论文标题
大型steklov特征值通过歧管上的同质化
Large Steklov eigenvalues via homogenisation on manifolds
论文作者
论文摘要
利用确定性均质理论精神的方法，我们获得了该歧管加权拉普拉斯特征值的一系列域序列的steklov特征值的收敛。通过去除渐近均匀分布的小型测量球来获得域，因为它们的半径趋于零。我们使用这种关系来构建具有较大steklov特征值的歧管。 在尺寸二，恒定重量等于1中，我们证明了Kokarev的上限为$8π$，对于第一个非零归一化的steklov特征值，在属0的可定向表面上是饱和的。对于其他拓扑类型和特征值指数，就拉普拉斯的最大化剂而言，我们还可以在特征值上获得最佳上限。对于前两个特征值，这些下限成为平等性。一个令人惊讶的结果是，首先是steklov特征功能，存在自由边界最小的表面，并严格大于$2π$。以前认为这是不可能的。我们提供了数值证据，表明一些已知的自由边界最小表面的实例具有这些特性，并且还显示了单位球中具有更大面积的单位球中新的自由边界最小表面的模拟。所有这些示例的第一个非零steklov特征值等于1，因为它们的对称性和拓扑结构，因此他们验证了Fraser和Li的一般猜想。 在三大尺寸中，我们证明了Colbois-el soufi-girouard的等值不平等很清晰，这意味着对加权拉普拉斯特征值的上限。我们还表明，在具有固定度量的任何流形中，都可以通过改变具有连接边界的重量域的构造，其第一个非零归一化的steklov特征值是任意的。
Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev's upper bound of $8π$ for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated. For other topological types and eigenvalue indices, we also obtain lower bounds on the best upper bound for the eigenvalue in terms of Laplace maximisers. For the first two eigenvalues, these lower bounds become equalities. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than $2π$. This was previously thought to be impossible. We provide numerical evidence that some of the already known examples of free boundary minimal surfaces have these properties and also exhibit simulations of new free boundary minimal surfaces of genus 0 in the unit ball with even larger area. The first nonzero Steklov eigenvalue of all these examples is equal to 1, as a consequence of their symmetries and topology, so that they verify a general conjecture by Fraser and Li. In dimension three and larger, we prove that the isoperimetric inequality of Colbois--El Soufi--Girouard is sharp and implies an upper bound for weighted Laplace eigenvalues. We also show that in any manifold with a fixed metric, one can construct by varying the weight a domain with connected boundary whose first nonzero normalised Steklov eigenvalue is arbitrarily large.
