Upcoming events and Extension technique: Difference between pages
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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 $\mathbb{R}^{n+1}_+$. | |||
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} | ||
==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}} | |||
Revision as of 22:11, 23 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 $\mathbb{R}^{n+1}_+$.
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}
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