Extension technique

The Dirichlet-to-Neumann operator for the upper half-plane maps the boundary value $U(x, 0)$ of a harmonic function $U(x, y)$ in the upper half-space $\R^{n+1}_+ = \R^n \times [0, \infty)$ to its outer normal derivative $-\partial_y U(x, 0)$. This operator coincides with the square root of the Laplace operator, $(-\Delta)^{1/2}$. Extension technique is a similar identification of non-local operators (most notably the fractional Laplacian $(-\Delta)^s$) as Dirichlet-to-Neumann operators for (possibly degenerate) elliptic equations. This construction is frequently used to turn nonlocal problems involving the fractional Laplacian into local problems in one more space dimension.

Fractional Laplacian

The extension problem for the fractional Laplacian $(-\Delta)^s$, $s \in (0, 1)$ takes the following form. 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.

The above extension technique is closely related to the concept of trace of a diffusion on a hyperplane.

Fractional powers of more general operators

In a similar way, extension problem for fractional powers $(-L)^s$ of a general self-adjoint elliptic operator $L$ in a domain $\Omega \subset \mathbb{R}^n$ can be constructed. In this case, the extension into a "cylinder" $\Omega \times [0, \infty)$ is considered. Let $U$ be a solution of \[ \partial_y (y^{1-2s} \partial_y U(x, y)) + L_x U(x, y) = 0 \] defined on $\Omega \times [0,\infty)$, with boundary conditions along $\partial \Omega \times [0,\infty)$ equal to the boundary conditions for $L$. Then \begin{equation} (-L)^s = C_{n,s} \lim_{y\rightarrow 0} y^{1-2s} U_y(x,y). \end{equation} If $L$ has a purely discrete spectrum on $\Omega$, the operator $(-L)^s$ has the same eigenfunctions as $L$, and its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $(-L)$.

For example, $L$ can be the Dirichlet Laplacian in $\Omega$. Note that $(-L)^s$ is not the same as the fractional Laplacian, except when $\Omega = \mathbb{R}^n$.

More general non-local operators

Let $L$ be as above, and consider the Dirichlet-to-Neumann operator $A$ related to the elliptic equation \[ \partial_y (w(y) \partial_y U(x, y)) + L_x U(x, y) = 0 \] in the upper half-plane. Then $A = f(-L)$ for some operator monotone function $f$. Conversely, for any operator monotone $f$, there is an appropriate extension problem for $f(-L)$. (This identification requires some conditions on $w(y)$ which ensure the extension problem is well-posed.)

The relation between $w$ and $f$ is equivalent to the Krein correspondence, and can be described as follows. For $\lambda \ge 0$, let $g_\lambda$ be the nonincreasing positive solution of the ODE \[ \partial_y (w(y) \partial_y g_\lambda(y)) = \lambda g_\lambda(y) \] for $y \ge 0$, satisfying $g_\lambda(0) = 1$. Furthermore, let $h$ be the nondecreasing solution of \[ \partial_y (w(y) \partial_y h(y)) = 0 \] satisfying $h(0) = 0$ and $h(1) = 1$. Then \[ f(\lambda) = \lim_{y \to 0^+} \frac{1 - g_\lambda(y)}{h(y)} . \] One can proof that $f$ defined above is operator monotone, and conversely, for any operator monotone $f$ one can find $w$ for which the above identity holds. Noteworthy, there are relatively few explicit pairs of corresponding $w$ and $f$.

Suppose now that $U(x, y)$ is a sufficiently regular solution of the extension problem \[ \partial_y (w(y) \partial_y U(x, y)) + L_x U(x, y) = 0 . \] For simplicity, suppose that the spectrum of $L$ is discrete. Let $\varphi$ be an eigenfunction of $-L$ with eigenvalue $\lambda$, and denote $U_\varphi(y) = \langle U(\cdot, y), \varphi \rangle$. Then $U_\varphi$ is a solution of the ODE \[ \partial_y (w(y) \partial_y U_\varphi(y)) + \lambda U_\varphi(y) = 0 . \] Hence, $U_\varphi(y) = U_\varphi(0) g_\lambda(y)$, and \[ -\lim_{y \to 0^+} \frac{U_\varphi(y) - U_\varphi(0)}{h(y)} = f(\lambda) U_\varphi(0) . \] It follows that \[ -\lim_{y \to 0^+} \frac{U(\cdot, y) - U_\varphi(\cdot, 0)}{h(y)} = f(-L) U(\cdot, 0) , \] or equivalently \[ -\lim_{y \to 0^+} \frac{\partial_y U(\cdot, y)}{h'(y)} = f(-L) U(\cdot, 0) . \] This proves that the Dirichlet-to-Neumann operator is indeed equal to $f(-L)$.

Operator monotone functions, often called complete Bernstein functions, form a subclass of Bernstein functions. Hence, existence of the extension problem is closely related to the concept of subordination. In particular, $f$ is operator monotone if and only if \[ f(\lambda) = a \lambda + \int_0^\infty \frac{r}{r + \lambda} \, \frac{\rho(dr)}{r} \] for some $a \ge 0$ and a measure $\rho(dr)$ such that $\int_0^\infty \min(r^{-1}, r^{-2}) \rho(dr) < \infty$.

If $L = \Delta$, then $A = f(-\Delta)$ for an operator monotone $f$ if and only if \[ -A u(x) = a \Delta u(x) + \int_{\R^n} (u(x + y) - u(x) - z \cdot \nabla u(x) \mathbf{1}_{|z| < 1}) k(z) \mathrm d z \] for some $a \ge 0$ and $k(z)$ of the form \begin{align*} k(z) &= \int_0^\infty \int_0^\infty (4 \pi t)^{-n/2} e^{-|z|^2 / (4 t)} e^{-t r} dt \rho(dr) \end{align*} This gives a fairly explicit description of translation-invariant non-local operators for which there an extension problem exists.

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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