论文标题
一类光谱分区问题的所有最小化器的规律性
Regularity of all minimizers of a class of spectral partition problems
论文作者
论文摘要
我们研究了有关单调和强制功能成本的相当广泛的最佳分区问题，涉及分区的差异特征值。我们为原始问题的自然放松版本以及本征函数的规律性和通用的自由边界条件显示了整个最小化版本的整个最小化器的鲜明规律性结果。除其他外，我们的结果涵盖了以下功能成本的案例\ [ （ω_1，\ dots，ω_m）\ mapSto \ sum_ {i = 1}^{m} {m} \ left（\ sum_ {j = 1}^{k_i}λ_{k_i}λ_{J} \ prod_ {i = 1}^{m} \ left（\ prod_ {j = 1}^{k_i}λ_{λ_{j}（ω_i）\ right），\ quad \ prod_ {i = 1} λ_{j}（ω_i）\ right）\] \]其中$（ω_1，\ dots，ω_m）$是分区的集合，$λ_{j}（ω_i）$是$ j $ - j $ -j $ -th laplace eigenvalue of Set $ω_i$与零dirichelet diriclet diriclet diriclet diriclet diriclet diriclet diriclet diriclet diriclet diriclet diriclet diriclet diriclet diricelet cormessions。
We study a rather broad class of optimal partition problems with respect to monotone and coercive functional costs that involve the Dirichlet eigenvalues of the partitions. We show a sharp regularity result for the entire set of minimizers for a natural relaxed version of the original problem, together with the regularity of eigenfunctions and a universal free boundary condition. Among others, our result covers the cases of the following functional costs \[ (ω_1, \dots, ω_m) \mapsto \sum_{i=1}^{m} \left( \sum_{j=1}^{k_i} λ_{j}(ω_i)^{p_i}\right)^{1/p_i}, \quad \prod_{i=1}^{m} \left( \prod_{j=1}^{k_i} λ_{j}(ω_i)\right), \quad \prod_{i=1}^{m} \left( \sum_{j=1}^{k_i} λ_{j}(ω_i)\right) \] where $(ω_1, \dots, ω_m)$ are the sets of the partition and $λ_{j}(ω_i)$ is the $j$-th Laplace eigenvalue of the set $ω_i$ with zero Dirichlet boundary conditions.
