On a Bernoulli problem with geometric constraints
Antoine Laurain; Yannick Privat
ESAIM: Control, Optimisation and Calculus of Variations (2012)
- Volume: 18, Issue: 1, page 157-180
- ISSN: 1292-8119
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top- [1] A. Acker, An extremal problem involving current flow through distributed resistance. SIAM J. Math. Anal.12 (1981) 169–172. Zbl0456.49007MR605427
- [2] C. Atkinson and C.R. Champion, Some boundary-value problems for the equation ∇·(|∇ϕ| N∇ϕ) = 0. Quart. J. Mech. Appl. Math. 37 (1984) 401–419. Zbl0567.73054MR760209
- [3] A. Beurling, On free boundary problems for the Laplace equation, Seminars on analytic functions 1. Institute for Advanced Studies, Princeton (1957). Zbl0077.11202
- [4] F. Bouchon, S. Clain and R. Touzani, Numerical solution of the free boundary Bernoulli problem using a level set formulation. Comput. Methods Appl. Mech. Eng.194 (2005) 3934–3948. Zbl1090.76048MR2149216
- [5] E.N. Dancer and D. Daners, Domain perturbation for elliptic equations subject to Robin boundary conditions. J. Differ. Equ.138 (1997) 86–132. Zbl0886.35063MR1458457
- [6] 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). Zbl1002.49029MR1855817
- [7] L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence (1998). Zbl0902.35002MR1625845
- [8] 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. Zbl0765.76005MR1260974
- [9] M. Flucher and M. Rumpf, Bernoulli’s free boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math.486 (1997) 165–204. Zbl0909.35154MR1450755
- [10] A. Friedman, Free boundary problem in fluid dynamics, in Variational methods for equilibrium problems of fluids, Trento 1983, Astérisque 118 (1984) 55–67. Zbl0588.76016MR761737
- [11] A. Friedman, Free boundary problems in science and technology. Notices Amer. Math. Soc.47 (2000) 854–861. Zbl1040.35145MR1776102
- [12] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman (Advanced Publishing Program), Boston (1985). Zbl0695.35060MR775683
- [13] J. Haslinger, T. Kozubek, K. Kunisch and G. Peichl, Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type. Comp. Optim. Appl.26 (2003) 231–251. Zbl1077.49030MR2013364
- [14] J. Haslinger, K. Ito, T. Kozubek, K. Kunisch and G. Peichl, On the shape derivative for problems of Bernoulli type. Interfaces in Free Boundaries11 (2009) 317–330. Zbl1178.49055MR2511644
- [15] A. Henrot and M. Pierre, Variation et optimisation de formes – Une analyse géométrique, Mathématiques & Applications 48. Springer, Berlin (2005). Zbl1098.49001MR2512810
- [16] A. Henrot and H. Shahgholian, 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. Zbl0955.35078MR1752296
- [17] A. Henrot and H. Shahgholian, 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. Zbl0977.35148MR1777029
- [18] A. Henrot and H. Shahgholian, The one phase free boundary problem for the p-Laplacian with non-constant Bernoulli boundary condition. Trans. Amer. Math. Soc.354 (2002) 2399–2416. Zbl0988.35174MR1885658
- [19] K. Ito, K. Kunisch and G.H. Peichl, Variational approach to shape derivatives for a class of Bernoulli problems. J. Math. Anal. Appl.314 (2006) 126–149. Zbl1088.49028MR2183542
- [20] C.T. Kelley, Iterative methods for optimization, Frontiers in Applied Mathematics 18. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999). Zbl0934.90082MR1678201
- [21] V.A. Kondratév, Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč.16 (1967) 209–292. Zbl0194.13405MR226187
- [22] C.M. Kuster, P.A. Gremaud and R. Touzani, Fast numerical methods for Bernoulli free boundary problems. SIAM J. Sci. Comput.29 (2007) 622–634. Zbl1136.65113MR2306261
- [23] J. Lamboley and A. Novruzi, Polygons as optimal shapes with convexity constraint. SIAM J. Control Optim.48 (2009) 3003–3025. Zbl1202.49053MR2599908
- [24] E. Lindgren and Y. Privat, A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition. Nonlinear Anal.67 (2007) 2497–2505. Zbl1123.35092MR2338115
- [25] J. Nocedal and S.J. Wright, Numerical optimization. Springer Series in Operations Research and Financial Engineering, Springer, New York, 2nd edition (2006). Zbl1104.65059MR2244940
- [26] J.R. Philip, n-diffusion. Austral. J. Phys.14 (1961) 1–13. Zbl0137.18402MR140343
- [27] J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization : Shape sensitivity analysis, Springer Series in Computational Mathematics 16. Springer-Verlag, Berlin (1992). Zbl0761.73003