Two thermal insulation problems and eigenvalues of boundary mean zero Laplacian
Yong H. Huang, Qinfeng Li, Qiuqi Li · arXiv (Cornell University) · 2020
In this paper, we study two shape optimization problems from thermal insulation background, both involve varying domains and the associated state functions. In the first problem, assuming that the heat source is radial, by computing second shape derivatives and referring to Stekloff eigenvalue problem, we obtain necessary and sufficient conditions such that ball shapes are stable shapes among smooth volume preserving perturbations. In the second problem, we prove that for any ball $B_R$ in $\mathbb{R}^n$, symmetry breaking of insulation material occurs exactly when $m\mu_2(B_R)0$ such that when $m<m_2$, the optimal distribution of insulation must vanish on some portion of boundary. This number $m_2$ is related to $\kappa_1(\Omega)$, the first eigenvalue of boundary mean zero Laplacian. Motivated by this and a possible new isoperimetric inequality from symmetry breaking phenomenon, we then study spectral properties of boundary mean zero Laplacian, which turns out to be closely related to the Neumann Laplacian eigenvalue problem when the domain is symmetric. Among many other results, we prove that $\kappa_1(\Omega) \le \mu_2(\Omega)$, and necessary and sufficient conditions for the equality holds are obtained. Explicit values of $\kappa_2$ on some special domains are also given. Open questions will also be posted.