Shape Optimization for Thermal Insulation Problems

Thermal insulation represents one of the major challenges for energy efficiency. Problems related to insulation are well-known and widely studied in mathematical physics. Neverthless, mathematics involved is still very tricky especially when one looks at shape optimization issues [1, 2], and sometimes the answers are not so intuitive [3].In this talk we will consider two domains: an internal (fixed) ball of radius r and an external domain whose geometry varies. Physically, we are considering a domain of given temperature, thermally insulated by surrounding it with a constant thickness of thermal insulator. Our question is related to the best (or worst) shape for the external domain, in terms of heat dispersion (of course, under prescribed geometrical constraints). Mathematically, our problem is composed by an elliptic PDE with Robin-Dirichlet boundary conditions. This work is still in progress and we want to share someremarks and open questions, in addition to the results obtained so far.REFERENCES[1] F. Della Pietra, C. Nitsch, C. Trombetti, An optimal insulation problem. Math. Ann., (2020). https://doi.org/10.1007/s00208-020-02058-6[2] D. Bucur, G. Buttazzo, C. Nitsch, Two optimization problems in thermal insulation. Notices Am. Math. Soc., 64(8): 830--835, 2017.[3] D. Bucur, G. Buttazzo, C. Nitsch, Symmetry breaking for a problem in optimal insulation. J.Math. Pures et Appl., 107(4): 451--463, 2017.