Extension technique - Revision history

imported>Pablo: /* Fractional powers of more general operators */

2015-08-05T19:19:08Z

Fractional powers of more general operators

← Older revision		Revision as of 14:19, 5 August 2015
Line 35:		Line 35:
	with boundary conditions along $\partial \Omega \times [0,\infty)$ equal to the boundary conditions for $L$. Then		with boundary conditions along $\partial \Omega \times [0,\infty)$ equal to the boundary conditions for $L$. Then
	\begin{equation}		\begin{equation}
	L^su(x) = C_{s} \lim_{y\rightarrow 0} y^{1-2s} U_y(x,y).		L^su(x) = -C_{s} \lim_{y\rightarrow 0} ~y^{1-2s} U_y(x,y).
	\end{equation}		\end{equation}
	Here $C_s$ is a constant depending only on $s$. Moreover, the solution can be explicitly written as		Here $C_s$ is a constant depending only on $s$. Moreover, the solution can be explicitly written as

imported>Pablo: /* Fractional powers of more general operators */

2015-08-05T19:18:03Z

Fractional powers of more general operators

← Older revision		Revision as of 14:18, 5 August 2015
Line 30:		Line 30:
	In a similar way, extension problem for fractional powers $L^s$ of a general self-adjoint nonnegative linear differential operators $L$ in a domain $\Omega \subset \mathbb{R}^n$ (or more generally, a Hilbert space) can be constructed. In this case, the extension into a "cylinder" $\Omega \times [0, \infty)$ is considered. Let $U$ be a solution of		In a similar way, extension problem for fractional powers $L^s$ of a general self-adjoint nonnegative linear differential operators $L$ in a domain $\Omega \subset \mathbb{R}^n$ (or more generally, a Hilbert space) can be constructed. In this case, the extension into a "cylinder" $\Omega \times [0, \infty)$ is considered. Let $U$ be a solution of
	$$\begin{cases}		$$\begin{cases}
	~~\partial_y~~ (y^{1-2s} ~~\partial_y U~~(x, y)~~) - L_x U~~(x, y) = 0,&\hbox{in}~\Omega\times(0,\infty),\\		- L_x U(x,y)+\frac{1-2s}{y}U_y(x,y)+U_{yy}(x,y) = 0,&\hbox{in}~\Omega\times(0,\infty),\\
	U(x,0)=u(x),&\hbox{on}~\Omega,		U(x,0)=u(x),&\hbox{on}~\Omega,
	\end{cases}$$		\end{cases}$$
Line 41:		Line 41:
	For details see <ref name="Stinga"/> and <ref name="Stinga-Torrea"/>.		For details see <ref name="Stinga"/> and <ref name="Stinga-Torrea"/>.

	If $L$ has a purely discrete spectrum on $\Omega$, the operator $L^s$ has the same eigenfunctions as $L$, and its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $L$. See for example <ref name="Caffarelli-Stinga"/> where this technique is applied to obtain global regularity estimates for fractional powers of divergence form elliptic operators with Dirichlet or Neumann boundary conditions.		See for example <ref name="Caffarelli-Stinga"/> where this technique is applied to obtain global regularity estimates for fractional powers of divergence form elliptic operators with Dirichlet or Neumann boundary conditions.

	For example, $L$ can be the Dirichlet Laplacian in $\Omega$.<ref name="CT"/> Note that $L^s$ is not the same as the fractional Laplacian, except when $\Omega = \mathbb{R}^n$.		If $L$ has a purely discrete spectrum on $\Omega$, the operator $L^s$ has the same eigenfunctions as $L$, and its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $L$. For example, $L$ can be the Dirichlet Laplacian in $\Omega$.<ref name="CT"/> Note that $L^s$ is not the same as the fractional Laplacian, except when $\Omega = \mathbb{R}^n$.

	==More general non-local operators==		==More general non-local operators==

imported>Pablo at 19:16, 5 August 2015

2015-08-05T19:16:48Z

← Older revision		Revision as of 14:16, 5 August 2015
Line 39:		Line 39:
	Here $C_s$ is a constant depending only on $s$. Moreover, the solution can be explicitly written as		Here $C_s$ is a constant depending only on $s$. Moreover, the solution can be explicitly written as
	$$U(x,y)=\frac{y^{2s}}{4^s\Gamma(s)}\int_0^\infty e^{-tL}u(x)e^{-y^2/(4t)}\,\frac{dt}{t^{1+s}}.$$		$$U(x,y)=\frac{y^{2s}}{4^s\Gamma(s)}\int_0^\infty e^{-tL}u(x)e^{-y^2/(4t)}\,\frac{dt}{t^{1+s}}.$$
	For details see		For details see <ref name="Stinga"/> and <ref name="Stinga-Torrea"/>.
	~~If $L$ has a purely discrete spectrum on $\Omega$, the operator $(-L)^s$ has the same eigenfunctions as $L$,~~ and ~~its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $(~~-~~L)$~~.

	For example, $L$ can be the Dirichlet Laplacian in $\Omega$.<ref name="CT"/> Note that $(-L)^s$ is not the same as the fractional Laplacian, except when $\Omega = \mathbb{R}^n$.		If $L$ has a purely discrete spectrum on $\Omega$, the operator $L^s$ has the same eigenfunctions as $L$, and its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $L$. See for example <ref name="Caffarelli-Stinga"/> where this technique is applied to obtain global regularity estimates for fractional powers of divergence form elliptic operators with Dirichlet or Neumann boundary conditions.

			For example, $L$ can be the Dirichlet Laplacian in $\Omega$.<ref name="CT"/> Note that $L^s$ is not the same as the fractional Laplacian, except when $\Omega = \mathbb{R}^n$.

	==More general non-local operators==		==More general non-local operators==
Line 93:		Line 94:
	<ref name="Stinga">{{Citation \| last1=Stinga \| first1=P. R.\| title=Fractional powers of second order partial differential operators: extension problem and regularity theory \| url=https://www.ma.utexas.edu/users/stinga/Stinga%20-%20PhD%20Thesis.pdf \| year=2010 \| journal=[[PhD. Thesis - Universidad Aut\'onoma de Madrid]]}}		<ref name="Stinga">{{Citation \| last1=Stinga \| first1=P. R.\| title=Fractional powers of second order partial differential operators: extension problem and regularity theory \| url=https://www.ma.utexas.edu/users/stinga/Stinga%20-%20PhD%20Thesis.pdf \| year=2010 \| journal=[[PhD. Thesis - Universidad Aut\'onoma de Madrid]]}}
	</ref><ref name="Stinga-Torrea">{{Citation \| last1=Stinga \| first1=P. R.\| last2=Torrea \| first2=J. L. \| title=Extension problem and Harnack's inequality for some fractional operators \| url=http://www.tandfonline.com/doi/abs/10.1080/03605301003735680#.VcJb9UWmR1Q \| year=2010 \| journal=[[Comm. Partial Differential Equations]] \| volume=35 \| issue=11 \| pages=2092-2122}}</ref>		</ref><ref name="Stinga-Torrea">{{Citation \| last1=Stinga \| first1=P. R.\| last2=Torrea \| first2=J. L. \| title=Extension problem and Harnack's inequality for some fractional operators \| url=http://www.tandfonline.com/doi/abs/10.1080/03605301003735680#.VcJb9UWmR1Q \| year=2010 \| journal=[[Comm. Partial Differential Equations]] \| volume=35 \| issue=11 \| pages=2092-2122}}</ref>
			</ref><ref name="Caffarelli-Stinga">{{Citation \| last1=Caffarelli \| first1=L. A. \| last2=Stinga \| first2=P. R. \| title=Fractional elliptic equations, Caccioppoli estimates and regularity \| url=http://www.sciencedirect.com/science/article/pii/S0294144915000153 \| year=2015 \| journal=[[Ann. Inst. H. Poincar\'e (C) Anal. Non Lin\'eaire]] \| pages=to appear}}</ref>
	}}		}}

	{{stub}}		{{stub}}

imported>Pablo at 19:11, 5 August 2015

2015-08-05T19:11:43Z

← Older revision		Revision as of 14:11, 5 August 2015
Line 91:		Line 91:
	<ref name="K">{{Citation \| last1=Kwaśnicki \| first1=M. \| title=Spectral analysis of subordinate Brownian motions on the half-line \| year=2011 \| journal=Studia Math. \| volume=206 \| pages=211–271 \| url=http://dx.doi.org/10.4064/sm206-3-2 \| doi=10.4064/sm206-3-2}}</ref>		<ref name="K">{{Citation \| last1=Kwaśnicki \| first1=M. \| title=Spectral analysis of subordinate Brownian motions on the half-line \| year=2011 \| journal=Studia Math. \| volume=206 \| pages=211–271 \| url=http://dx.doi.org/10.4064/sm206-3-2 \| doi=10.4064/sm206-3-2}}</ref>
	<ref name="KSV">{{Citation \| last1=Kim \| first1=P. \| last2=Song \| first2=R. \| last3=Vondraček \| first3=Z. \| title=On harmonic functions for trace processes \| url=http://dx.doi.org/10.1002/mana.200910008 \| doi=10.1002/mana.200910008 \| year=2011 \| journal=Math. Nachr. \| volume=284 \| pages=1889–1902}}</ref>		<ref name="KSV">{{Citation \| last1=Kim \| first1=P. \| last2=Song \| first2=R. \| last3=Vondraček \| first3=Z. \| title=On harmonic functions for trace processes \| url=http://dx.doi.org/10.1002/mana.200910008 \| doi=10.1002/mana.200910008 \| year=2011 \| journal=Math. Nachr. \| volume=284 \| pages=1889–1902}}</ref>
			<ref name="Stinga">{{Citation \| last1=Stinga \| first1=P. R.\| title=Fractional powers of second order partial differential operators: extension problem and regularity theory \| url=https://www.ma.utexas.edu/users/stinga/Stinga%20-%20PhD%20Thesis.pdf \| year=2010 \| journal=[[PhD. Thesis - Universidad Aut\'onoma de Madrid]]}}
			</ref><ref name="Stinga-Torrea">{{Citation \| last1=Stinga \| first1=P. R.\| last2=Torrea \| first2=J. L. \| title=Extension problem and Harnack's inequality for some fractional operators \| url=http://www.tandfonline.com/doi/abs/10.1080/03605301003735680#.VcJb9UWmR1Q \| year=2010 \| journal=[[Comm. Partial Differential Equations]] \| volume=35 \| issue=11 \| pages=2092-2122}}</ref>
	}}		}}

	{{stub}}		{{stub}}

imported>Pablo: /* Fractional powers of more general operators */

2015-08-05T19:10:33Z

Fractional powers of more general operators

← Older revision		Revision as of 14:10, 5 August 2015
Line 37:		Line 37:
	L^su(x) = C_{s} \lim_{y\rightarrow 0} y^{1-2s} U_y(x,y).		L^su(x) = C_{s} \lim_{y\rightarrow 0} y^{1-2s} U_y(x,y).
	\end{equation}		\end{equation}
			Here $C_s$ is a constant depending only on $s$. Moreover, the solution can be explicitly written as
			$$U(x,y)=\frac{y^{2s}}{4^s\Gamma(s)}\int_0^\infty e^{-tL}u(x)e^{-y^2/(4t)}\,\frac{dt}{t^{1+s}}.$$
			For details see
	If $L$ has a purely discrete spectrum on $\Omega$, the operator $(-L)^s$ has the same eigenfunctions as $L$, and its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $(-L)$.		If $L$ has a purely discrete spectrum on $\Omega$, the operator $(-L)^s$ has the same eigenfunctions as $L$, and its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $(-L)$.

imported>Pablo: /* Fractional powers of more general operators */

2015-08-05T19:09:05Z

Fractional powers of more general operators

← Older revision		Revision as of 14:09, 5 August 2015
Line 29:		Line 29:
	==Fractional powers of more general operators==		==Fractional powers of more general operators==
	In a similar way, extension problem for fractional powers $L^s$ of a general self-adjoint nonnegative linear differential operators $L$ in a domain $\Omega \subset \mathbb{R}^n$ (or more generally, a Hilbert space) can be constructed. In this case, the extension into a "cylinder" $\Omega \times [0, \infty)$ is considered. Let $U$ be a solution of		In a similar way, extension problem for fractional powers $L^s$ of a general self-adjoint nonnegative linear differential operators $L$ in a domain $\Omega \subset \mathbb{R}^n$ (or more generally, a Hilbert space) can be constructed. In this case, the extension into a "cylinder" $\Omega \times [0, \infty)$ is considered. Let $U$ be a solution of
	\[ \partial_y (y^{1-2s} \partial_y U(x, y)) - L_x U(x, y) = 0 \]		$$\begin{cases}
	~~defined on $~~\Omega \times [0,\infty)$, with boundary conditions along $\partial \Omega \times [0,\infty)$ equal to the boundary conditions for $L$. Then		\partial_y (y^{1-2s} \partial_y U(x, y)) - L_x U(x, y) = 0,&\hbox{in}~\Omega\times(0,\infty),\\
			U(x,0)=u(x),&\hbox{on}~\Omega,
			\end{cases}$$
			with boundary conditions along $\partial \Omega \times [0,\infty)$ equal to the boundary conditions for $L$. Then
	\begin{equation}		\begin{equation}
	L^s = C_{s} \lim_{y\rightarrow 0} y^{1-2s} U_y(x,y).		L^su(x) = C_{s} \lim_{y\rightarrow 0} y^{1-2s} U_y(x,y).
	\end{equation}		\end{equation}
	If $L$ has a purely discrete spectrum on $\Omega$, the operator $(-L)^s$ has the same eigenfunctions as $L$, and its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $(-L)$.		If $L$ has a purely discrete spectrum on $\Omega$, the operator $(-L)^s$ has the same eigenfunctions as $L$, and its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $(-L)$.

imported>Pablo: /* Fractional powers of more general operators */

2015-08-05T19:07:18Z

Fractional powers of more general operators

← Older revision		Revision as of 14:07, 5 August 2015
Line 28:		Line 28:

	==Fractional powers of more general operators==		==Fractional powers of more general operators==
	In a similar way, extension problem for fractional powers $(-L)^s$ of a general self-adjoint ~~elliptic operator~~ $L$ in a domain $\Omega \subset \mathbb{R}^n$ can be constructed. In this case, the extension into a "cylinder" $\Omega \times [0, \infty)$ is considered. Let $U$ be a solution of		In a similar way, extension problem for fractional powers $L^s$ of a general self-adjoint nonnegative linear differential operators $L$ in a domain $\Omega \subset \mathbb{R}^n$ (or more generally, a Hilbert space) can be constructed. In this case, the extension into a "cylinder" $\Omega \times [0, \infty)$ is considered. Let $U$ be a solution of
	\[ \partial_y (y^{1-2s} \partial_y U(x, y)) + L_x U(x, y) = 0 \]		\[ \partial_y (y^{1-2s} \partial_y U(x, y)) - L_x U(x, y) = 0 \]
	defined on $\Omega \times [0,\infty)$, with boundary conditions along $\partial \Omega \times [0,\infty)$ equal to the boundary conditions for $L$. Then		defined on $\Omega \times [0,\infty)$, with boundary conditions along $\partial \Omega \times [0,\infty)$ equal to the boundary conditions for $L$. Then
	\begin{equation}		\begin{equation}
	(-L)^s = C_{n,s} \lim_{y\rightarrow 0} y^{1-2s} U_y(x,y).		L^s = C_{s} \lim_{y\rightarrow 0} y^{1-2s} U_y(x,y).
	\end{equation}		\end{equation}
	If $L$ has a purely discrete spectrum on $\Omega$, the operator $(-L)^s$ has the same eigenfunctions as $L$, and its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $(-L)$.		If $L$ has a purely discrete spectrum on $\Omega$, the operator $(-L)^s$ has the same eigenfunctions as $L$, and its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $(-L)$.

imported>Mateusz: /* More general non-local operators */

2012-07-19T08:36:52Z

More general non-local operators

← Older revision		Revision as of 03:36, 19 July 2012
Line 63:		Line 63:
	This proves that the Dirichlet-to-Neumann operator is indeed equal to $f(-L)$. The proof in the continuous spectrum case is similar.<ref name="K"/><ref name="KSV"/>		This proves that the Dirichlet-to-Neumann operator is indeed equal to $f(-L)$. The proof in the continuous spectrum case is similar.<ref name="K"/><ref name="KSV"/>

	[[Operator monotone function]]s, often called complete Bernstein functions, form a subclass of [[Bernstein function]]s. Hence, existence of the extension problem is closely related to the concept of [[subordination]]. Operators of the form $f(-\Delta)$ for an operator monotone $f$ admit an explicit [[Operator monotone function#Operator monotone functions of the Laplacian\|description]]. This gives a fairly explicit condition for the existence of the extension problem for a ~~givend~~ translation-invariant non-local operator.		[[Operator monotone function]]s, often called complete Bernstein functions, form a subclass of [[Bernstein function]]s. Hence, existence of the extension problem is closely related to the concept of [[subordination]]. Operators of the form $f(-\Delta)$ for an operator monotone $f$ admit an explicit [[Operator monotone function#Operator monotone functions of the Laplacian\|description]]. This gives a fairly explicit condition for the existence of the extension problem for a given translation-invariant non-local operator.

	==Relationship with Scattering operators==		==Relationship with Scattering operators==

imported>Mateusz at 08:35, 19 July 2012

2012-07-19T08:35:48Z

← Older revision		Revision as of 03:35, 19 July 2012
Line 63:		Line 63:
	This proves that the Dirichlet-to-Neumann operator is indeed equal to $f(-L)$. The proof in the continuous spectrum case is similar.<ref name="K"/><ref name="KSV"/>		This proves that the Dirichlet-to-Neumann operator is indeed equal to $f(-L)$. The proof in the continuous spectrum case is similar.<ref name="K"/><ref name="KSV"/>

	[[Operator monotone function]]s, often called complete Bernstein functions, form a subclass of [[Bernstein function]]s. Hence, existence of the extension problem is closely related to the concept of [[subordination]]. ~~In particular, $f$ is operator monotone if and only if~~		[[Operator monotone function]]s, often called complete Bernstein functions, form a subclass of [[Bernstein function]]s. Hence, existence of the extension problem is closely related to the concept of [[subordination]]. Operators of the form $f(-\Delta)$ for an operator monotone $f$ admit an explicit [[Operator monotone function#Operator monotone functions of the Laplacian\|description]]. This gives a fairly explicit condition for the existence of the extension problem for a givend translation-invariant non-local operator.
	~~\[ f(\lambda) = a \lambda + b + \int_0^\infty \frac{r}{r + \lambda} \, \frac{\rho(dr)}{r} \]~~
	~~for some $a, b \ge 0$ and a measure $\rho(dr)$ such that $\int_0^\infty \min(r^{-1}, r^{-2}) \rho(dr) < \infty$.~~

	If $~~L = \Delta$, then $A =~~ f(-\Delta)$ for an operator monotone $f$ ~~if and only if~~
	\[ ~~-A u(x) = a \Delta u(x) + \int_{\R^n} (u(x + y) - u(x) - z \cdot \nabla u(x) \mathbf{1}_{\|z\| < 1}) k(z) \mathrm d z \]~~
	~~for some $a \ge 0$ and $k(z)$~~ of the ~~form~~
	~~\begin{align*}~~
	~~k(z) &= \int_0^\infty \int_0^\infty (4 \pi t)^{-n/2} e^{-~~\|~~z\|^2 / (4 t)} e^{-t r} dt \rho(dr)~~
	~~\end{align*}~~
	This gives a fairly explicit ~~description~~ of translation-invariant non-local ~~operators for which there an extension problem exists~~.

	==Relationship with Scattering operators==		==Relationship with Scattering operators==

imported>Mateusz: /* More general non-local operators */

2012-07-19T08:17:28Z

More general non-local operators

← Older revision		Revision as of 03:17, 19 July 2012
Line 64:		Line 64:

	[[Operator monotone function]]s, often called complete Bernstein functions, form a subclass of [[Bernstein function]]s. Hence, existence of the extension problem is closely related to the concept of [[subordination]]. In particular, $f$ is operator monotone if and only if		[[Operator monotone function]]s, often called complete Bernstein functions, form a subclass of [[Bernstein function]]s. Hence, existence of the extension problem is closely related to the concept of [[subordination]]. In particular, $f$ is operator monotone if and only if
	\[ f(\lambda) = a \lambda + \int_0^\infty \frac{r}{r + \lambda} \, \frac{\rho(dr)}{r} \]		\[ f(\lambda) = a \lambda + b + \int_0^\infty \frac{r}{r + \lambda} \, \frac{\rho(dr)}{r} \]
	for some $a \ge 0$ and a measure $\rho(dr)$ such that $\int_0^\infty \min(r^{-1}, r^{-2}) \rho(dr) < \infty$.		for some $a, b \ge 0$ and a measure $\rho(dr)$ such that $\int_0^\infty \min(r^{-1}, r^{-2}) \rho(dr) < \infty$.

	If $L = \Delta$, then $A = f(-\Delta)$ for an operator monotone $f$ if and only if		If $L = \Delta$, then $A = f(-\Delta)$ for an operator monotone $f$ if and only if