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

Denis Vinokurov

TL;DR

The paper develops a general framework for higher-dimensional eigenvalue optimization within a conformal class, proving existence of maximizers for a family of normalized functionals and linking critical points to $p$-harmonic maps into spheres. The core method uses topological tensor products to overcome uncontrolled eigenvalue multiplicities and to formulate convergence in $H^{1,p}(M)\widehat{\otimes}_\pi H^{1,p}(M)$, which is shown to correspond to $\ell^2$-valued maps. A thorough bubbling analysis and a detailed regularity theory show that, for $p$ near $m$, maximizers are smooth, while for $p<m$ no bubbling occurs; when $p=m$, the maximizers decompose into deserved $m$-harmonic components on spheres. The results extend known two-dimensional theories to higher dimensions, illuminate the role of $p$-harmonic maps in spectral optimization, and provide a robust, tensor-product–driven toolkit for eigenvalue problems with complex multiplicities and density dependences.

Abstract

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 smooth, 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.

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

TL;DR

The paper develops a general framework for higher-dimensional eigenvalue optimization within a conformal class, proving existence of maximizers for a family of normalized functionals and linking critical points to -harmonic maps into spheres. The core method uses topological tensor products to overcome uncontrolled eigenvalue multiplicities and to formulate convergence in , which is shown to correspond to -valued maps. A thorough bubbling analysis and a detailed regularity theory show that, for near , maximizers are smooth, while for no bubbling occurs; when , the maximizers decompose into deserved -harmonic components on spheres. The results extend known two-dimensional theories to higher dimensions, illuminate the role of -harmonic maps in spectral optimization, and provide a robust, tensor-product–driven toolkit for eigenvalue problems with complex multiplicities and density dependences.

Abstract

We prove existence results for optimization problems for the th Laplace eigenvalue on closed Riemannian manifolds of dimension , 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 -harmonic maps into spheres, where . For sufficiently close to , the maximizers are smooth, whereas for 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.
Paper Structure (26 sections, 50 theorems, 183 equations)

This paper contains 26 sections, 50 theorems, 183 equations.

Table of Contents

  1. Introduction and main results
  2. Eigenvalue optimization in higher dimensions
  3. Examples on selected manifolds
  4. Ideas of the proofs of existence and regularity
  5. Acknowledgements
  6. Preliminaries
  7. Topological tensor products of Banach spaces
  8. A zoo of convergences
  9. Two main tensor products
  10. Tensor products of Hilbert spaces
  11. Vector-valued Sobolev spaces
  12. Applications of tensor products to Sobolev spaces
  13. Clarke subdifferential
  14. Weighted eigenvalues and weighted Sobolev spaces
  15. Proofs of the examples from Section \ref{['sec:examples']}
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $(M,g)$ be a closed connected Riemannian manifold of dimension $m \geq 2$, $p\in [2,m]$, $d = 3 + \lfloor p+2\sqrt{p-1}\rfloor$, $(\alpha,\mu) \in L^{p/(p-2)}_+(M)\times L^1_+(M)$, and we fix $\alpha \equiv 1$ if $p=2$. Then $\overline{\lambda}_{k,p}(\alpha,\mu) \leq \Lambda_{k,p}(g)$. If equali for some spectrally stable (see Definition def:stable-map) $p$-harmonic map $u \in H^{1,p}(M,\mathb

Theorems & Definitions (104)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 94 more