H 2 convergence of solutions of a biharmonic problem on a truncated convex sector near the angle π

Abdelkader Tami; Mounir Tlemcani

Applications of Mathematics (2021)

  • Volume: 66, Issue: 3, page 383-395
  • ISSN: 0862-7940

Abstract

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We consider a biharmonic problem Δ 2 u ω = f ω with Navier type boundary conditions u ω = Δ u ω = 0 , on a family of truncated sectors Ω ω in ℝ 2 of radius r , 0 < r < 1 and opening angle ω , ω ∈ ( 2 π / 3 , π ] when ω is close to π . The family of right-hand sides ( f ω ) ω ∈ ( 2 π / 3 , π ] is assumed to depend smoothly on ω in L 2 ( Ω ω ) . The main result is that u ω converges to u π when ω → π with respect to the H 2 -norm. We can also show that the H 2 -topology is optimal for such a convergence result.

How to cite

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Tami, Abdelkader, and Tlemcani, Mounir. "$H^2$ convergence of solutions of a biharmonic problem on a truncated convex sector near the angle $\pi $." Applications of Mathematics 66.3 (2021): 383-395. <http://eudml.org/doc/297647>.

@article{Tami2021,
abstract = {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.},
author = {Tami, Abdelkader, Tlemcani, Mounir},
journal = {Applications of Mathematics},
keywords = {sector; convex; biharmonic; elliptic; singularity; convergence; Sobolev space},
language = {eng},
number = {3},
pages = {383-395},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$H^2$ convergence of solutions of a biharmonic problem on a truncated convex sector near the angle $\pi $},
url = {http://eudml.org/doc/297647},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Tami, Abdelkader
AU - Tlemcani, Mounir
TI - $H^2$ convergence of solutions of a biharmonic problem on a truncated convex sector near the angle $\pi $
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 3
SP - 383
EP - 395
AB - 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.
LA - eng
KW - sector; convex; biharmonic; elliptic; singularity; convergence; Sobolev space
UR - http://eudml.org/doc/297647
ER -

References

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