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I wish to find the functional whose minimisation yields the follwoing equation on the vector function u $(\lambda + \mu) \nabla (\nabla \cdot u) + \mu \nabla^{2} u = 0$, the Navier equation of linear elasticity. I know that this equation has a vaiational principle from physical reasons, but struggle to find it. The term containing the laplacian is easier to handle, I can not even prove simmetry of the weak form for the first one: how to integrate by parts (grad div u) v to obtain a term symmetric in u e v? Many thanks

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By a corollary of the divergence theorem: $$\int_\Omega \nabla(\nabla\cdot u)\cdot v = - \int_\Omega \nabla\cdot u\,\nabla\cdot v + \int_{\partial\Omega}(\nabla\cdot g)v\cdot n.$$

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  • $\begingroup$ That is right indeed! Thank you very much $\endgroup$ – Buco Nov 13 '13 at 9:11
  • $\begingroup$ Well, if you think that answer is correct, then marking it so helps this site (such that this question will not appear as unanswered anymore)… $\endgroup$ – Dirk Nov 14 '13 at 12:17
  • $\begingroup$ Where did the $g$ come from? $\endgroup$ – ddddDDDD May 7 '17 at 17:47
  • $\begingroup$ It's from a boundary condition (not present in the question, but in the general form of the theorem). $\endgroup$ – Dirk May 8 '17 at 4:32

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