We consider an overdetermined problem of Serrin-type with respect to an operator in divergence form with piecewise constant coefficients. We give sufficient condition for unique solvability near radially symmetric configurations by means of a perturbation argument relying on shape derivatives and the implicit function theorem. This problem is also treated numerically, by means of a steepest descent algorithm based on a Kohn–Vogelius functional.
Keywords: Two-phase, overdetermined problem, Serrin problem, shape derivative, implicit function theorem, Kohn–Vogelius functional, augmented Lagrangian
@article{COCV_2020__26_1_A65_0,
author = {Cavallina, Lorenzo and Yachimura, Toshiaki},
title = {On a two-phase {Serrin-type} problem and its numerical computation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2020},
publisher = {EDP Sciences},
volume = {26},
doi = {10.1051/cocv/2019048},
mrnumber = {4151427},
zbl = {1450.35189},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2019048/}
}
TY - JOUR AU - Cavallina, Lorenzo AU - Yachimura, Toshiaki TI - On a two-phase Serrin-type problem and its numerical computation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2019048/ DO - 10.1051/cocv/2019048 LA - en ID - COCV_2020__26_1_A65_0 ER -
%0 Journal Article %A Cavallina, Lorenzo %A Yachimura, Toshiaki %T On a two-phase Serrin-type problem and its numerical computation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2019048/ %R 10.1051/cocv/2019048 %G en %F COCV_2020__26_1_A65_0
Cavallina, Lorenzo; Yachimura, Toshiaki. On a two-phase Serrin-type problem and its numerical computation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 65. doi: 10.1051/cocv/2019048
[1] and , Sobolev spaces. Second edition. Vol. 140 of Pure and Applied Mathematics (Amsterdam). Elsevier/ Press, Amsterdam (2003). | MR | Zbl
[2] , Uniqueness theorems for surfaces in the large V. Vestnik Leningrad Univ. 13 (1958) 5–8. [English translation: Trans. Am. Math. Soc. 21 (1962) 412–415]. | MR | Zbl
[3] and , Structural optimization with freefem++. Struct. Multidisc. Optim. 32 (2006) 173–181. | MR | Zbl | DOI
[4] and , Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325 (1981) 105–144. | MR | Zbl
[5] , , , and , A Dirichlet–Neumann cost functional approach for the Bernoulli problem. J. Eng. Math. 81 (2013) 157–176. | MR | Zbl | DOI
[6] , On free boundary problems for the Laplace equation, Seminars on analytic functions I, Institute Advanced Studies Seminars Princeton (1957) 248–263. Collected works of Arne Beurling I. Birkhäuser, Boston (1989) 250–265. | Zbl
[7] , and , An overdetermined problem with non-constant boundary condition. Interfaces Free Bound. 16 (2014) 215–241. | MR | Zbl | DOI
[8] , 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 | DOI
[9] , , and , A free boundary problem for the Stokes equations. ESAIM: COCV 23 (2017) 195–215. | MR | Zbl | Numdam
[10] , , and , On the stability of the Serrin problem. J. Diff. Equ. 245 (2008) 1566–1583. | MR | Zbl | DOI
[11] , Stability analysis of the two-phase torsional rigidity near a radial configuration. Appl. Anal. 98 (2019) 1889–1900. | MR | Zbl | DOI
[12] , Analysis of two-phase shape optimization problems by means of shape derivatives. Doctoral dissertation, Tohoku University, Sendai, Japan. Preprint (2018). | arXiv
[13] , and , Two-phase heat conductors with a surface of the constant flow property. To appear in: J. Geom. Anal. (2019). Available from: . | DOI | MR | Zbl
[14] , and , An extremal eigenvalue problem for a two-phase conductor in a ball. Appl. Math. Optim. 60 (2009) 173–184. | MR | Zbl | DOI
[15] , and , Minimization of the ground state for two phase conductors in low contrast regime. SIAM J. Appl. Math. 72 (2012) 1238–1259. | MR | Zbl | DOI
[16] , , and , Geometrical shape optimization in fluid mechanics using Freefem++. Struct. Multidisc. Optim. 58 (2018) 2761–2788. | MR | DOI
[17] and , Extremal eigenvalue problems for two-phase conductors. Arch. Ratl. Mech. Anal. 136 (1996) 101–117. | MR | Zbl | DOI
[18] , Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J. Control Optim. 45 (2006) 343–367. | MR | Zbl | DOI
[19] and , Shapes and Geometries: Analysis Differential Calculus, and Optimization. SIAM, Philadelphia (2001). | MR | Zbl
[20] and , On a Kohn–Vogelius like formulation of free boundary problems. Comput. Optim. Appl. 52 (2012) 69–85. | MR | Zbl | DOI
[21] and , Bernoulli’s free boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math. 486 (1997) 165–204. | MR | Zbl
[22] and , Mesh generation, application to Finite Elements. Wiley & Sons (2008). | MR | Zbl | DOI
[23] , , , and , On the shape derivative for problems of Bernoulli type. Interfaces Free Bound. 11 (2009) 317–330. | MR | Zbl | DOI
[24] , New development in freefem++. J. Numer. Math. 20 (2012) 251–265. | MR | Zbl | DOI
[25] and , Shape variation and optimization (a geometrical analysis). Vol. 28 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich (2018). | MR | Zbl
[26] and , The one phase free boundary problem for the -Laplacian with non-constant Bernoulli boundary condition. Trans. Am. Math. Soc. 354 (2002) 2399–2416. | MR | Zbl | DOI
[27] , and , Variational approach to shape derivatives for a class of Bernoulli problems. J. Math. Anal. Appl. 314 (2006) 126–149. | MR | Zbl | DOI
[28] , and , A bilevel shape optimization problem for the exterior Bernoulli free boundary value problem. Interfaces Free Bound. 16 (2014) 459–487. | MR | Zbl | DOI
[29] and , Relaxation of a Variational Method for Impedance Computed Tomography. Commun. Pure Appl. Math. 40 (1987) 745–777. | MR | Zbl | DOI
[30] , Global minimizer of the ground state for two phase conductors in low contrast regime. ESAIM: COCV 20 (2014) 362–388. | MR | Zbl | Numdam
[31] and , On a Bernoulli problem with geometric constraints. ESAIM: COCV 18 (2012) 157–180. | MR | Zbl | Numdam
[32] and , Droplet footprint control. SIAM J. Control Optim. 53 (2015) 771–799. | MR | Zbl | DOI
[33] and , Serrin’s problem and Alexandrov’s Soap Bubble Theorem: stability via integral identities. Preprint (2017). To appear in: Indiana Univ. Math. J. 2020. | arXiv | MR
[34] and , On the control of coefficients in partial differential equations. In Topics in the mathematical modelling of composite materials. In Vol. 31 of Progr. Nonlinear Differential Equations Appl. Birkhäuser Boston, Boston, MA (1997) 1–8. | MR | Zbl
[35] and , Calculus of variations and homogenization. In Topics in the mathematical modelling of composite materials. Vol. 31 of Progr. Nonlinear Differential Equations Appl. Birkhäuser Boston, Boston, MA (1997) 139–173. | MR | Zbl
[36] , Topics in Nonlinear Functional Analysis, Revised reprint ofthe 1974 original. In Vol. 6 of Courant Lecture Notes in Mathematics. American Mathematical Society, Providence, RI (2001). | MR | Zbl | DOI
[37] and , The classical overdetermined Serrin problem. Complex Variables Elliptic Equ. 63 (2018) 1107–1122. | MR | Zbl | DOI
[38] and , Numerical Optimization. Springer (2006). | MR | Zbl
[39] , A symmetry problem in potential theory. Arch. Rat. Mech. Anal. 43 (1971) 304–318. | MR | Zbl | DOI
[40] and , Introduction to Shape Optimization: Shape Sensitivity Analysis. In Vol. 10 of Springer Series in Computational Mathematics. Springer–Verlag, Berlin (1992). | MR | Zbl
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This research was partially supported by the Challenging Exploratory Research No.16K13768 of Japan Society for the Promotion of Science and the Grant-in-Aid for JSPS Fellows No. 18J11430 and No. 19J12344.
