# Equivalence of linear elasticity and biharmonic equations: variational formulation

Wikipedia tells me that the equations for linear elasticity and biharmonic equations have the same solution for Dirichlet boundary condition. How do you show the equivalence in the variational formulation?

The article on Wikipedia does not mention boundary conditions for the biharmonic equation. It is therefore impossible to establish a variational formulation, cfr. this answer. Please also note that Dirichlet b.c. for the biharmonic equation are written in terms of $u_i$ and $\frac{\partial u_i}{\partial n}$ (or some other first order derivative, depending on the problem), while Dirichlet b.c. for linear elasticity are written in terms of $u_i$ alone. It is therefore dubious to claim that
Moreover I had no time to check if the derivation of the biharmonic equation given on Wikipedia is correct (and I do not have access to the classical literature right now) but a lot of ad hoc assumptions are made (e.g. $\lambda = -\beta^2$) so I suspect that the supposed equivalence is far from being general.