Dirichlet-to-Neumann operator on the upper half-plane

Core idea

Let $u$ be the Poisson extension of sufficiently regular boundary data $f$ on $\mathbb R$ . The Dirichlet-to-Neumann operator sends $f$ to the boundary normal derivative

\Lambda f=-\partial_yu(\,\cdot\,+i0).

With this outward-normal convention,

\Lambda f=H(f')=(-\partial_x^2)^{1/2}f,

where $H$ is the Hilbert transform.

Fourier symbol

The Poisson extension satisfies $\widehat u(\xi,y)=e^{-2\pi|\xi|y}\widehat f(\xi)$ . Hence $\widehat{\Lambda f}(\xi)=2\pi|\xi|\widehat f(\xi)$ for the $2\pi$ Fourier normalization.

Sign convention

Using the inward derivative $+\partial_y$ replaces $\Lambda$ by $-\Lambda$ . Statements involving (H[-f']) use that alternative convention.

References

Luis Caffarelli and Luis Silvestre, “An extension problem related to the fractional Laplacian,” Communications in PDE 32 (2007), 1245–1260. DOI record.