Positivity for the clamped plate equation under high tension

Abstract

In this article, we consider positivity issues for the clamped plate equation with high tension $$\gamma >0$$ γ > 0 . This equation is given by $$\Delta ^2u - \gamma \Delta u=f$$ Δ 2 u - γ Δ u = f under clamped boundary conditions. Here, we show that given a positive f, i.e. upwards pushing, we find a $$\gamma _0>0$$ γ 0 > 0 such that for all $$\gamma \ge \gamma _0$$ γ ≥ γ 0 the bending u is indeed positive. This $$\gamma _0$$ γ 0 only depends on the domain and the ratio of the $$L^1$$ L 1 and $$L^\infty $$ L ∞ norm of f. In contrast to a recent result by Cassani and Tarsia, our approach is valid in all dimensions.