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

Abdelkader Tami; Mounir Tlemcani

Applications of Mathematics (2021)

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

Abstract

top
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.

How to cite

top

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},
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
UR - http://eudml.org/doc/297647
ER -

References

top
  1. Blum, H., Rannacher, R., 10.1002/mma.1670020416, Math. Methods Appl. Sci. 2 (1980), 556-581. (1980) Zbl0445.35023MR0595625DOI10.1002/mma.1670020416
  2. Costabel, M., Dauge, M., 10.1017/S0308210500021272, Proc. R. Soc. Edinb., Sect. A 123 (1993), 109-155. (1993) Zbl0791.35032MR1204855DOI10.1017/S0308210500021272
  3. Costabel, M., Dauge, M., 10.1017/S0308210500021284, Proc. R. Soc. Edinb., Sect. A 123 (1993), 157-184. (1993) Zbl0791.35033MR1204855DOI10.1017/S0308210500021284
  4. Dauge, M., 10.1007/BFb0086682, Lecture Notes in Mathematics 1341. Springer, Berlin (1988). (1988) Zbl0668.35001MR0961439DOI10.1007/BFb0086682
  5. Dauge, M., Nicaise, S., Bourlard, M., Lubuma, J. M.-S., 10.1051/m2an/1990240100271, RAIRO, Modélisation Math. Anal. Numér. 24 (1990), 27-52 French. (1990) Zbl0691.35023MR1034897DOI10.1051/m2an/1990240100271
  6. Gazzola, F., Grunau, H.-C., Sweers, G., 10.1007/978-3-642-12245-3, Lecture Notes in Mathematics 1991. Springer, Berlin (2010). (2010) Zbl1239.35002MR2667016DOI10.1007/978-3-642-12245-3
  7. Grisvard, P., Alternative de Fredholm rélative au problème de Dirichlet dans un polygone ou un polyèdre, Boll. Unione Mat. Ital., IV. Ser. 5 (1972), 132-164 French. (1972) Zbl0277.35035MR0312068
  8. Grisvard, P., 10.1137/1.9781611972030, Monograhs and Studies in Mathematics 24. Pitman, Boston (1985). (1985) Zbl0695.35060MR0775683DOI10.1137/1.9781611972030
  9. Kondrat'ev, V. A., Boundary problems for elliptic equation in domains with conical or angular points, Trans. Mosc. Math. Soc. 16 (1967), 227-313 translation from Tr. Mosk. Mat. O.-va 16 1967 209-292. (1967) Zbl0194.13405MR0226187
  10. Maz'ya, V. G., Plamenevskij, B. A., 10.1002/mana.19780810103, Transl., Ser. 2, Am. Math. Soc. 123 (1984), 1-56 translation from Math. Nachr. 81 1978 25-82. (1984) Zbl0554.35035MR0492821DOI10.1002/mana.19780810103
  11. Maz'ya, V. G., Plamenevskij, B. A., $L_p$-estimates of solutions of elliptic boundary value problems in domains with edges, Trans. Mosc. Math. Soc. 1 (1980), 49-97 translation from Tr. Mosk. Mat. O.-va 37 1978 49-93. (1980) Zbl0453.35025MR0514327
  12. Maz'ya, V. G., Rossmann, J., 10.1002/mana.19921550115, Math. Nachr. 155 (1992), 199-220. (1992) Zbl0794.35039MR1231265DOI10.1002/mana.19921550115
  13. Nicaise, S., 10.1002/mma.1670170104, Maths. Methods Appl. Sci. 17 (1994), 21-39. (1994) Zbl0820.35041MR1257586DOI10.1002/mma.1670170104
  14. Nicaise, S., Sändig, A.-M., 10.1002/mma.1670170602, Math. Methods Appl. Sci. 17 (1994), 395-429. (1994) Zbl0824.35014MR1274152DOI10.1002/mma.1670170602
  15. Nicaise, S., Sändig, A.-M., 10.1002/mma.1670170603, Math. Methods Appl. Sci. 17 (1994), 431-450. (1994) Zbl0824.35015MR1274152DOI10.1002/mma.1670170603
  16. Stylianou, A., Comparison and Sign Preserving Properties of Bilaplace Boundary Value Problems in Domains with Corners. PhD Thesis, Universität Köln, München (2010). (2010) Zbl1297.35006
  17. Tami, A., Etude d'un problème pour le bilaplacien dans une famille d'ouverts du plan, PhD Thesis. Aix-Marseille University, Marseille, 2016. Available at https://www.theses.fr/2016AIXM4362\kern0pt French. 
  18. Tami, A., 10.21136/AM.2019.0057-19, Appl. Math., Praha 64 (2019), 485-499. (2019) Zbl07144725MR4022159DOI10.21136/AM.2019.0057-19

NotesEmbed ?

top

You must be logged in to post comments.