We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k-th perimeter-normalized Steklov eigenvalue is \(8\pi k\), which is the best upper bound for the \(k^{\text {th}}\) area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realizing a weighted Neumann problem as a limit of Steklov problems on perforated domains. For \(k=1\), the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.

我们将一系列变分特征值与紧凑黎曼流形上的任何氡测度相关联。对于特定的度量选择，我们恢复了拉普拉斯、斯特克洛夫和其他经典特征值问题。在论文的第一部分中，我们研究了变分特征值的性质并建立了一般连续性结果，这表明对于在适当 Sobolev 空间的对偶中收敛的一系列测度，相关的特征值也收敛。论文的第二部分致力于形状优化的各种应用。主题是研究 Steklov 特征值的尖锐等周不等式，而不对边界的连通分量的数量做任何假设。特别是，我们解决了平面域的每个 Steklov 特征值的等周问题：第k个周长归一化 Steklov 特征值是\(8\pi k\)，这是球体上拉普拉斯算子的\(k^{\text {th}}\)区域归一化特征值的最佳上限。证明涉及将加权诺依曼问题实现为穿孔域上 Steklov 问题的限制。对于\(k=1\)，最大化序列的连接边界组件的数量必须趋于无穷大，我们提供了连接组件数量的定量下限。我们分析的一个令人惊讶的结果是，任何具有固定周长的平面域的最大化序列都必须折叠成一个点。