Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
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:
[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. DOI 10.1002/mma.1670020416 | MR 0595625 | Zbl 0445.35023
[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. DOI 10.1017/S0308210500021272 | MR 1204855 | Zbl 0791.35032
[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. DOI 10.1017/S0308210500021284 | MR 1204855 | Zbl 0791.35033
[4] Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Smoothness and Asymptotics of Solutions. Lecture Notes in Mathematics 1341. Springer, Berlin (1988). DOI 10.1007/BFb0086682 | MR 0961439 | Zbl 0668.35001
[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. DOI 10.1051/m2an/1990240100271 | MR 1034897 | Zbl 0691.35023
[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). DOI 10.1007/978-3-642-12245-3 | MR 2667016 | Zbl 1239.35002
[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. MR 0312068 | Zbl 0277.35035
[8] Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Monograhs and Studies in Mathematics 24. Pitman, Boston (1985). DOI 10.1137/1.9781611972030 | MR 0775683 | Zbl 0695.35060
[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. MR 0226187 | Zbl 0194.13405
[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. DOI 10.1002/mana.19780810103 | MR 0492821 | Zbl 0554.35035
[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. MR 0514327 | Zbl 0453.35025
[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. DOI 10.1002/mana.19921550115 | MR 1231265 | Zbl 0794.35039
[13] Nicaise, S.: Polygonal interface problems for the biharmonic operator. Maths. Methods Appl. Sci. 17 (1994), 21-39. DOI 10.1002/mma.1670170104 | MR 1257586 | Zbl 0820.35041
[14] Nicaise, S., Sändig, A.-M.: General interface problems I. Math. Methods Appl. Sci. 17 (1994), 395-429. DOI 10.1002/mma.1670170602 | MR 1274152 | Zbl 0824.35014
[15] Nicaise, S., Sändig, A.-M.: General interface problems II. Math. Methods Appl. Sci. 17 (1994), 431-450. DOI 10.1002/mma.1670170603 | MR 1274152 | Zbl 0824.35015
[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
[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.: The elliptic problems in a family of planar open sets. Appl. Math., Praha 64 (2019), 485-499. DOI 10.21136/AM.2019.0057-19 | MR 4022159 | Zbl 07144725
Partner of
EuDML logo