| Title:
|
The elliptic problems in a family of planar open sets (English) |
| Author:
|
Tami, Abdelkader |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
64 |
| Issue:
|
5 |
| Year:
|
2019 |
| Pages:
|
485-499 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We propose, on a model case, a new approach to classical results obtained by V. A. Kondrat'ev (1967), P. Grisvard (1972), (1985), H. Blum and R. Rannacher (1980), V. G. Maz'ya (1980), (1984), (1992), S. Nicaise (1994a), (1994b), (1994c), M. Dauge (1988), (1990), (1993a), (1993b), A. Tami (2016), and others, describing the singularities of solutions of an elliptic problem on a polygonal domain of the plane that may appear near a corner. It provides a more precise description of how the solutions decompose, puts into evidence the analogy of such decompositions with standard Taylor expansions, and gives uniform estimates with respect to the angle parameter. This last property allows the treatment of families of elliptic problems on families of open sets. (English) |
| Keyword:
|
biharmonic operator |
| Keyword:
|
elliptic problems |
| Keyword:
|
nonsmooth boundaries |
| Keyword:
|
uniform singularity estimates |
| Keyword:
|
Sobolev spaces |
| MSC:
|
35B40 |
| MSC:
|
35B45 |
| MSC:
|
35J25 |
| MSC:
|
35J40 |
| MSC:
|
35J75 |
| MSC:
|
35Q99 |
| idZBL:
|
07144725 |
| idMR:
|
MR4022159 |
| DOI:
|
10.21136/AM.2019.0057-19 |
| . |
| Date available:
|
2019-10-16T10:57:52Z |
| Last updated:
|
2021-11-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147846 |
| . |
| 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] Grisvard, P.: Alternative de Fredholm relative 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:
|
[7] Grisvard, P.: Elliptic Problems in Nonsmooth Domains.Monographs and Studies in Mathematics 24, Pitman Advanced Publishing Program, Pitman Publishing, Boston (1985). Zbl 0695.35060, MR 0775683, 10.1137/1.9781611972030 |
| Reference:
|
[8] Kondrat'ev, V. A.: Boundary problems for elliptic equations in domains with conical or angular points.Trans. Mosc. Math. Soc. 16 (1967), 227-313 Translated from Trudy Moskov. Mat. Obšč. 16 1967 209-292. Zbl 0194.13405, MR 0226187 |
| Reference:
|
[9] 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. (1980), 49-97. Zbl 0453.35025, MR 0514327 |
| Reference:
|
[10] Maz'ya, V. G., Plamenevskij, B. A.: Estimates in $L_p$ and in Hölder classes and the Miranda-Agmon maximum principle for the solutions of elliptic boundary value problems in domains with singular points on the boundary.Transl., Ser. 2, Am. Math. Soc. 123 (1984), 1-56 Translated from Math. Nachr. 81 1978 25-82. Zbl 0554.35035, MR 0492821, 10.1002/mana.19780810103 |
| Reference:
|
[11] Maz'ya, V., 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:
|
[12] Nicaise, S.: Polygonal interface problems for the biharmonic operator.Math. Methods Appl. Sci. 17 (1994), 21-39. Zbl 0820.35041, MR 1257586, 10.1002/mma.1670170104 |
| Reference:
|
[13] 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:
|
[14] 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:
|
[15] Tami, A.: Etude d'un problème pour le bilaplacien dans une famille d'ouverts du plan.Ph.D. Thesis, Aix-Marseille University France (2016). Available at https://www.theses.fr/224126822 French. |
| . |
