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 $\mathbb{R}^{n+1}_+$. 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.

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

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 $(-\Delta)$. 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.

References

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