Positivity for a Hinged Convex Plate with Stress
Guido H. Sweers, Katerina Vassi · SIAM Journal on Mathematical Analysis · 2018
The boundary value problem for the Kirchhoff--Love model of a hinged elastic plate with stress is as follows: $ \Delta^{2} u - \tau \Delta u = f $ in $\Omega\subset\mathbb{R}^2$, $u= \Delta u -(1- \sigma ) \kappa u_{ u} =0$ on $\partial \Omega$ with weight $f \in L^{2}(\Omega)$, Poisson ratio $\sigma\in (-1,1)$, stress coefficient $\tau \geq 0$, and boundary curvature $\kappa$. We will prove that this problem is positivity preserving on convex domains, meaning $f\geq 0$ implies $u\geq 0$. The proof relies on optimal estimates for a weighted first Steklov eigenvalue and on an application of the Kreĭn--Rutman theorem for an auxiliary problem. The case $\tau =0$ has been studied by Parini and Stylianou [ SIAM J. Math. Anal., 41 (2009), pp. 2031--2037].