Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-24T12:21:52.219Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Bernoulli problem with geometric constraints

Published online by Cambridge University Press:  02 December 2010

Antoine Laurain  and
Yannick Privat
Antoine Laurain
Affiliation:
Karl-Franzens-University of Graz, Department of Mathematics and Scientific Computing, Heinrichstrasse 36, 8010 Graz, Austria. [email protected]
Yannick Privat
Affiliation:
IRMAR, ENS Cachan Bretagne, Univ. Rennes 1, CNRS, UEB, av. Robert Schuman, 35170 Bruz, France; [email protected]
Get access

Abstract

A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane  {x1 = 0}  where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints.

Keywords

Free boundary problemBernoulli conditionshape optimization
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 1 , January 2012 , pp. 157 - 180
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acker, A., An extremal problem involving current flow through distributed resistance. SIAM J. Math. Anal. 12 (1981) 169–172. Google Scholar
Atkinson, C. and Champion, C.R., Some boundary-value problems for the equation ∇·(|∇ϕ| N∇ϕ) = 0. Quart. J. Mech. Appl. Math. 37 (1984) 401–419. Google Scholar
A. Beurling, On free boundary problems for the Laplace equation, Seminars on analytic functions 1. Institute for Advanced Studies, Princeton (1957).
Bouchon, F., Clain, S. and Touzani, R., Numerical solution of the free boundary Bernoulli problem using a level set formulation. Comput. Methods Appl. Mech. Eng. 194 (2005) 3934–3948. Google Scholar
Dancer, E.N. and Daners, D., Domain perturbation for elliptic equations subject to Robin boundary conditions. J. Differ. Equ. 138 (1997) 86–132. Google Scholar
M.C. Delfour and J.-P. Zolésio, Shapes and geometries – Analysis, differential calculus, and optimization, Advances in Design and Control 4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001).
L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence (1998).
A. Fasano, Some free boundary problems with industrial applications, in Shape optimization and free boundaries (Montreal, PQ, 1990), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 380, Kluwer Acad. Publ., Dordrecht (1992) 113–142.
Flucher, M. and Rumpf, M., Bernoulli’s free boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math. 486 (1997) 165–204. Google Scholar
Friedman, A., Free boundary problem in fluid dynamics, in Variational methods for equilibrium problems of fluids, Trento 1983, Astérisque 118 (1984) 55–67. Google Scholar
Friedman, A., Free boundary problems in science and technology. Notices Amer. Math. Soc. 47 (2000) 854–861. Google Scholar
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman (Advanced Publishing Program), Boston (1985).
Haslinger, J., Kozubek, T., Kunisch, K. and Peichl, G., Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type. Comp. Optim. Appl. 26 (2003) 231–251. Google Scholar
Haslinger, J., Ito, K., Kozubek, T., Kunisch, K. and Peichl, G., On the shape derivative for problems of Bernoulli type. Interfaces in Free Boundaries 11 (2009) 317–330. Google Scholar
A. Henrot and M. Pierre, Variation et optimisation de formes – Une analyse géométrique, Mathématiques & Applications 48. Springer, Berlin (2005).
Henrot, A. and Shahgholian, H., Existence of classical solutions to a free boundary problem for the p-Laplace operator. I. The exterior convex case. J. Reine Angew. Math. 521 (2000) 85–97. Google Scholar
Henrot, A. and Shahgholian, H., Existence of classical solutions to a free boundary problem for the p-Laplace operator. II. The interior convex case. Indiana Univ. Math. J. 49 (2000) 311–323. Google Scholar
Henrot, A. and Shahgholian, H., The one phase free boundary problem for the p-Laplacian with non-constant Bernoulli boundary condition. Trans. Amer. Math. Soc. 354 (2002) 2399–2416. Google Scholar
Ito, K., Kunisch, K. and Peichl, G.H., Variational approach to shape derivatives for a class of Bernoulli problems. J. Math. Anal. Appl. 314 (2006) 126–149. Google Scholar
C.T. Kelley, Iterative methods for optimization, Frontiers in Applied Mathematics 18. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999).
Kondratév, V.A., Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč. 16 (1967) 209–292. Google Scholar
Kuster, C.M., Gremaud, P.A. and Touzani, R., Fast numerical methods for Bernoulli free boundary problems. SIAM J. Sci. Comput. 29 (2007) 622–634. Google Scholar
Lamboley, J. and Novruzi, A., Polygons as optimal shapes with convexity constraint. SIAM J. Control Optim. 48 (2009) 3003–3025. Google Scholar
Lindgren, E. and Privat, Y., A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition. Nonlinear Anal. 67 (2007) 2497–2505. Google Scholar
J. Nocedal and S.J. Wright, Numerical optimization. Springer Series in Operations Research and Financial Engineering, Springer, New York, 2nd edition (2006).
Philip, J.R., n-diffusion. Austral. J. Phys. 14 (1961) 1–13. Google Scholar
J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization : Shape sensitivity analysis, Springer Series in Computational Mathematics 16. Springer-Verlag, Berlin (1992).