When does the ̄rst eigenfunction of the clamped plate equation have a ̄xed sign?

For most `nice' elliptic boundary value problems there is a general expectation that the ̄rst eigenfunction is unique and of ̄xed sign. And indeed, for second order elliptic di®erential equations with Dirichlet boundary conditions such a result holds as a consequence of the maximum principle. It is well known that such a maximum principle does not have a direct generalization to higher order elliptic problems. Nevertheless, the hypothesis that the principal eigenfunction for the biharmonic Dirichlet problem is of ̄xed sign does appear in earlier papers, see for example [32] from 1950. Let us be more precise.