两个最优化问题的集中破缺
Concentration breaking on two optimization problems
摘要：
本文利用Serrin的超定结果、微扰论以及不同边界条件下Laplacian特征值的比较，研究了Lipschitz区域上两个绝热问题的边界浓度破缺现象.由于这两个问题中的泛函都不是$C^1$，另一个关键因素是获得这两个问题在Lipschitz域上的全局Hölder正则性。第一个问题给出了破缺阈值对域的精确依赖性，第二个问题得到了球域上的破缺值，它与$2π$在$2$维上有联系.
英文摘要：
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 Hölder 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 $2π$ in dimension $2$.
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