Fractional Laplacian - Revision history

imported>Pablo: /* Via the heat semigroup */

2015-08-05T19:04:37Z

Via the heat semigroup

← Older revision		Revision as of 14:04, 5 August 2015
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	v(x,0)=f(x),&\hbox{for}~x\in\mathbb{R}^n.		v(x,0)=f(x),&\hbox{for}~x\in\mathbb{R}^n.
	\end{cases}$$		\end{cases}$$
	~~See~~ <ref name="Stinga"/> and <ref name="Stinga-Torrea"/>.		This semigroup language approach was introduced in <ref name="Stinga"/> and <ref name="Stinga-Torrea"/>.

	=== From functional calculus ===		=== From functional calculus ===

imported>Pablo: /* As a singular integral */

2015-08-05T19:02:15Z

As a singular integral

← Older revision		Revision as of 14:02, 5 August 2015
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	\[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {\|x-y\|^{n+2s}} \mathrm d y .\]		\[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {\|x-y\|^{n+2s}} \mathrm d y .\]

	Where $c_{n,s}$ is a constant depending on dimension and $s$. ~~Indeed, it~~ is ~~enough to~~ write down the heat kernel in the semigroup formula		Where $c_{n,s}$ is a constant depending on dimension and $s$.
	\eqref{semigroup}, use that $e^{t\Delta}1\equiv1$ and Fubini's Theorem, ~~see~~
			One way of deducing this formula is the following: write down the heat kernel in the semigroup formula \eqref{semigroup}, use that $e^{t\Delta}1\equiv1$ and apply Fubini's Theorem. See <ref name="Stinga"/> and <ref name="Stinga-Torrea"/> for the details of this computation. Notice that this derivation gives the explicit value of the constant:
			$$c_{n,s}=\frac{4^s\Gamma(n/2+s)}{\pi^{n/2}\|\Gamma(-s)\|}.$$

	This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.		This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.

imported>Pablo: /* As a singular integral */

2015-08-05T18:59:22Z

As a singular integral

← Older revision		Revision as of 13:59, 5 August 2015
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	\[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {\|x-y\|^{n+2s}} \mathrm d y .\]		\[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {\|x-y\|^{n+2s}} \mathrm d y .\]

	Where $c_{n,s}$ is a constant depending on dimension and $s$. Indeed, it is enough to write down the heat kernel in the semigroup formula, use that $e^{t\Delta}1\equiv1$ and Fubini's Theorem, see		Where $c_{n,s}$ is a constant depending on dimension and $s$. Indeed, it is enough to write down the heat kernel in the semigroup formula
			\eqref{semigroup}, use that $e^{t\Delta}1\equiv1$ and Fubini's Theorem, see

	This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.		This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.

imported>Pablo: /* Via the heat semigroup */

2015-08-05T18:59:00Z

Via the heat semigroup

← Older revision		Revision as of 13:59, 5 August 2015
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	$$\lambda^s=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{-t\lambda}-1\big)\,\frac{dt}{t^{1+s}},$$		$$\lambda^s=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{-t\lambda}-1\big)\,\frac{dt}{t^{1+s}},$$
	valid for $\lambda\geq0$ and $0<s<1$. By taking $\lambda=\|\xi\|^2$ and using the Fourier transform definition of the fractional Laplacian we get		valid for $\lambda\geq0$ and $0<s<1$. By taking $\lambda=\|\xi\|^2$ and using the Fourier transform definition of the fractional Laplacian we get
	$$(-\Delta)^sf(x)=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{t\Delta}f(x)-f(x)\big)\,\frac{dt}{t^{1+s}}.$$		\begin{equation}\label{semigroup}
			(-\Delta)^sf(x)=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{t\Delta}f(x)-f(x)\big)\,\frac{dt}{t^{1+s}}.
			\end{equation}
	Here $v(x,t)=e^{t\Delta}f(x)$ is the solution to the heat equation on $\mathbb{R}^n$ with initial temperature $f$, namely,		Here $v(x,t)=e^{t\Delta}f(x)$ is the solution to the heat equation on $\mathbb{R}^n$ with initial temperature $f$, namely,
	$$\begin{cases}		$$\begin{cases}

imported>Pablo: /* As a singular integral */

2015-08-05T18:58:30Z

As a singular integral

← Older revision		Revision as of 13:58, 5 August 2015
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	\[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {\|x-y\|^{n+2s}} \mathrm d y .\]		\[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {\|x-y\|^{n+2s}} \mathrm d y .\]

	Where $c_{n,s}$ is a constant depending on dimension and $s$.		Where $c_{n,s}$ is a constant depending on dimension and $s$. Indeed, it is enough to write down the heat kernel in the semigroup formula, use that $e^{t\Delta}1\equiv1$ and Fubini's Theorem, see

	This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.		This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.

imported>Pablo at 18:56, 5 August 2015

2015-08-05T18:56:53Z

← Older revision		Revision as of 13:56, 5 August 2015
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	<ref name="RS-K">{{Citation \| last1=Ros-Oton \| first1=X. \| last2=Serra \| first2=J. \| title=Boundary regularity for fully nonlinear integro-differential equations \| year=2014 \| journal=[[preprint arXiv]] }}</ref>		<ref name="RS-K">{{Citation \| last1=Ros-Oton \| first1=X. \| last2=Serra \| first2=J. \| title=Boundary regularity for fully nonlinear integro-differential equations \| year=2014 \| journal=[[preprint arXiv]] }}</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 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>
	}}		}}

imported>Pablo at 18:56, 5 August 2015

2015-08-05T18:56:26Z

← Older revision		Revision as of 13:56, 5 August 2015
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	<ref name="Grubb">{{Citation \| last1=Grubb \| first1=G. \| title=Fractional Laplacians on domains, a development of Hormander's theory of $mu$-transmission pseudodifferential operators \| url=http://arxiv.org/abs/1310.0951 \| year=2014 \| journal=[[arXiv]] \| pages=1-43 }}</ref>		<ref name="Grubb">{{Citation \| last1=Grubb \| first1=G. \| title=Fractional Laplacians on domains, a development of Hormander's theory of $mu$-transmission pseudodifferential operators \| url=http://arxiv.org/abs/1310.0951 \| year=2014 \| journal=[[arXiv]] \| pages=1-43 }}</ref>
	<ref name="RS-K">{{Citation \| last1=Ros-Oton \| first1=X. \| last2=Serra \| first2=J. \| title=Boundary regularity for fully nonlinear integro-differential equations \| year=2014 \| journal=[[preprint arXiv]] }}</ref>		<ref name="RS-K">{{Citation \| last1=Ros-Oton \| first1=X. \| last2=Serra \| first2=J. \| title=Boundary regularity for fully nonlinear integro-differential equations \| year=2014 \| journal=[[preprint arXiv]] }}</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">{{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>
			}}

imported>Pablo at 18:53, 5 August 2015

2015-08-05T18:53:06Z

← Older revision		Revision as of 13:53, 5 August 2015
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	<ref name="Grubb">{{Citation \| last1=Grubb \| first1=G. \| title=Fractional Laplacians on domains, a development of Hormander's theory of $mu$-transmission pseudodifferential operators \| url=http://arxiv.org/abs/1310.0951 \| year=2014 \| journal=[[arXiv]] \| pages=1-43 }}</ref>		<ref name="Grubb">{{Citation \| last1=Grubb \| first1=G. \| title=Fractional Laplacians on domains, a development of Hormander's theory of $mu$-transmission pseudodifferential operators \| url=http://arxiv.org/abs/1310.0951 \| year=2014 \| journal=[[arXiv]] \| pages=1-43 }}</ref>
	<ref name="RS-K">{{Citation \| last1=Ros-Oton \| first1=X. \| last2=Serra \| first2=J. \| title=Boundary regularity for fully nonlinear integro-differential equations \| year=2014 \| journal=[[preprint arXiv]] }}</ref>		<ref name="RS-K">{{Citation \| last1=Ros-Oton \| first1=X. \| last2=Serra \| first2=J. \| title=Boundary regularity for fully nonlinear integro-differential equations \| year=2014 \| journal=[[preprint arXiv]] }}</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">{{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>

imported>Pablo at 18:51, 5 August 2015

2015-08-05T18:51:44Z

← Older revision		Revision as of 13:51, 5 August 2015
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	=== Via the heat semigroup ===		=== Via the heat semigroup ===
	Recall the numerical formula		Recall the numerical formula
	$$\lambda^s=\frac{1}{\Gamma(-s)}\int_0^\~~infy~~\big(e^{-t\lambda}-1\big)\,\frac{dt}{t^{1+s}},$$		$$\lambda^s=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{-t\lambda}-1\big)\,\frac{dt}{t^{1+s}},$$
	valid for $\lambda\geq0$ and $0<s<1$. By taking $\lambda=\|\xi\|^2$ and using the Fourier transform definition of the fractional Laplacian we get		valid for $\lambda\geq0$ and $0<s<1$. By taking $\lambda=\|\xi\|^2$ and using the Fourier transform definition of the fractional Laplacian we get
	$$(-\Delta)^sf(x)=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{t\Delta}f(x)-f(x)\big)\,\frac{dt}{t^{1+s}}.$$		$$(-\Delta)^sf(x)=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{t\Delta}f(x)-f(x)\big)\,\frac{dt}{t^{1+s}}.$$

imported>Pablo at 18:51, 5 August 2015

2015-08-05T18:51:16Z

@@ Line 15: / Line 15: @@
 This formula is the simplest to understand and it is useful for problems in the whole space. On the other hand, it is hard to obtain local estimates from it.
 === From functional calculus ===
@@ Line 121: / Line 133: @@
 <ref name="RS-K">{{Citation  | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=Boundary regularity for fully nonlinear integro-differential equations | year=2014 | journal=[[preprint arXiv]] }}</ref>
 }}