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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?

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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

linear elasticity and biharmonic equations have the same solution for Dirichlet boundary condition

if one does not provides some practical mean for defining a well posed problem for the biharmonic equation equivalent to the elastic problem.

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.

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