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I'm having trouble trying to deal with the following PDE system:

$\Omega$ is an open bounded set in $\mathbb{R}^n$,

$\mu \sum_{j=1}^{n}{\partial^2u_{i}/\partial x_{j}^{2}} +(\lambda +\mu) \sum_{j=1}^{n}\partial^{2}u_{j}/\partial x_{i} \partial x_{j} = f_{i}$, $f_i \in L^2$, $i=1,2,...,n$, in $\Omega$,

$u=0$ on $\partial\Omega$

I'm trying to prove the existence and uniqueness of it in the following cases:

a) $\lambda>0$, and $\lambda +\mu \geq 0$

b) $\lambda>0$, and $\lambda +2 \mu>0$

It seems that Lax-Milgram Theorem is to be needed but the relevant estimates are hard to prove...Could anyone shed some light on it?

Thanks in advance.

alby
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    See Theorem 8.6.1 in "Uniqueness theorems in linear elasticity" by Knops and Payne. – Biswajit Banerjee Dec 12 '15 at 20:12
  • @BiswajitBanerjee: I was just going to ask that isn't it the elasticity equations and your comment bumped up! :) – Hosein Rahnama Dec 12 '15 at 20:15
  • @Biswajit Banerjee Thanks a lot! But I think the theorem states something different...it actually uses the solution of some other system constructed for the problem to prove the uniqueness, but it seems that the newly constructed system is identical to the original one in this case, and we have not yet obtained the existence of the solution...so how can this method be applied...? – alby Dec 13 '15 at 03:27
  • I'm a novice in functional analysis and can't help with the details. But I can recall that Korn's inequality is invoked at some stage of the proof which I saw a long time ago in B. Daya Reddy's "Introductory Functional Analysis". – Biswajit Banerjee Dec 13 '15 at 04:44

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