Extension technique: Difference between revisions

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The fractional Laplace operator $(-\Delta)^s$ on $\mathbb{R}^n$ is the Dirichlet-to-Neumann operator of a degenerate elliptic equation on the upper half-space $\mathbb{R}^{n+1}_+$.
The fractional Laplace operator $(-\Delta)^s$ on $\mathbb{R}^n$ is the Dirichlet-to-Neumann operator of a degenerate elliptic equation on the upper half-space $\mathbb{R}^{n+1}_+$.<ref name="CS"/> It is frequently used to localize problems involving the fractional Laplacian.  


Let  
Let  
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\int y^{1-2s} |\nabla U|^2 dx dy  
\int y^{1-2s} |\nabla U|^2 dx dy  
\end{equation}
\end{equation}
The weight $y^{1-2s}$, for $0<s<1$, lies inside the Muckenhoupt $A_2$ class of weights. It is known that degenerate 2nd order elliptic PDEs with these weights satisfy many of the usual properties of uniformly elliptic PDEs, such as the maximum principle, the [[De Giorgi-Nash-Moser]] regularity theory, the [[boundary Harnack inequality]], the Wiener criterion for regularity of a boundary point, all have suitable analogues.<ref name="FKS"/><ref name="FKJ1"/><ref name="FKJ2"/>
The translation invariance of the operator in the $x$-directions can be applied to obtain higher regularity results and Liouville type properties.<ref name="CSS"/>
==References==
==References==
*{{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}}
{{reflist|refs=
<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="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="FKS">{{Citation | last1=Fabes | first1=Eugene B. | last2=Kenig | first2=Carlos E. | last3=Serapioni | first3=Raul P. | title=The local regularity of solutions of degenerate elliptic equations | url=http://dx.doi.org/10.1080/03605308208820218 | doi=10.1080/03605308208820218 | year=1982 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=7 | issue=1 | pages=77–116}}</ref>
<ref name="FKJ1">{{Citation | last1=Fabes | first1=Eugene B. | last2=Kenig | first2=Carlos E. | last3=Jerison | first3=David | title=Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) | publisher=Wadsworth | series=Wadsworth Math. Ser. | year=1983 | chapter=Boundary behavior of solutions to degenerate elliptic equations | pages=577–589}}</ref>
<ref name="FKJ2">{{Citation | last1=Fabes | first1=Eugene B. | last2=Jerison | first2=David | last3=Kenig | first3=Carlos E. | title=The Wiener test for degenerate elliptic equations | url=http://www.numdam.org/item?id=AIF_1982__32_3_151_0 | year=1982 | journal=[[Annales de l'Institut Fourier|Université de Grenoble. Annales de l'Institut Fourier]] | issn=0373-0956 | volume=32 | issue=3 | pages=151–182}}</ref>
}}

Revision as of 11:55, 31 May 2011

The fractional Laplace operator $(-\Delta)^s$ on $\mathbb{R}^n$ is the Dirichlet-to-Neumann operator of a degenerate elliptic equation on the upper half-space $\mathbb{R}^{n+1}_+$.[1] It is frequently used to localize problems involving the fractional Laplacian.

Let $$U:\mathbb{R}^n \times \mathbb{R}_+ \longrightarrow \mathbb{R}$$ be a function satisfying \begin{equation} \label{eqn:Main} \nabla \cdot (y^{1-2s} \nabla U(x,y)) = 0 \end{equation} on the upper half-space, lying inside the appropriately weighted Sobolev space $\dot{H}(1-2s,\mathbb{R}^{n+1}_+)$. Then if we let $u(x) = U(x,0)$, we have \begin{equation} \label{eqn:Neumann} (-\Delta)^s u(x) = -C_{n,s} \lim_{y\rightarrow 0} y^{1-2s} \partial_y U(x,y). \end{equation} The energy associated with the operator in \eqref{eqn:Main} is \begin{equation} \label{eqn:Energy} \int y^{1-2s} |\nabla U|^2 dx dy \end{equation}

The weight $y^{1-2s}$, for $0<s<1$, lies inside the Muckenhoupt $A_2$ class of weights. It is known that degenerate 2nd order elliptic PDEs with these weights satisfy many of the usual properties of uniformly elliptic PDEs, such as the maximum principle, the De Giorgi-Nash-Moser regularity theory, the boundary Harnack inequality, the Wiener criterion for regularity of a boundary point, all have suitable analogues.[2][3][4]

The translation invariance of the operator in the $x$-directions can be applied to obtain higher regularity results and Liouville type properties.[5]

References

  1. 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 
  2. Fabes, Eugene B.; Kenig, Carlos E.; Serapioni, Raul P. (1982), "The local regularity of solutions of degenerate elliptic equations", Communications in Partial Differential Equations 7 (1): 77–116, doi:10.1080/03605308208820218, ISSN 0360-5302, http://dx.doi.org/10.1080/03605308208820218 
  3. Fabes, Eugene B.; Kenig, Carlos E.; Jerison, David (1983), "Boundary behavior of solutions to degenerate elliptic equations", Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, pp. 577–589 
  4. Fabes, Eugene B.; Jerison, David; Kenig, Carlos E. (1982), "The Wiener test for degenerate elliptic equations", Université de Grenoble. Annales de l'Institut Fourier 32 (3): 151–182, ISSN 0373-0956, http://www.numdam.org/item?id=AIF_1982__32_3_151_0 
  5. 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