Variational Principle for an Elliptic equation

Question

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

Answer 1

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