Concentration breaking on two optimization problems

In the present paper, we study the boundary concentration-breaking phenomena on two thermal insulation problems considered on Lipschitz domains, based on Serrin’s overdetermined result, the perturbation argument, and a comparison of Laplacian eigenvalues with different boundary conditions. Since neither of the functionals in the two problems is C¹, another key ingredient is to obtain the global Hölder regularity of minimizers of both problems on Lipschitz domains. Also, the exact dependence on the domain of breaking thresholds is given in the first problem, and the breaking values are obtained in the second problem on ball domains, which are related to 2π in dimension 2.