Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer 1

up vote 3 down vote accepted

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.

share|improve this answer
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 '10 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. –  Andrey Rekalo Dec 6 '10 at 9:07
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.