  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 results, perturbation argument and comparison of
Laplacian eigenvalues with different boundary conditions. Since neither of the
functionals in the two problems is $C^1$, another key ingredient is to obtain
the global H\"older regularity of minimizers to both problems on Lipschitz
domains. Also, exact dependence on domain of breaking thresholds is also given
in the first problem, and the breaking values are obtained in the second
problem on ball domains, which are related to $2\pi$ in dimension $2$.
