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.
Mots clés : free boundary problem, Bernoulli condition, shape optimization
@article{COCV_2012__18_1_157_0,
author = {Laurain, Antoine and Privat, Yannick},
title = {On a {Bernoulli} problem with geometric constraints},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {157--180},
publisher = {EDP-Sciences},
volume = {18},
number = {1},
year = {2012},
doi = {10.1051/cocv/2010049},
mrnumber = {2887931},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2010049/}
}
TY - JOUR AU - Laurain, Antoine AU - Privat, Yannick TI - On a Bernoulli problem with geometric constraints JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 157 EP - 180 VL - 18 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010049/ DO - 10.1051/cocv/2010049 LA - en ID - COCV_2012__18_1_157_0 ER -
%0 Journal Article %A Laurain, Antoine %A Privat, Yannick %T On a Bernoulli problem with geometric constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 157-180 %V 18 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2010049/ %R 10.1051/cocv/2010049 %G en %F COCV_2012__18_1_157_0
Laurain, Antoine; Privat, Yannick. On a Bernoulli problem with geometric constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 157-180. doi : 10.1051/cocv/2010049. http://www.numdam.org/articles/10.1051/cocv/2010049/
[1] , An extremal problem involving current flow through distributed resistance. SIAM J. Math. Anal. 12 (1981) 169-172. | MR | Zbl
[2] and , Some boundary-value problems for the equation ∇·(|∇ϕ| N∇ϕ) = 0. Quart. J. Mech. Appl. Math. 37 (1984) 401-419. | MR | Zbl
[3] , On free boundary problems for the Laplace equation, Seminars on analytic functions 1. Institute for Advanced Studies, Princeton (1957). | Zbl
[4] , and , Numerical solution of the free boundary Bernoulli problem using a level set formulation. Comput. Methods Appl. Mech. Eng. 194 (2005) 3934-3948. | MR | Zbl
[5] and , Domain perturbation for elliptic equations subject to Robin boundary conditions. J. Differ. Equ. 138 (1997) 86-132. | MR | Zbl
[6] and , Shapes and geometries - Analysis, differential calculus, and optimization, Advances in Design and Control 4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001). | MR | Zbl
[7] , Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence (1998). | MR | Zbl
[8] , 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. | MR | Zbl
[9] and , Bernoulli's free boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math. 486 (1997) 165-204. | EuDML | MR | Zbl
[10] , Free boundary problem in fluid dynamics, in Variational methods for equilibrium problems of fluids, Trento 1983, Astérisque 118 (1984) 55-67. | Numdam | MR | Zbl
[11] , Free boundary problems in science and technology. Notices Amer. Math. Soc. 47 (2000) 854-861. | MR | Zbl
[12] , Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman (Advanced Publishing Program), Boston (1985). | MR | Zbl
[13] , , and , Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type. Comp. Optim. Appl. 26 (2003) 231-251. | MR | Zbl
[14] , , , and , On the shape derivative for problems of Bernoulli type. Interfaces in Free Boundaries 11 (2009) 317-330. | MR | Zbl
[15] and , Variation et optimisation de formes - Une analyse géométrique, Mathématiques & Applications 48. Springer, Berlin (2005). | MR | Zbl
[16] and , 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. | MR | Zbl
[17] and , 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. | MR | Zbl
[18] and , The one phase free boundary problem for the p-Laplacian with non-constant Bernoulli boundary condition. Trans. Amer. Math. Soc. 354 (2002) 2399-2416. | MR | Zbl
[19] , and , Variational approach to shape derivatives for a class of Bernoulli problems. J. Math. Anal. Appl. 314 (2006) 126-149. | MR | Zbl
[20] , Iterative methods for optimization, Frontiers in Applied Mathematics 18. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999). | MR | Zbl
[21] , Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč. 16 (1967) 209-292. | MR | Zbl
[22] , and , Fast numerical methods for Bernoulli free boundary problems. SIAM J. Sci. Comput. 29 (2007) 622-634. | MR | Zbl
[23] and , Polygons as optimal shapes with convexity constraint. SIAM J. Control Optim. 48 (2009) 3003-3025. | MR | Zbl
[24] and , A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition. Nonlinear Anal. 67 (2007) 2497-2505. | MR | Zbl
[25] and , Numerical optimization. Springer Series in Operations Research and Financial Engineering, Springer, New York, 2nd edition (2006). | MR | Zbl
[26] , n-diffusion. Austral. J. Phys. 14 (1961) 1-13. | MR | Zbl
[27] and , Introduction to shape optimization : Shape sensitivity analysis, Springer Series in Computational Mathematics 16. Springer-Verlag, Berlin (1992). | Zbl
Cité par Sources :
