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• W3173157067 abstract "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 $mmu_2(B_R)<frac{1}{2}frac{P^2(B_R)}{|B_R|}(=2pi,, mbox{when},n=2)$, where $m$ is the total amount of material $m$, and $mu_2$ is the first nonzero eigenvalue of Neumann Laplacian. Based on this, we then obtain local minimality of ball shape when $n=2$ and $m$ is bigger than the symmetry breaking threshold. Also, in the first problem, by relating to Serrin's overdetermined symmetry result, we prove that concentration breaking phenomenon occurs for any nonradial domains. In the second problem, we show that for any domain $Omega$, there is $m_2>0$ 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." @default.
• W3173157067 created "2021-07-05" @default.
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• W3173157067 date "2020-10-31" @default.
• W3173157067 modified "2023-09-27" @default.
• W3173157067 title "Two thermal insulation problems and eigenvalues of boundary mean zero Laplacian" @default.
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• W3173157067 hasPublicationYear "2020" @default.
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