Free boundary problems: Difference between revisions
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=Studied problems for $(-\Delta)^s$= | =Studied problems for $(-\Delta)^s$= | ||
Here are some free boundary problems for the fractional | Here are some free boundary problems for the [[fractional Laplacian]] operator $(-\Delta)^s$ that have been studied to some extent. | ||
The [[extension technique]], representing the fractional Laplace operator as a suitable Dirichlet-to-Neumann operator on the boundary of the upper half-space in one more dimension, figures prominently in the study of many problems below. | The [[extension technique]],<ref name="CS"/> representing the fractional Laplace operator as a suitable Dirichlet-to-Neumann operator on the boundary of the upper half-space in one more dimension, figures prominently in the study of many problems below. | ||
==[[Fractional obstacle problem]]== | ==[[Fractional obstacle problem]]== | ||
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 $|x|\rightarrow \infty$. The problem can be formulated as a variational problem as well, either through the extension or through a | 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 | 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 limit of the Almgren frequency formula.<ref name="S"/><ref name="CSS"/> | ||
==[[Fractional Alt-Caffarelli problem]]== | ==[[Fractional Alt-Caffarelli problem]]== | ||
The problem is to seek the minimizer of an energy which is the sum of the Dirichlet form | The problem is to seek the minimizer of an energy which is the sum of the [[Dirichlet form]] corresponding to the fractional Laplacian and the measure of the positivity set, taking the minimizer $u$ among nonnegative functions. It is usually formulated in terms of the extension. | ||
Like its 2nd order analogue, solutions to problem are known to have optimal regularity of Holder class $C^s$, and to be $s$-harmonic away from the zero set. They have nondegeneracy of order $s$ as well, which is to say that | Like its 2nd order analogue, solutions to problem are known to have optimal regularity of Holder class $C^s$, and to be $s$-harmonic away from the zero set. They have nondegeneracy of order $s$ as well, which is to say that | ||
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where $\Lambda$ is the zero set of the $u$. | where $\Lambda$ is the zero set of the $u$. | ||
Unlike its 2nd order analogue, Hausdorff measure estimates of the free boundary are not known, but there are Lebesgue density estimates of the zero and free set in the neighborhood of a free boundary point. The free boundary is $C^{1,\alpha}$ about points where it is suitably flat. | Unlike its 2nd order analogue, Hausdorff measure estimates of the free boundary are not known, but there are Lebesgue density estimates of the zero and free set in the neighborhood of a free boundary point. The free boundary is $C^{1,\alpha}$ about points where it is suitably flat.<ref name="CRS"/><ref name="DR"/> | ||
==[[ | ==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="CRS">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Roquejoffre | first2=Jean-Michel | last3=Sire | first3=Yannick | title=Variational problems for free boundaries for the fractional Laplacian | url=http://dx.doi.org/10.4171/JEMS/226 | doi=10.4171/JEMS/226 | year=2010 | journal=Journal of the European Mathematical Society (JEMS) | issn=1435-9855 | volume=12 | issue=5 | pages=1151–1179}}</ref> | |||
<ref name="DR">{{Citation | last1=De Silva | first1=Daniela | last2=Roquejoffre | first2=Jean-Michel | title=Regularity in a one-phase free boundary problem for the fractional Laplacian | url=http://arxiv.org/abs/1102.3086v1 | year=2011}}</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> | |||
}} | |||
Latest revision as of 20:36, 15 April 2015
Studied problems for $(-\Delta)^s$
Here are some free boundary problems for the fractional Laplacian operator $(-\Delta)^s$ that have been studied to some extent.
The extension technique,[1] representing the fractional Laplace operator as a suitable Dirichlet-to-Neumann operator on the boundary of the upper half-space in one more dimension, figures prominently in the study of many problems below.
Fractional obstacle problem
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 limit of the Almgren frequency formula.[2][3]
Fractional Alt-Caffarelli problem
The problem is to seek the minimizer of an energy which is the sum of the Dirichlet form corresponding to the fractional Laplacian and the measure of the positivity set, taking the minimizer $u$ among nonnegative functions. It is usually formulated in terms of the extension.
Like its 2nd order analogue, solutions to problem are known to have optimal regularity of Holder class $C^s$, and to be $s$-harmonic away from the zero set. They have nondegeneracy of order $s$ as well, which is to say that $$ u(X) \geq c d(X,\Lambda)^s $$ where $\Lambda$ is the zero set of the $u$.
Unlike its 2nd order analogue, Hausdorff measure estimates of the free boundary are not known, but there are Lebesgue density estimates of the zero and free set in the neighborhood of a free boundary point. The free boundary is $C^{1,\alpha}$ about points where it is suitably flat.[4][5]
References
- ↑ Caffarelli, Luis; Silvestre, Luis (2007), "An extension problem related to the fractional Laplacian", Communications in Partial Differential Equations 32 (7): 1245–1260, doi:10.1080/03605300600987306, ISSN 0360-5302, http://dx.doi.org.ezproxy.lib.utexas.edu/10.1080/03605300600987306
- ↑ Silvestre, Luis (2007), "Regularity of the obstacle problem for a fractional power of the Laplace operator", Communications on Pure and Applied Mathematics 60 (1): 67–112, doi:10.1002/cpa.20153, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20153
- ↑ Caffarelli, Luis; Salsa, Sandro; Silvestre, Luis (2008), "Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian", Inventiones Mathematicae 171 (2): 425–461, doi:10.1007/s00222-007-0086-6, ISSN 0020-9910, http://dx.doi.org/10.1007/s00222-007-0086-6
- ↑ Caffarelli, Luis A.; Roquejoffre, Jean-Michel; Sire, Yannick (2010), "Variational problems for free boundaries for the fractional Laplacian", Journal of the European Mathematical Society (JEMS) 12 (5): 1151–1179, doi:10.4171/JEMS/226, ISSN 1435-9855, http://dx.doi.org/10.4171/JEMS/226
- ↑ De Silva, Daniela; Roquejoffre, Jean-Michel (2011), Regularity in a one-phase free boundary problem for the fractional Laplacian, http://arxiv.org/abs/1102.3086v1