# Invertibility of the Laplacian

Consider the Laplacian as an operator $\Delta\colon W^{2,p}(\Omega)\subset L^p(\Omega)\to L^p(\Omega)$ subject to homogeneous Robin boundary conditions, where $\Omega\subset \mathbf R^n$ is either bounded with smooth boundary or a halfspace and $1< p< \infty$. Is there any reference giving information about the invertibility of $\Delta$ in these spaces?

A careful exposition can be found in "Analytic Semigroups and Semilinear Initial Boundary Value Problems" by Kazuaki Taira.

He considers the mixed boundary value problem $$\begin{cases} A u=f & \mbox{in }\Omega,\\\ Bu:=a\left.\frac{\partial u}{\partial \nu}+bu\right|_{\partial\Omega}=\phi & \mbox{on }\partial\Omega,\end{cases}\qquad\qquad\qquad(*)$$ where $A$ is a second-order uniformly elliptic differential operator with $C^{\infty}$ coefficients and $\Omega\subset\mathbb R^n$ is a bounded domain with smooth boundary. The functions $a$, $b$ are assumed to be smooth and to satisfy some natural non-degeneracy conditions.

He shows that, for $1< p < \infty$ and $s > 1+1/p$, the mapping $$(A,B):H^{s,p}(\Omega)\to H^{s-2,p}(\Omega)\oplus B^{s-1-1/p,p}(\partial\Omega)$$ is an algebraic and topological isomorphism (Theorem 1, p. 4). Here $H^{s,p}(\Omega)$ is the standard Sobolev space and $B^{s-1/p,p}(\partial\Omega)$ is the Besov space of the traces (or boundary values) of functions $u\in H^{s,p}(\Omega)$.

This implies, for any $f\in H^{s-2,p}(\Omega)$ and $\phi\in B^{s-1-1/p,p}(\partial\Omega)$, the existence and uniqueness of a solution $u\in H^{s,p}(\Omega)$ to problem ($*$).

Edit. In your case $s=2$ and $\phi\equiv 0$ so one does not even have to worry about the fractional order Sobolev spaces and the Besov space of traces on the boundary.

• Thank you very much for your reference. I just got the book, and noticed that the author is only dealing with bounded domains. Are there any references which deal with the unbounded-domain case, especially halfspace?
– Marc
Dec 6, 2010 at 8:36
• Unfortunately, I don't know such a reference. However, I believe that the proof from Taira's book (i.e. the reduction of the BVP to a problem for the corresponding pseudo-differential operator on the boundary and proving regularity estimates for the operator) should work mutatis mutandis in the case of halfspace. Dec 6, 2010 at 9:07