On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle

Abstract

Let u(r,θ) be biharmonic and bounded in the circular sector $\text{¦θ¦ <}\text{π}\text{4}\text{, 0 < r < ρ (ρ > 1)}$ and vanish together with $\text{δ}\text{u}\text{δ}\text{θ}$ when $\text{¦θ¦ =}\text{π}\text{4}$. We consider the transform $\text{u}\text{̂}\text{(p,θ) = ∝}{}_0{}^1\text{r}{}^{\mathrm{p\; -\; 1}}\text{u(r,θ)dr}$. We show that for any fixed θ₀ u(p,θ₀) is meromorphic with no real poles and cannot be entire unless u(r, θ₀) ≡ 0. It follows then from a theorem of Doetsch that u(r, θ₀) either vanishes identically or oscillates as r → 0.