4
$\begingroup$

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

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


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)

$\endgroup$
1
  • $\begingroup$ I saw this paper yesterday. Not sure whether you need it or not. If you have an answer please let me know. I am interested in it as well. Anyway, here is the link $\endgroup$
    – JumpJump
    Commented Aug 15, 2015 at 11:53

1 Answer 1

2
$\begingroup$

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.

$\endgroup$

Your Answer

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

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