Applications of Mathematics, Vol. 66, No. 3, pp. 383-395, 2021
$H^2$ convergence of solutions of a biharmonic problem on a truncated convex sector near the angle $\pi$
Abdelkader Tami, Mounir Tlemcani
Received October 27, 2019. Published online March 5, 2021.
Abstract: We consider a biharmonic problem $\Delta^2u_{\omega}=f_\omega$ 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 $r$, $0<r<1$ 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.
Keywords: sector, convex, biharmonic, elliptic, singularity, convergence, Sobolev space
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Affiliations: Abdelkader Tami (corresponding author), Mounir Tlemcani, Département de Mathématiques, Université des Sciences et de la Technologie d'Oran Mohamed-Boudiaf USTOMB, El Mnaouar, BP 1505, Bir El Djir 31000, Oran, Algeria, e-mail: abdelkader21fr@gmail.com, abdelkader.tami@univ-usto.dz, mounir.tlemcani@gmail.com
