Optimization of an eigenvalue arising in optimal insulation with a lower bound

• Published in: Advances in continuous and discrete models Vol. 2025; no. 1; p. 95
• Main Authors: Bartels, Sören, Buttazzo, Giuseppe, Keller, Hedwig
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 13.05.2025; Springer Nature B.V
• Subjects: Analysis; Approximation; Broken symmetry; Convexity; Decay rate; Difference and Functional Equations; Eigenvalues; Frontiers in Dynamics; Functional Analysis; Insulation; Lower bounds; Mathematics; Mathematics and Statistics; Optimization; Optimization and Control; Ordinary Differential Equations; Partial Differential Equations; Shape optimization; Symmetry; 65N25; 65N12; Numerical scheme; 49R05; Optimal insulation; Shape optimization; 52-08; 49Q10; 35J25; Symmetry breaking
• ISSN: 2731-4235; 1687-1839; 2731-4235; 1687-1847
• DOI: 10.1186/s13662-025-03955-3

An eigenvalue problem arising in optimal insulation related to the minimization of the heat-decay rate of an insulated body is adapted to enforce a positive lower bound imposed on the distribution of insulating material. We prove the existence of optimal domains among a class of convex shapes and propose a numerical scheme to approximate the eigenvalue. The stability of the shape optimization among convex, bounded domains in R 3 is proven for an approximation with polyhedral domains under a nonconformal convexity constraint. We prove that on the ball, symmetry breaking of the optimal insulation can be expected in general. To observe how the lower bound affects the breaking of symmetry in the optimal insulation and the shape optimization, the eigenvalue and optimal domains are approximated for several values of mass m and lower bounds ℓ min ≥ 0 . The numerical experiments suggest that, in general, symmetry breaking still arises, unless m is close to a critical value m 0 , and ℓ min is large enough such that almost all of the mass m is fixed through the lower bound. For ℓ min = 0 , the numerical results are consistent with previous numerical experiments on shape optimization restricted to rotationally symmetric, convex domains.