Dependence of the constant in Korn's inequality on the domain

Question

Let $\Omega \subset \mathbb{R}^N$ be an open, connected set with Lipschitz boundary, $N \geqslant 2$ and

$$ \mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i j} ( v) \varepsilon_{i j} ( v) \mathrm{d} x := \int_{\Omega} \sum_{i, j} \left( \frac{v_{i, j} + v_{j, i}}{2} \right)^2 \mathrm{d} x $$

be defined in $H^1 ( \Omega, \mathbb{R}^N)$. Korn's inequality says that there exists a constant $c$ such that

$$ \mathcal{E} ( v) + \| v \|_{L^2}^2 \geqslant c \| v \|^2_{H^1} $$

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Of all the proofs I've seen, none mentioned exactly the dependence of the constant $c$ of the characteristics of the domain $\Omega$. So my question is:

How does the constant $c$ depend on the characteristics of the domain $\Omega$ (Lipschitz constant, etc)?

Can you point me to a reference which treats this subject in more detail?

(I am interested particularly in the case $N=3$ so if there are some results which work only for this case I don't mind)

Answer 1

I have not worked out the details, but this may be a way to find an answer:

The kernel of your operator $\mathcal E$ consists of rigid body motions, namely translations, reflexions and rotations. This is a finite dimensional space and for a given domain, you can compute the constant for such functions and try to find its minimum. Then you decompose any other function into a rigid body motion and one that is orthogonal to get the result.