Biharmonic eigenvalue problems and \(L^ p\) estimates (Q921244)

We study the spectra of the biharmonic operators (i) \(Lu=\Delta^ 2u\), with \(u=0\), \(\Delta u=0\) on the boundary, (ii) \(Lu=\Delta^ 2u\), with \(\partial u/\partial n=0\), \((\partial /\partial n)(\Delta u)=0\) on the boundary, and (iii) \(Lu=\Delta^ 2u\), with \(u=0\), \(\partial u/\partial n=0\) on the boundary. We then obtain \(W^{4,p}\) estimates for the solution u of the associated boundary value problems: (i) \(\Delta^ 2u- \lambda^ 2_ 1u=f\), with \(u=\Delta u=0\) on the boundary, (ii) \(\Delta^ 2u=f\), with \(\partial u/\partial n=(\partial /\partial n)(\Delta u)=0\) on the boundary and (iii) \(\Delta^ 2u-\mu_ 1u=f\), with \(u=(\partial u/\partial n)(\Delta u)=0\) on the boundary. Here \(\lambda^ 2_ 1\) and \(\mu_ 1\) denote the first eigenvalues of the associated eigenvalue problems.