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Shape optimization of a Dirichlet type energy for semilinear elliptic partial differential equations. (English) Zbl 1467.49034

Summary: Minimizing the so-called “Dirichlet energy” with respect to the domain under a volume constraint is a standard problem in shape optimization which is now well understood. This article is devoted to a prototypal non-linear version of the problem, where one aims at minimizing a Dirichlet-type energy involving the solution to a semilinear elliptic PDE with respect to the domain, under a volume constraint. One of the main differences with the standard version of this problem rests upon the fact that the criterion to minimize does not write as the minimum of an energy, and thus most of the usual tools to analyze this problem cannot be used. By using a relaxed version of this problem, we first prove the existence of optimal shapes under several assumptions on the problem parameters. We then analyze the stability of the ball, expected to be a good candidate for solving the shape optimization problem, when the coefficients of the involved PDE are radially symmetric.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
35J61 Semilinear elliptic equations
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[1] D. Bucur and G. Buttazzo, Variational methods in shape optimization problems, Vol. 65 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston, MA (2005). · Zbl 1117.49001
[2] G. Buttazzo and G. Dal Maso An existence result for a class of shape optimization problems. Arch. Rational Mech. Anal. 122 (1993) 183-195. · Zbl 0811.49028
[3] Y. Chen, L. Wu, A.M. Society and B. Hu, Second Order Elliptic Equations and Elliptic Systems. Translations of Mathematical Monographs. American Mathematical Society (1998). · Zbl 0902.35003
[4] R. Cominetti and J.-P. Penot, Tangent sets to unilateral convex sets. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) 1631-1636. · Zbl 0866.49026
[5] M. Dambrine and J. Lamboley. Stability in shape optimization with second variation. J. Diff. Equ. 267 (2019) 3009-3045. · Zbl 1416.49046
[6] M. Dambrine and M. Pierre, About stability of equilibrium shapes. ESAIM: M2AN 34 (2000) 811-834. · Zbl 0966.49023 · doi:10.1051/m2an:2000105
[7] M.C. Delfour and J.P. Zolésio, Shapes and geometries, Vol. 22 of Advances in Design and Control, second edition. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2011). · Zbl 1251.49001
[8] A. Evgrafov, The limits of porous materials in the topology optimization of Stokes flows. Appl. Math. Optim. 52 (2005) 263-277. · Zbl 1207.49004
[9] R.A. Feijóo, A.A. Novotny, E. Taroco and C. Padra, The topological derivative for the Poisson’s problem. Math. Models Methods Appl. Sci. 13 (2003) 1825-1844. · Zbl 1063.49030
[10] M. Hayouni, Lipschitz continuity of the state function in a shape optimization problem. J. Convex Anal. 6 (1999) 71-90. · Zbl 0948.49021
[11] A. Henrot, Extremum problems for eigenvalues of elliptic operators, in Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006). · Zbl 1109.35081
[12] A. Henrot (Ed.), Shape Optimization and Spectral Theory. De Gruyter Open, Warsaw (2017). · Zbl 1369.49004
[13] A. Henrot and M. Pierre, Variation et optimisation de formes: une analyse géométrique, Vol. 48. Springer Science & Business Media (2006). · Zbl 1098.49001
[14] A. Henrot, M. Pierre and M. Rihani, Positivity of the shape Hessian and instability of some equilibrium shapes. Mediterr. J. Math. 1 (2004) 195-214. · Zbl 1072.49028
[15] M. Iguernane, S. Nazarov, J.-R. Roche, J. Sokolowski and K. Szulc, Topological derivatives for semilinear elliptic equations. Int. J. Appl. Math. Comput. Sci. 19 (2009) 191-205. · Zbl 1167.49039
[16] B. Kawohl, Rearrangements and convexity of level sets in PDE, Vol. 1150 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1985). · Zbl 0593.35002
[17] B. Kawohl, O. Pironneau, L. Tartar and J.-P. Zolésio, Optimal shape design, Centro Internazionale Matematico Estivo (C.I.M.E.), Florence (2000). Lectures given at the Joint C.I.M./C.I.M.E. Summer School held in Tróia, June 1-6, 1998, edited by A. Cellina and A. Ornelas, Fondazione CIME/CIME Foundation Subseries. Vol. 1740 of Lecture Notes in Mathematics. Springer-Verlag, Berlin. · Zbl 0954.00031
[18] I. Mazari, G. Nadin and Y. Privat, Optimal location of resources maximizing the total population size in logistic models. J. Math. Appl. 134 (2020) 1-35. · Zbl 1433.92038
[19] A.A. Novotny and J. Sokołowski, Topological Derivatives in Shape Optimization. Springer, Berlin, Heidelberg (2013). · Zbl 1276.35002
[20] E. Russ, B. Trey and B. Velichkov, Existence and regularity of optimal shapes for elliptic operators with drift. Calc. Variations Partial Differ. Equ. 58 (2019) 199. · Zbl 1428.49044
[21] G. Talenti, Nonlinear elliptic equations, rearrangements of functions and orlicz spaces. Ann. Math. Pura Appl. 120 (1979) 159-184. · Zbl 0419.35041
[22] B. Velichkov, Existence and Regularity Results for Some Shape Optimization Problems. Scuola Normale Superiore (2015). · Zbl 1316.49006
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