We consider a biharmonic problem ${\Delta }^2{u}_{\omega }=f_{\omega }$ with Navier type boundary conditions $u_{\omega }=\Delta u_{\omega }=0$, on a family of truncated sectors ${\Omega }_{\omega }$ in ${ℝ}^2$ of radius $r$, $0<r<1$ and opening angle $\omega$, $\omega \in (2\pi /3,\pi ]$ when $\omega$ is close to $\pi$. The family of right-hand sides ${\left(f_{\omega }\right)}_{\omega \in (2\pi /3,\pi ]}$ is assumed to depend smoothly on $\omega$ in $L^2\left({\Omega }_{\omega }\right)$. The main result is that $u_{\omega }$ converges to $u_{\pi }$ when $\omega \to \pi$ with respect to the $H^2$-norm. We can also show that the $H^2$-topology is optimal for such a convergence result.

Let $A$ be the $L^p$ realization ($1\&lt;p\&lt;\infty$) of a differential operator $P(D_x,D_t)$ on ${ℝ}^n\times {ℝ}^{+}$ with general boundary conditions $B_k(D_x,D_t)u(x,0)=0$ ($1\le k\le m$). Here $P$ is a homogeneous polynomial of order $2m$ in $n+1$ complex variables that satisfies a suitable ellipticity condition, and for $1\le k\le m$$B_k$ is a homogeneous polynomial of order...