Counter-example of Necas inequality

Question

Here is Necas inequality :

Let $\Omega$ be open Lipschitz domain of $\mathbb R^d$. Then there is $C>0$ that depends only on $\Omega$ such that: $$\left \| P \right \| _{L_2(\Omega)} \leqslant C \left \| P \right \| _{\chi(\Omega)} \quad\forall P \in L_2^{0}(\Omega). $$

with the notation

$\\ L_2^{0}(\Omega)=\{P\in L^2(\Omega);\; \int_{\Omega}P(x)dx=0\}$

$\chi(\Omega):=\{P \in H^{-1}(\Omega) ,\nabla P \in (H^{-1}(\Omega))^N\}$ and $\left \| P \right \| _{\chi(\Omega)} = \left \| P \right \|_{H^{-1}(\Omega)}+\left \| \nabla P \right \| _{(H^{-1}(\Omega))^N}$ .

My question is: How can I find a Counter-example of Necas inequality, for an irregular domain (non Lipschitz domain)?

thank you for your help.

Answer 1

Mine is not the best answer but it's a start... Necas' inequality is used to prove Korn's inequality. In turn Korn's inequality implies Poincare's inequality (see Theorem 2.2 in this paper, so if Poincare's inequality fails, so does Necas'. The standard example of irregular domains where Poincare;s inequality fails is a domain made of rooms and corridors. You can find it either in Mazya's book on Sobolev spaces or in Leoni's book on Sobolev spaces. The original paper where this example was first introduced is L. E. Fraenkel On Regularity of the Boundary in the Theory of Sobolev Spaces, 1979.