$H^2$ convergence of solutions of a biharmonic problem on a truncated convex sector near the angle $\pi $

Tami, Abdelkader ; Tlemcani, Mounir

Applications of Mathematics, Tome 66 (2021) no. 3, pp. 383-395.

Résumé

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 $\mathbb {R}^2$ of radius $r$, $0$ and opening angle $\omega $, $\omega \in (2\pi /3,\pi ]$ when $\omega $ is close to $\pi $. The family of right-hand sides $(f_\omega )_{\omega \in (2\pi /3,\pi ]}$ is assumed to depend smoothly on $\omega $ in $L^{2}(\Omega _{\omega })$. The main result is that $u_{\omega }$ converges to $u_\pi $ when $ \omega \rightarrow \pi $ with respect to the $H^2$-norm. We can also show that the $H^2$-topology is optimal for such a convergence result.

Détail

DOI : 10.21136/AM.2021.0284-19

Classification : 35B40, 35B45, 35J25, 35J40, 35J75, 35Q99
Keywords: sector; convex; biharmonic; elliptic; singularity; convergence; Sobolev space