# How to prove the second Korn inequality?

$$\textbf{Theorem}.1$$ (The first Korn inequality) Suppose that $$\Omega$$ is a bounded domain in $$\mathbb{R}^d$$ with Lipschitz boundary. Then\ $$\begin{eqnarray} \sqrt{2}\left\|\triangledown u\right\|_{L^2(\Omega)}\leq \left\|\triangledown u+(\triangledown u)^T\right\|_{L^2(\Omega)} \end{eqnarray}$$ for any $$u\in H_{0}^{1}(\Omega;\mathbb{R}^d)$$, where $$(\triangledown u)^T$$ denotes the transpose of $$\triangledown u$$.

$$\textbf{Theorem}.2$$ (The second Korn inequality) Suppose that $$\Omega$$ is a bounded domain in $$\mathbb{R}^d$$ with Lipschitz boundary. If $$u\in H^{1}(\Omega,\mathbb{R}^d)$$ is a function with the property that $$u\perp R$$ in $$H^{1}(\Omega;\mathbb{R}^d)$$, then
$$\begin{eqnarray} \int_{\Omega}|\triangledown u|^2dx\leq C\int_{\Omega}|\triangledown u+(\triangledown u)^T|^2dx \end{eqnarray}$$ where $$R=\left\{\phi=Bx+b:B\in\mathbb{R}^{d\times d} \text{ is skew-symmetric and }b\in\mathbb{R}^d\right\}$$ and $$C$$ is a constant.

I recently see the two theorems in a book about elliptic equations. I tried to get the estimate for the second inequality by direct computation which works in the proof of the first Korn inequality, but for this inequality, I cannot combine the condition $$u\perp R$$ with the final results. Can you give me some hints or references?

• not my field and on a quick glance I couldnt find a matching statement, but this claims to have proofs (and it has references too) : fuchsbraun.homepage.t-online.de/media/… (J. A. NITSCHE - On Korn’s second inequality) Commented Sep 4, 2021 at 5:21
• @Calvin Khor Sorry, I cannot open your web link. Can you give me another? Commented Sep 4, 2021 at 8:24
• Sure, searching on zbMATH: https://zbmath.org/?q=J.+A.+NITSCHE+On+Korn’s+second+inequality one finds two links that should be open-access: one is on the EUDML, and one is on some site called esaim-m2an (DOI: doi.org/10.1051/m2an/1981150302371). There is also this on numdam. Hopefully one of these work for you Commented Sep 4, 2021 at 8:29

You can find a full proof (to my knowledge the simpler one currently known) in the paper [1] and in the book [2], chapter I, §2.1 pp. 14-21. The original proof of Arthur Korn is so long and involved that K.O. Friedrichs, who gave a much simpler yet sophisticated proof, had doubts on his validity: starting from the work of Friedrichs, several authors gave their (in general quite complex) proofs, until Olga Oleĭnik gave a much shorter and simpler one (despite being still not elementary).

New edit. While ordering my library, I noted reference [1b]: in this paper Oleĭnik an Kondratiev prove the classical second Korn inequality for bounded domains satisfying the cone condition (theorem 1, a three page proof) and for certain classes of unbounded domains. They also prove that the constants in the inequality are sharp in some precise sense.

References

[1] Vladimir Alexandrovitch Kondratiev, Olga Arsenievna Oleĭnik, "On Korn’s inequalities" (English), Comptes Rendus de l’Académie des Sciences, Série I, 308, No. 16, pp. 483-487 (1989), MR0995908, Zbl 0698.35067.

[1b] Vladimir Alexandrovitch Kondrat’ev, Olga Arsenievna Oleĭnik, "Hardy’s and Korn’s type inequalities and their applications". (English) Rendiconti di Matematica e delle sue Applicazioni, VII Serie 10, No. 3, 641-666 (1990), MR1080319, Zbl 0767.35020, also found in the commemorative book Scritti matematici. Dedicati a Maria Adelaide Sneider, Università "La Sapienza", 415-440 (1990).

[2] Olga Arsenievna Oleĭnik, Alexei Stanislavovich Shamaev, Grigorii Andronikovich Yosifian, Mathematical problems in elasticity and homogenization. (English) Studies in Mathematics and its Applications. 26. Amsterdam-London-New York-Tokyo: North- Holland, pp. xiii+398 (1992), ISBN: 0-444-88441-6, MR1195131, Zbl 0768.73003.

Let us assume Korn's inequality in the usual form $$\int_{\Omega} |\nabla u|^2 \leq C\left (Q(u)+\int_\Omega |u|^2 \right ),$$ with $$Q(u)=\int_\Omega |\nabla u +\nabla u^{T}|^2$$, for every $$u \in H^1(\Omega)$$ (you find it in the paper indicated by @Calvin Khor, for example, but the proof is not so elementary as in the case of $$H^1_0(\Omega)$$). Then the claim follows by contradiction. If not true, we find a sequence or vector fields $$(u_n) \subset R^\perp$$ such that $$1=\|u_n\| \geq n \,Q(u_n).$$ Then $$(u_n)$$ is bounded in $$H^1$$ and we may assume that is converges to $$u_0$$ weakly in $$H^1$$ and strongly in $$L^2$$, with $$u_0 \in R^\perp$$. Then $$Q(u_0)=0$$ which means that $$\nabla u_0=-(\nabla u_0)^T$$ and then $$u_0 \in R$$. Finally, $$u_0 \in R \cap R^\perp$$ gives $$u_0=0$$ but $$\|u_0\|=1$$.