arXiv subjects

Denis Vinokurov

Publications and source records attributed to Denis Vinokurov.

6 recordsLinked to original sources

arXiv2026

Eigenvalue optimization via a first-variation formula

We compute the Clarke subdifferential of the $k$th eigenvalue functional on the space of self-adjoint operators, obtaining a first-variation formula that remains valid even when the eigenvalue lies at the edge of the essential spectrum. This formula provides an effective tool for describing the structure of critical points in eigenvalue optimization problems and can also yield simple proofs of the existence of optimizers. We illustrate these advantages through applications to the optimization of weighted Laplace and Steklov eigenvalues. In particular, we characterize all optimal weights, thereby answering some open questions posed by Kokarev, and give a short proof that such weights exist.

math.SP↗

arXiv2026

Geometric bounds for Steklov and weighted Neumann eigenvalues on Euclidean domains

We obtain sharp upper bounds for the first two nonzero Steklov eigenvalues among bounded domains in Euclidean spaces of dimension $d \geq 7$ under a natural normalization involving volume and boundary measure. These bounds are derived from a characterization of optimal domains and weights for the first two nonzero weighted Neumann eigenvalues. In dimensions $3 \leq d \leq 6$, we obtain strict upper bounds. We further establish strict upper bounds for all higher Steklov eigenvalues on planar simply connected domains with continuous boundary, extending previous results which, beyond the second nonzero eigenvalue, were known only for smooth planar domains.

math.SP↗

arXiv2026

Eigenvalue optimization in higher dimensions and $p$-harmonic maps

We prove existence results for optimization problems for the $k$th Laplace eigenvalue on closed Riemannian manifolds of dimension $m \geq 3$, depending on the choice of normalization. One such normalization leads to eigenvalue optimization within a conformal class, for which existence of maximizers was previously known only in dimension two. We also prove that all absolutely continuous maximizers of the normalized eigenvalue functionals are always induced by $p$-harmonic maps into spheres, where $p \in [2,m]$. For $p$ sufficiently close to $m$, the maximizers are always H\"older-continuous, whereas for $p<m$ no bubbling occurs. A key tool in our analysis is the application of techniques from the theory of topological tensor products, which appear to be well suited for studying eigenvalue-related optimization problems.

math.SP↗

arXiv2025

Conformal optimization of eigenvalues on surfaces with symmetries

Given a conformal action of a discrete group on a Riemann surface, we study the maximization of Laplace and Steklov eigenvalues within a conformal class, considering metrics invariant under the group action. We establish natural conditions for the existence and regularity of maximizers. Our method simplifies previously known techniques for proving existence and regularity results in conformal class optimization. Finally, we provide a complete solution to the equivariant maximization problem for Laplace eigenvalues on the sphere and Steklov eigenvalues on the disk, resolving open questions posed by Arias-Marco et al. (2024) regarding the sharpness of the Hersch-Payne-Schiffer inequality and the maximization of Steklov eigenvalues by the standard disk among planar simply connected domains with $n\text{-rotational}$ symmetry.

math.SP↗

arXiv2025

Maximizing higher eigenvalues in dimensions three and above

We study the problem of maximizing the $k$-th eigenvalue functional over the class of absolutely continuous measures on a closed Riemannian manifold of dimension $m\geq 3$. Extending the work of Karpukhin and Stern on the first eigenvalue, we prove that, for every $k\geq 1$, the supremum is attained by a measure induced by a harmonic map into a finite-dimensional sphere. The map is smooth outside a closed singular set of Hausdorff dimension at most $m-7$, and is therefore smooth when $3 \leq m \leq 6$. We further prove that this dimension bound is optimal: for every $m \geq 7$ and every integer $0\leq d \leq m-7$, there exists a maximizing harmonic map on the $m$-dimensional round sphere whose singular set has Hausdorff dimension $d$.

math.SP↗

arXiv2022

The first eigenvalue of the Laplacian on orientable surfaces

The famous Yang-Yau inequality provides an upper bound for the first eigenvalue of the Laplacian on an orientable Riemannian surface solely in terms of its genus $γ$ and the area. Its proof relies on the existence of holomorhic maps to $\mathbb{CP}^1$ of low degree. Very recently, A.~Ros was able to use certain holomorphic maps to $\mathbb{CP}^2$ in order to give a quantitative improvement of the Yang-Yau inequality for $γ=3$. In the present paper, we generalize Ros' argument to make use of holomorphic maps to $\mathbb{CP}^n$ for any $n>0$. As an application, we obtain a quantitative improvement of the Yang-Yau inequality for all genera $γ>3$ except for $γ= 4,6,8,10,14$.

math.DG↗