Embedding of domain of fractional power of Laplacian into Sobolev space for cylindrical domains

Question

On a bounded domain $\Omega \subset \mathbb R^d, d\geq 2$ with smooth boundary, it is well known that for the Dirichlet Laplacian $\Delta_D$, $D((-\Delta_D)^\frac12) = H^1_0(\Omega)$.

I'm interested in the same problem on a cylindrical domain, that means $\Omega = \mathbb R \times (0,1) \subset \mathbb R^2$ with Dirichlet boundary conditions. I found a reference [1] which asserts that in this case, the Dirichlet Laplacian satisfies the maximal $L^p$-regularity property and fractional powers of $-\Delta_D$ are well-defined.

However, I would further be interested in knowing whether one can prove an embedding $D((-\Delta_D)^\frac12) \hookrightarrow H^1_0(\Omega)$ by means of a bound of the form $$\|\nabla u\|_{L^2(\Omega)} \leq C\|(-\Delta_D)^\frac12 u\|_{L^2(\Omega)}.$$ Is there any known reference for such a result on a cylindrical domain?

Thanks in advance!

[1] Tobias Nau and Jürgen Saal. “R-sectoriality of Cylindrical Boundary Value Problems”.

Answer 1

This Kato square root property is indeed true from abstract principles and without any regularity on the domain since $-\Delta_D$ is selfadjoint. (For $H^1_0(\Omega)$ being the closure of test functions on $\Omega$ with respect to the $H^1(\Omega)$ norm.)

This can be seen by using Kato's own form approach; in fact it is the very content of his second representation theorem that you can find in Chapter VI.§2.6, Theorem 2.23, in his seminal book Perturbation Theory for Linear Operators .

The property will also stay true for more general elliptic operators as long as they stay selfadjoint. For non-selfadjoint elliptic operators, now, this is where the fun starts.. but amazing progress has been made, for which I would like to point to Bechtel/Egert/Haller-Dintelmann: The Kato square root problem on locally uniform domains.