On the differential equation uxxxx + uyyyy = ¿ for an anisotropic stiff material

We study the differential operator $L=\frac{\partial^4}{\partial x^4}+\frac{\partial^4}{\partial y^4}$ and investigate positivity preserving properties in the sense that $f\geq 0$ implies that solutions $u$ of $Lu-\lambda u=f$ are nonnegative. Since the operator is of fourth order we have no maximum principle at our disposal. The operator models the deformation of an anisotropic stiff material like a wire fabric, and it has to be complemented by appropriate boundary conditions. Our operator was introduced by Jacob II Bernoulli as the operator that supposedly models the vibrations of an elastic plate. This model was later revised by Kirchhoff, because the operator and its solutions were anisotropic. Modern materials, however, are often anisotropic, and therefore the old model of Bernoulli deserves an updated investigation. It turns out that even our apparently simple model problem contains some hard analytical challenges. Key words. orthotropic plate, anisotropic operator, vibrations, spectrum, fourth order elliptic, clamped and hinged boundary condition, positivity of the operator, Green's function, Kirchhoff plate AMS Subject Classifications. 35J35 , 35J40 , 35P10 , 74E10 , 74G55 , 74G40 , 74H45 , 74K10 , 74K20