Previous |  Up |  Next

Article

Title: $H^2$ convergence of solutions of a biharmonic problem on a truncated convex sector near the angle $\pi $ (English)
Author: Tami, Abdelkader
Author: Tlemcani, Mounir
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 66
Issue: 3
Year: 2021
Pages: 383-395
Summary lang: English
.
Category: math
.
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. (English)
Keyword: sector
Keyword: convex
Keyword: biharmonic
Keyword: elliptic
Keyword: singularity
Keyword: convergence
Keyword: Sobolev space
MSC: 35B40
MSC: 35B45
MSC: 35J25
MSC: 35J40
MSC: 35J75
MSC: 35Q99
DOI: 10.21136/AM.2021.0284-19
.
Date available: 2021-05-20T13:34:42Z
Last updated: 2021-06-07
Stable URL: http://hdl.handle.net/10338.dmlcz/148900
.
Reference: [1] Blum, H., Rannacher, R.: On the boundary value problem of the biharmonic operator on domains with angular corners.Math. Methods Appl. Sci. 2 (1980), 556-581. Zbl 0445.35023, MR 0595625, 10.1002/mma.1670020416
Reference: [2] Costabel, M., Dauge, M.: General edge asymptotics of solutions of second-order elliptic boundary value problems I.Proc. R. Soc. Edinb., Sect. A 123 (1993), 109-155. Zbl 0791.35032, MR 1204855, 10.1017/S0308210500021272
Reference: [3] Costabel, M., Dauge, M.: General edge asymptotics of solutions of second-order elliptic boundary value problems II.Proc. R. Soc. Edinb., Sect. A 123 (1993), 157-184. Zbl 0791.35033, MR 1204855, 10.1017/S0308210500021284
Reference: [4] Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Smoothness and Asymptotics of Solutions.Lecture Notes in Mathematics 1341. Springer, Berlin (1988). Zbl 0668.35001, MR 0961439, 10.1007/BFb0086682
Reference: [5] Dauge, M., Nicaise, S., Bourlard, M., Lubuma, J. M.-S.: Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques I.: Résultats généraux pour le problème de Dirichlet.RAIRO, Modélisation Math. Anal. Numér. 24 (1990), 27-52 French. Zbl 0691.35023, MR 1034897, 10.1051/m2an/1990240100271
Reference: [6] Gazzola, F., Grunau, H.-C., Sweers, G.: Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains.Lecture Notes in Mathematics 1991. Springer, Berlin (2010). Zbl 1239.35002, MR 2667016, 10.1007/978-3-642-12245-3
Reference: [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. Zbl 0277.35035, MR 0312068
Reference: [8] Grisvard, P.: Elliptic Problems in Nonsmooth Domains.Monograhs and Studies in Mathematics 24. Pitman, Boston (1985). Zbl 0695.35060, MR 0775683, 10.1137/1.9781611972030
Reference: [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. Zbl 0194.13405, MR 0226187
Reference: [10] Maz'ya, V. G., Plamenevskij, B. A.: Estimates in $L_p$ and in Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary.Transl., Ser. 2, Am. Math. Soc. 123 (1984), 1-56 translation from Math. Nachr. 81 1978 25-82. Zbl 0554.35035, MR 0492821, 10.1002/mana.19780810103
Reference: [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. Zbl 0453.35025, MR 0514327
Reference: [12] Maz'ya, V. G., Rossmann, J.: On a problem of Babuška. (Stable asymptotics of the solution to the Dirichlet problem for elliptic equations of second order in domains with angular points).Math. Nachr. 155 (1992), 199-220. Zbl 0794.35039, MR 1231265, 10.1002/mana.19921550115
Reference: [13] Nicaise, S.: Polygonal interface problems for the biharmonic operator.Maths. Methods Appl. Sci. 17 (1994), 21-39. Zbl 0820.35041, MR 1257586, 10.1002/mma.1670170104
Reference: [14] Nicaise, S., Sändig, A.-M.: General interface problems I.Math. Methods Appl. Sci. 17 (1994), 395-429. Zbl 0824.35014, MR 1274152, 10.1002/mma.1670170602
Reference: [15] Nicaise, S., Sändig, A.-M.: General interface problems II.Math. Methods Appl. Sci. 17 (1994), 431-450. Zbl 0824.35015, MR 1274152, 10.1002/mma.1670170603
Reference: [16] Stylianou, A.: Comparison and Sign Preserving Properties of Bilaplace Boundary Value Problems in Domains with Corners. PhD Thesis.Universität Köln, München (2010). Zbl 1297.35006
Reference: [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.
Reference: [18] Tami, A.: The elliptic problems in a family of planar open sets.Appl. Math., Praha 64 (2019), 485-499. Zbl 07144725, MR 4022159, 10.21136/AM.2019.0057-19
.

Fulltext not available (moving wall 24 months)

Partner of
EuDML logo