$ denote the usual extremal Pucci operators, which have the formula		If $\mathcal L$ consists of purely second order operators of the form $Lu = \mathrm{tr} \, A \cdot D^2 u$ with $\lambda I \leq A \leq \Lambda I$, then $M^+_{\mathcal L}$ and $M^-_{\mathcal L}$ denote the usual extremal Pucci operators, which have the formula
	\begin{align*}		\begin{align*}
	P^+(D^2 u) &= \Lambda \mathrm{tr}(D^2u^+) - \lambda \mathrm{tr}(D^2u^-)\\		P^+(D^2 u) &= \Lambda \ \mathrm{tr}(D^2u^+) - \lambda \ \mathrm{tr}(D^2u^-)\\
	P^-(D^2 u) &= \lambda \mathrm{tr}(D^2u^+) - \Lambda \mathrm{tr}(D^2u^-)		P^-(D^2 u) &= \lambda \ \mathrm{tr}(D^2u^+) - \Lambda \ \mathrm{tr}(D^2u^-)
	\end{align*}		\end{align*}

imported>Luis at 17:59, 29 May 2011

2011-05-29T17:59:40Z

← Older revision		Revision as of 12:59, 29 May 2011
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	If $\mathcal L$ consists of purely second order operators of the form $Lu = \mathrm{tr} \, A \cdot D^2 u$ with $\lambda I \leq A \leq \Lambda I$, then $M^+_{\mathcal L}$ and $M^-_{\mathcal L}$ denote the usual extremal Pucci operators, which have the formula		If $\mathcal L$ consists of purely second order operators of the form $Lu = \mathrm{tr} \, A \cdot D^2 u$ with $\lambda I \leq A \leq \Lambda I$, then $M^+_{\mathcal L}$ and $M^-_{\mathcal L}$ denote the usual extremal Pucci operators, which have the formula
	\begin{align*}		\begin{align*}
	P^+(D^2 u) &= \Lambda \mathrm(D^2u^+) - \lambda \mathrm(D^2u^-)\\		P^+(D^2 u) &= \Lambda \mathrm{tr}(D^2u^+) - \lambda \mathrm{tr}(D^2u^-)\\
	P^-(D^2 u) &= \lambda \mathrm(D^2u^+) - \Lambda \mathrm(D^2u^-)		P^-(D^2 u) &= \lambda \mathrm{tr}(D^2u^+) - \Lambda \mathrm{tr}(D^2u^-)
	\end{align*}		\end{align*}

imported>Luis at 20:10, 27 May 2011

2011-05-27T20:10:18Z

← Older revision		Revision as of 15:10, 27 May 2011
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	Given a family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:		Given any family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:
	\begin{align*}		\begin{align*}
	M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} \, L u(x) \\		M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} \, L u(x) \\
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	\end{align*}		\end{align*}

	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, which have the formula		If $\mathcal L$ consists of purely second order operators of the form $Lu = \mathrm{tr} \, A \cdot D^2 u$ with $\lambda I \leq A \leq \Lambda I$, then $M^+_{\mathcal L}$ and $M^-_{\mathcal L}$ denote the usual extremal Pucci operators, which have the formula
	\begin{align*}		\begin{align*}
	P^+(D^2 u) &= \Lambda \mathrm(D^2u^+) - \lambda \mathrm(D^2u^-)\\		P^+(D^2 u) &= \Lambda \mathrm(D^2u^+) - \lambda \mathrm(D^2u^-)\\

imported>Luis: Created page with "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^+_\mathc..."

2011-05-27T20:09:06Z

Created page with "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^+_\mathc..."

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Given a family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:
\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*}

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, which have the formula
\begin{align*}
P^+(D^2 u) &= \Lambda \mathrm(D^2u^+) - \lambda \mathrm(D^2u^-)\\
P^-(D^2 u) &= \lambda \mathrm(D^2u^+) - \Lambda \mathrm(D^2u^-)
\end{align*}

If $\mathcal{L}$ consists of all [[linear integro-differential operator|symmetric purely integro-differential operators, uniformly elliptic of order $s$]], then the extremal operators have the formula<ref name="S"/>
\begin{align*}
M^+\, u &= \int_{\R^n} \left( \Lambda \delta u(x,y)^+ - \lambda \delta u(x,y)^- \right) \frac{(2-s)}{|y|^{n+s}} \mathrm d y \\
M^-\, u &= \int_{\R^n} \left( \lambda \delta u(x,y)^+ - \Lambda \delta u(x,y)^- \right) \frac{(2-s)}{|y|^{n+s}} \mathrm d y
\end{align*}
where $\delta u(x,y) = (u(x+y) + u(x-y) - 2u(x))$. These two extremal operator are sometimes called "the ''monster'' Pucci operators".

== References ==
{{reflist|refs=
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder estimates for solutions of integro-differential equations like the fractional Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 | doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 | pages=1155–1174}}</ref>
}}