Compactness of 𝑀-uniform domains and optimal thermal insulation problems

References

[1] E. Acerbi and G. Buttazzo, Reinforcement problems in the calculus of variations, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 4, 273–284. 10.1016/s0294-1449(16)30380-8Search in Google Scholar

[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., The Clarendon, Oxford, 2000. 10.1093/oso/9780198502456.001.0001Search in Google Scholar

[3] G. A. Beer, The Hausdorff metric and convergence in measure, Michigan Math. J. 21 (1974), 63–64. 10.1307/mmj/1029001209Search in Google Scholar

[4] G. A. Beer, Starshaped sets and the Hausdorff metric, Pacific J. Math. 61 (1975), no. 1, 21–27. 10.2140/pjm.1975.61.21Search in Google Scholar

[5] A. Boulkhemair and A. Chakib, On the uniform Poincaré inequality, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1439–1447. 10.1080/03605300600910241Search in Google Scholar

[6] A. Braides, Approximation of Free-Discontinuity Problems, Lecture Notes in Math. 1694, Springer, Berlin, 1998. 10.1007/BFb0097344Search in Google Scholar

[7] H. Brézis, L. A. Caffarelli and A. Friedman, Reinforcement problems for elliptic equations and variational inequalities, Ann. Mat. Pura Appl. (4) 123 (1980), 219–246. 10.1007/BF01796546Search in Google Scholar

[8] D. Bucur, Minimization of the 𝑘-th eigenvalue of the Dirichlet Laplacian, Arch. Ration. Mech. Anal. 206 (2012), no. 3, 1073–1083. 10.1007/s00205-012-0561-0Search in Google Scholar

[9] D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Progr. Nonlinear Differential Equations Appl. 65, Birkhäuser, Boston, 2005. 10.1007/b137163Search in Google Scholar

[10] D. Bucur, G. Buttazzo and C. Nitsch, Symmetry breaking for a problem in optimal insulation, J. Math. Pures Appl. (9) 107 (2017), no. 4, 451–463. 10.1016/j.matpur.2016.07.006Search in Google Scholar

[11] D. Bucur, G. Buttazzo and C. Nitsch, Two optimization problems in thermal insulation, Notices Amer. Math. Soc. 64 (2017), no. 8, 830–835. 10.1090/noti1557Search in Google Scholar

[12] D. Bucur and A. Giacomini, A variational approach to the isoperimetric inequality for the Robin eigenvalue problem, Arch. Ration. Mech. Anal. 198 (2010), no. 3, 927–961. 10.1007/s00205-010-0298-6Search in Google Scholar

[13] D. Bucur and A. Giacomini, The Saint-Venant inequality for the Laplace operator with Robin boundary conditions, Milan J. Math. 83 (2015), no. 2, 327–343. 10.1007/s00032-015-0243-0Search in Google Scholar

[14] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Textb. Math., CRC Press, Boca Raton, 2015. 10.1201/b18333Search in Google Scholar

[15] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Math. 85, Cambridge University, Cambridge, 1986. Search in Google Scholar

[16] F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Anal. Math. 36 (1979), 50–74. 10.1007/BF02798768Search in Google Scholar

[17] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math. 80, Birkhäuser, Basel, 1984. 10.1007/978-1-4684-9486-0Search in Google Scholar

[18] P. Harjulehto, Traces and Sobolev extension domains, Proc. Amer. Math. Soc. 134 (2006), no. 8, 2373–2382. 10.1090/S0002-9939-06-08228-1Search in Google Scholar

[19] A. Henrot, Shape Optimization and Spectral Theory, De Gruyter, Warsaw, 2017. 10.1515/9783110550887Search in Google Scholar

[20] A. Henrot and M. Pierre, Variation et optimisation de formes, Math. Appl. (Berlin 48, Springer, Berlin, 2005. 10.1007/3-540-37689-5Search in Google Scholar

[21] Y. Huang, Q. F. Li and Q. Q. Li, Stability breaking, concentration breaking and asymptotic analysis in two thermal insulation problems, preprint (2020), https://arxiv.org/abs/2011.00291. Search in Google Scholar

[22] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237–266. 10.4310/jdg/1214438998Search in Google Scholar

[23] E. Järvenpää, M. Järvenpää, A. Käenmäki, T. Rajala, S. Rogovin and V. Suomala, Packing dimension and Ahlfors regularity of porous sets in metric spaces, Math. Z. 266 (2010), no. 1, 83–105. 10.1007/s00209-009-0555-2Search in Google Scholar

[24] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), no. 1–2, 71–88. 10.1007/BF02392869Search in Google Scholar

[25] Q. F. Li and C. Y. Wang, On a variational problem of nematic liquid crystal droplets, preprint (2019). 10.29007/c737Search in Google Scholar

[26] F. H. Lin and C. C. Poon, On nematic liquid crystal droplets, Elliptic and Parabolic Methods in Geometry (Minneapolis 1994), A K Peters, Wellesley (1996), 91–121. Search in Google Scholar

[27] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems. An introduction to geometric measure theory, Cambridge Stud. Adv. Math. 135, Cambridge University, Cambridge, 2012. 10.1017/CBO9781139108133Search in Google Scholar

[28] O. Martio, John domains, bi-Lipschitz balls and Poincaré inequality, Rev. Roumaine Math. Pures Appl. 33 (1988), no. 1–2, 107–112. Search in Google Scholar

[29] N. G. Meyers and W. P. Ziemer, Integral inequalities of Poincaré and Wirtinger type for BV functions, Amer. J. Math. 99 (1977), no. 6, 1345–1360. 10.2307/2374028Search in Google Scholar

[30] F. D. Pietra, C. Nitsch, R. Scala and C. Trombetti, An optimization problem in thermal insulation with Robin boundary conditions, preprint (2020), https://arxiv.org/abs/2008.02193. Search in Google Scholar

[31] D. Ruiz, On the uniformity of the constant in the Poincaré inequality, Adv. Nonlinear Stud. 12 (2012), no. 4, 889–903. 10.1515/ans-2012-0413Search in Google Scholar

[32] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University, Princeton, 1970. 10.1515/9781400883882Search in Google Scholar

[33] S. K. Vodop’janov, V. M. Gol’dšteĭn and T. G. Latfullin, A criterion for the extension of functions of the class L 2 1 from unbounded plane domains, Sibirsk. Mat. Zh. 20 (1979), no. 2, 416–419, 464. 10.1007/BF00970040Search in Google Scholar