Fractional obstacle problem - Revision history

imported>Luis: Redirected page to Obstacle problem for the fractional Laplacian

2012-01-28T18:50:20Z

Redirected page to Obstacle problem for the fractional Laplacian

← Older revision		Revision as of 13:50, 28 January 2012
Line 1:		Line 1:
	The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $\|u\|\rightarrow 0$ as $\|x\|\rightarrow \infty$ if $\Omega$ is, say, all of $\mathbb{R}^n$. The problem can be formulated as a variational problem as well, either through the extension or directly through a Dirichlet-like nonlocal energy on $\mathbb{R}^n$.		#REDIRECT [[obstacle problem for the fractional Laplacian]]

	Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the [[~~Almgren frequency formula]].<ref name="S"/><ref name="CSS"/><ref name="CS"/>~~

	~~==References==~~
	~~{{reflist\|refs=~~
	~~<ref name="CSS">{{Citation \| last1=Caffarelli \| first1=Luis \| last2=Salsa \| first2=Sandro \| last3=Silvestre \| first3=Luis \| title=Regularity estimates for the solution and the free boundary of the~~ obstacle problem for the fractional Laplacian ~~\| url=http://dx.doi.org/10.1007/s00222-007-0086-6 \| doi=10.1007/s00222-007-0086-6 \| year=2008 \| journal=[[Inventiones Mathematicae]] \| issn=0020-9910 \| volume=171 \| issue=2 \| pages=425–461}}</ref>~~
	<ref name="CS">{{Citation \| last1=Caffarelli \| first1=Luis \| last2=Silvestre \| first2=Luis \| title=An extension problem related to the fractional Laplacian \| url=http://dx.doi.org.ezproxy.lib.utexas.edu/10.1080/03605300600987306 \| doi=10.1080/03605300600987306 \| year=2007 \| journal=Communications in Partial Differential Equations \| issn=0360-5302 \| volume=32 \| issue=7 \| pages=1245–1260}}</ref>
	<ref name="S">{{Citation \| last1=Silvestre \| first1=Luis \| title=Regularity of the obstacle problem for a fractional power of the Laplace operator \| url=http://dx.doi.org/10.1002/cpa.20153 \| doi=10.1002/cpa.20153 \| year=2007 \| journal=[[Communications on Pure and Applied Mathematics]] ~~\| issn=0010-3640 \| volume=60 \| issue=1 \| pages=67–112}}</ref>~~
	}}

imported>Nestor at 06:16, 2 June 2011

2011-06-02T06:16:48Z

← Older revision		Revision as of 01:16, 2 June 2011
Line 1:		Line 1:
	The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $\|u\|\rightarrow 0$ as $\|x\|\rightarrow \infty$ if $\Omega$ is, say, all of $\mathbb{R}^n$. The problem can be formulated as a variational problem as well, either through the extension or directly through a Dirichlet-like nonlocal energy on $\mathbb{R}^n$.		The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $\|u\|\rightarrow 0$ as $\|x\|\rightarrow \infty$ if $\Omega$ is, say, all of $\mathbb{R}^n$. The problem can be formulated as a variational problem as well, either through the extension or directly through a Dirichlet-like nonlocal energy on $\mathbb{R}^n$.

	Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the [[Almgren frequency formula]].<ref name="S"/><ref name="CSS"/>		Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the [[Almgren frequency formula]].<ref name="S"/><ref name="CSS"/><ref name="CS"/>

	==References==		==References==

imported>RayAYang: just copy over - expand later

2011-05-31T20:32:54Z

just copy over - expand later

New page

The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$ if $\Omega$ is, say, all of $\mathbb{R}^n$. The problem can be formulated as a variational problem as well, either through the extension or directly through a Dirichlet-like nonlocal energy on $\mathbb{R}^n$.

Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the [[Almgren frequency formula]].<ref name="S"/><ref name="CSS"/>

==References==
{{reflist|refs=
<ref name="CSS">{{Citation | last1=Caffarelli | first1=Luis | last2=Salsa | first2=Sandro | last3=Silvestre | first3=Luis | title=Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian | url=http://dx.doi.org/10.1007/s00222-007-0086-6 | doi=10.1007/s00222-007-0086-6 | year=2008 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=171 | issue=2 | pages=425–461}}</ref>
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=An extension problem related to the fractional Laplacian | url=http://dx.doi.org.ezproxy.lib.utexas.edu/10.1080/03605300600987306 | doi=10.1080/03605300600987306 | year=2007 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=32 | issue=7 | pages=1245–1260}}</ref>
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | url=http://dx.doi.org/10.1002/cpa.20153 | doi=10.1002/cpa.20153 | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</ref>
}}