no. 6
Shape optimization problems with Robin conditions on the free boundary
Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 6, pp. 1539-1568.

We provide a free discontinuity approach to a class of shape optimization problems involving Robin conditions on the free boundary. More precisely, we identify a large family of domains on which such problems are well posed in a way that the extended problem can be considered a relaxed version of the corresponding one on regular domains, we prove existence of a solution and obtain some qualitative information on the optimal sets.

DOI : 10.1016/j.anihpc.2015.07.001
Mots clés : Free boundary problems, Shape optimization problems, Robin boundary conditions, Free discontinuity problems, Functions of bounded variation 
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     author = {Bucur, Dorin and Giacomini, Alessandro},
     title = {Shape optimization problems with {Robin} conditions on the free boundary},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1539--1568},
     publisher = {Elsevier},
     volume = {33},
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     year = {2016},
     doi = {10.1016/j.anihpc.2015.07.001},
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     zbl = {1352.49045},
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Bucur, Dorin; Giacomini, Alessandro. Shape optimization problems with Robin conditions on the free boundary. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 6, pp. 1539-1568. doi : 10.1016/j.anihpc.2015.07.001. http://www.numdam.org/articles/10.1016/j.anihpc.2015.07.001/

[1] Alt, H.W.; Caffarelli, L.A. Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., Volume 325 (1981), pp. 105–144 | MR | Zbl

[2] Ambrosio, L.; Fusco, N.; Pallara, D. Functions of Bounded Variation and Free Discontinuity Problems, Oxf. Math. Monogr., The Clarendon Press, Oxford University Press, New York, 2000 | MR | Zbl

[3] Babadjian, J.-F.; Giacomini, A. Existence of strong solutions for quasi-static evolution in brittle fracture, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume XIII (2014), pp. 925–974 | MR | Zbl

[4] Braides, A. Approximation of Free-Discontinuity Problems, Lect. Notes Math., vol. 1694, Springer-Verlag, Berlin, 1998 | DOI | MR | Zbl

[5] Bucur, D.; Giacomini, A. Faber–Krahn inequalities for the Robin–Laplacian: a free discontinuity approach, Arch. Ration. Mech. Anal., Volume 218 (2015) no. 2, pp. 757–824 | DOI | MR | Zbl

[6] Bucur, D.; Luckhaus, S. Monotonicity formula and regularity for general free discontinuity problems, Arch. Ration. Mech. Anal., Volume 211 (2014) no. 2, pp. 489–511 | DOI | MR | Zbl

[7] Carriero, M.; Leaci, A. Existence theorem for a Dirichlet problem with free discontinuity set, Nonlinear Anal., Volume 15 (1990) no. 7, pp. 661–677 | DOI | MR | Zbl

[8] Cortesani, G.; Toader, R. A density result in SBV with respect to non-isotropic energies, Nonlinear Anal., Ser. B: Real World Appl., Volume 38 (1999) no. 5, pp. 585–604 | MR | Zbl

[9] De Giorgi, E.; Ambrosio, L. New functionals in the calculus of variations, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur. (8), Volume 82 (1988) no. 2, pp. 199–210 (1989) | MR | Zbl

[10] De Giorgi, E.; Carriero, M.; Leaci, A. Existence theorem for a minimum problem with free discontinuity set, Arch. Ration. Mech. Anal., Volume 108 (1989) no. 3, pp. 195–218 | MR | Zbl

[11] Fonseca, I.; Fusco, N. Regularity results for anisotropic image segmentation models, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 24 (1997) no. 3, pp. 463–499 | Numdam | MR | Zbl

[12] Fusco, N.; Mingione, G.; Trombetti, C. Regularity of minimizers for a class of anisotropic free discontinuity problems, J. Convex Anal., Volume 8 (2001) no. 2, pp. 349–367 | MR | Zbl

[13] Siaudeau, A. Ahlfors-régularité des quasi-minima de Mumford–Shah, J. Math. Pures Appl. (9), Volume 82 (2003) no. 12, pp. 1697–1731 | DOI | MR | Zbl

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