4
$\begingroup$

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with the boundary of class $C^{1,1}$. Let $A$ be a selfadjoint operator acting in $L^2(\Omega;\mathbb{C}^n)$. The operator $A$ is given by the differential expression $b(D)^* g^0 b(D)$ with the Dirichlet boundary condition. The domain of $A$ is $H^2(\Omega;\mathbb{C}^n)\cap H^1_0(\Omega;\mathbb{C}^n)$. Here $g^0$ is a constant Hermitian positive definite $(m\times m)$-matrix; $b(D)= \sum_{l=1}^d b_l D_l$ is a first order differential operator, where $b_l$ are constant $(m\times n)$-matrices. It is assumed that $m \ge n$ and the symbol $b(\xi) = \sum_{l=1}^d b_l \xi_l$ is subject to the condition $$ {\rm rank}\, b(\xi) =n,\ \ 0\ne \xi \in \mathbb{R}^d, $$ which ensures that $A$ is strongly elliptic.

It is known that $A$ is continuous from $H^s(\Omega;\mathbb{C}^n)$ to $H^{s+2}(\Omega;\mathbb{C}^n)$ for integer $s\ge 0$ if $\partial \Omega \in C^{s+1,1}$. This follows from the results about regularity of solutions of strongly elliptic systems. We can refer to [McLean, Chapter 4].

QUESTION: is this true for non-integer $s>0$? The (optimal) condition of smoothness of $\partial \Omega$ is of interest. What is the reference?

The same question for the case of the Neumann boundary condition is interesting too.

[McLean] McLean W., Strongly elliptic systems and boundary integral equations, Cambridge: Cambridge Univ. Press, 2000.

$\endgroup$
3
  • $\begingroup$ You can substitute a Sobolev type regularity for the assumption that $\partial\Omega$ is $C^{s+1,1}$. Mazya's book is a good reference for regularity results under optimal assumptions. $\endgroup$ Jan 21, 2015 at 13:55
  • $\begingroup$ @Michael Renardy: thank you! Which Mazya's book do you mean? $\endgroup$ Jan 22, 2015 at 23:03
  • $\begingroup$ @MichaelRenardy: Michael, I know that this an old post, but would you please elaborate on which Mazya reference you have in mind? I think it might be relevant for me. Thank you. $\endgroup$ May 5, 2018 at 13:45

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.