Article
MSC:
35B40,
35B45,
35J25,
35J40,
35J75,
35Q99 | MR 4263157 | Zbl 07361061 | DOI: 10.21136/AM.2021.0284-19
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Keywords:
sector; convex; biharmonic; elliptic; singularity; convergence; Sobolev space
sector; convex; biharmonic; elliptic; singularity; convergence; Sobolev space
Summary:
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 $r$, $0<r<1$ 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.
References:
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