A shape optimization approach for a class of free boundary problems of Bernoulli type

Abdesslam Boulkhemair; Abdeljalil Nachaoui; Abdelkrim Chakib

Applications of Mathematics (2013)

  • Volume: 58, Issue: 2, page 205-221
  • ISSN: 0862-7940

Abstract

top
We are interested in an optimal shape design formulation for a class of free boundary problems of Bernoulli type. We show the existence of the optimal solution of this problem by proving continuity of the solution of the state problem with respect to the domain. The main tools in establishing such a continuity are a result concerning uniform continuity of the trace operator with respect to the domain and a recent result on the uniform Poincaré inequality for variable domains.

How to cite

top

Boulkhemair, Abdesslam, Nachaoui, Abdeljalil, and Chakib, Abdelkrim. "A shape optimization approach for a class of free boundary problems of Bernoulli type." Applications of Mathematics 58.2 (2013): 205-221. <http://eudml.org/doc/252510>.

@article{Boulkhemair2013,
abstract = {We are interested in an optimal shape design formulation for a class of free boundary problems of Bernoulli type. We show the existence of the optimal solution of this problem by proving continuity of the solution of the state problem with respect to the domain. The main tools in establishing such a continuity are a result concerning uniform continuity of the trace operator with respect to the domain and a recent result on the uniform Poincaré inequality for variable domains.},
author = {Boulkhemair, Abdesslam, Nachaoui, Abdeljalil, Chakib, Abdelkrim},
journal = {Applications of Mathematics},
keywords = {shape optimization; Bernoulli; free boundary problem; exterior Bernoulli problem; optimal solution; state problem; continuity of the state problem; uniform tubular neighbourhood; diffeomorphism; uniform trace theorem; uniform Poincaré inequality; shape optimization; free boundary problem; exterior Bernoulli problem; optimal solution; state problem; uniform tubular neighbourhood; diffeomorphism; uniform trace theorem; uniform Poincaré inequality},
language = {eng},
number = {2},
pages = {205-221},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A shape optimization approach for a class of free boundary problems of Bernoulli type},
url = {http://eudml.org/doc/252510},
volume = {58},
year = {2013},
}

TY - JOUR
AU - Boulkhemair, Abdesslam
AU - Nachaoui, Abdeljalil
AU - Chakib, Abdelkrim
TI - A shape optimization approach for a class of free boundary problems of Bernoulli type
JO - Applications of Mathematics
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 2
SP - 205
EP - 221
AB - We are interested in an optimal shape design formulation for a class of free boundary problems of Bernoulli type. We show the existence of the optimal solution of this problem by proving continuity of the solution of the state problem with respect to the domain. The main tools in establishing such a continuity are a result concerning uniform continuity of the trace operator with respect to the domain and a recent result on the uniform Poincaré inequality for variable domains.
LA - eng
KW - shape optimization; Bernoulli; free boundary problem; exterior Bernoulli problem; optimal solution; state problem; continuity of the state problem; uniform tubular neighbourhood; diffeomorphism; uniform trace theorem; uniform Poincaré inequality; shape optimization; free boundary problem; exterior Bernoulli problem; optimal solution; state problem; uniform tubular neighbourhood; diffeomorphism; uniform trace theorem; uniform Poincaré inequality
UR - http://eudml.org/doc/252510
ER -

References

top
  1. Acker, A., 10.1137/0512017, SIAM J. Math. Anal. 12 169-172 (1981). (1981) Zbl0456.49007MR0605427DOI10.1137/0512017
  2. Acker, A., Meyer, R., A free boundary problem for the p -Laplacian: uniqueness, convexity, and successive approximation of solutions, Electron. J. Differ. Equ. 8 1-20 (1995). (1995) Zbl0820.35037
  3. Alt, H. W., Caffarelli, L. A., Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 105-144 (1981). (1981) Zbl0449.35105MR0618549
  4. Beurling, A., On free-boundary problems for the Laplace equations, Sem. analytic functions 1 248-263 (1958). (1958) Zbl0099.08302
  5. Boulkhemair, A., Chakib, A., 10.1080/03605300600910241, Commun. Partial Differ. Equations 32 1439-1447 (2007). (2007) Zbl1128.26010MR2354499DOI10.1080/03605300600910241
  6. Boulkhemair, A., Chakib, A., Nachaoui, A., Uniform trace theorem and application to shape optimization, Appl. Comput. Math. 7 192-205 (2008). (2008) Zbl1209.49060MR2537339
  7. Cardaliaguet, P., Tahraoui, R., Some uniqueness results for Bernoulli interior free-boundary problems in convex domains, Electron. J. Differ. Equ. 102 1-16 (2002). (2002) Zbl1029.35227MR1949576
  8. Chenais, D., 10.1016/0022-247X(75)90091-8, J. Math. Anal. Appl. 52 189-219 1975. Zbl0317.49005MR0385666DOI10.1016/0022-247X(75)90091-8
  9. Crouzeix, M., Variational approach of a magnetic shaping problem, Eur. J. Mech, B 10 527-536 (1991). (1991) Zbl0741.76089MR1139607
  10. Descloux, J., Stability of the solutions of the bidimensional magnetic shaping problem in absence of surface tension, Eur. J. Mech, B 10 513-526 (1991). (1991) Zbl0741.76025MR1139606
  11. Fasano, A., Some free boundary problems with industrial applications, Shape optimization and free boundaries, Proc. NATO ASI, Sémin. Math. Supér., Montréal/Can. 1990 113-142 (1992). (1992) Zbl0765.76005MR1260974
  12. Flucher, M., Rumpf, M., Bernoulli's free-boundary problem, qualitative theory and numerical approximation, J. Reine Angew. Math. 486 165-205 (1997). (1997) Zbl0909.35154MR1450755
  13. Friedman, A., Free boundary problem in fluid dynamics, variational methods for equilibrium problems of fluids, Astérisque 118 55-67 (1984). (1984) MR0761737
  14. Haslinger, J., Kozubek, T., Kunisch, K., Peichl, G., 10.1023/A:1026095405906, Comput. Optim. Appl. 26 231-251 (2003). (2003) Zbl1077.49030MR2013364DOI10.1023/A:1026095405906
  15. Haslinger, J., Kozubek, T., Kunisch, K., Peichl, G., 10.1016/j.jmaa.2003.10.038, J. Math. Anal. Appl. 290 665-685 (2004). (2004) Zbl1034.49042MR2033050DOI10.1016/j.jmaa.2003.10.038
  16. Henrot, A., Pierre, M., 10.1007/3-540-37689-5, Springer Berlin (2005), French. (2005) MR2512810DOI10.1007/3-540-37689-5
  17. Ito, K., Kunisch, K., Peichl, G., 10.1016/j.jmaa.2005.03.100, J. Math. Anal. Appl. 314 126-149 (2006). (2006) Zbl1088.49028MR2183542DOI10.1016/j.jmaa.2005.03.100
  18. Kinderlehrer, D., Nirenberg, L., Regularity in free boundary problems, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 4 373-391 (1977). (1977) Zbl0352.35023MR0440187
  19. Lewy, H., 10.1090/S0002-9939-1952-0049399-9, Proc. Am. Math. Soc. 3 111-113 (1952). (1952) Zbl0046.41706MR0049399DOI10.1090/S0002-9939-1952-0049399-9
  20. Pironneau, O., Optimal Shape Design for Elliptic Systems. Springer Series in Computational Physics, Springer New York (1984). (1984) MR0725856

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.