Extension technique
The fractional Laplacian $(-\Delta)^s$ on $\mathbb{R}^n$ can be obtained as the Dirichlet-to-Neumann operator of a degenerate elliptic equation on the upper half-space $\mathbb{R}^{n+1}_+$.[1] This construction is frequently used to turn nonlocal problems involving the fractional Laplacian into local problems in one more space dimension.
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, and the Wiener criterion for regularity of a boundary point.[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]
Extensions to the extension
The extension into a "cylinder" lying above a domain $\Omega \subset \mathbb{R}^n$ can also yields a relationship with a fractional order operator. Consider the equation $\nabla \cdot (y^{1-2s} \nabla U) = 0$ defined on the cylinder $\Omega \times [0,\infty)$, with Dirichlet conditions $U = 0$ along $\partial \Omega \times [0,\infty)$. For this equation, we also have the relationship that \begin{equation} A_s = C_{n,s} \lim_{y\rightarrow 0} y^{1-2s} U_y(x,y) \end{equation} with $A_s$ a fractional order operator. $A_s$ has the same eigenfunctions as $(-\Delta)$ on $\Omega$, and its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $(-\Delta)$.[6] This operator is not the same as the fractional Laplacian, except when $\Omega = \mathbb{R}^n$.
Relationship with Scattering operators
There is an identification between the fractional Laplacian defined by the extension and the fractional Paneitz operator from Scattering Theory when the order of the operator is less than 1.[7]
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
- ↑ 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
- ↑ 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
- ↑ 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
- ↑ 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
- ↑ Tan, Jinggang; Cabré, Xavier (2010), "Positive solutions of nonlinear problems involving the square root of the Laplacian", Advances in Mathematics 224 (5): 2052–2093, doi:10.1016/j.aim.2010.01.025, ISSN 0001-8708, http://dx.doi.org/10.1016/j.aim.2010.01.025
- ↑ González, Maria del Mar; Chang, Sun-Yung Alice (2011), "Fractional Laplacian in conformal geometry", Advances in Mathematics 226 (2): 1410–1432, doi:10.1016/j.aim.2010.07.016, ISSN 0001-8708, http://dx.doi.org/10.1016/j.aim.2010.07.016
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