Inequalities between Dirichlet and Neumann eigenvalues in large dimensions

Technical Summary: Inequalities between Dirichlet and Neumann eigenvalues in large dimensions

Problem Statement
Let $\Omega$ be a bounded domain in $\mathbb{R}^d$. The paper investigates the spectral gap between the eigenvalues of the Laplace operator under Dirichlet boundary conditions ($\lambda_k$) and Neumann boundary conditions ($\mu_k$). It is a standard result, derived from the min-max principle, that $\mu_k \leq \lambda_k$ for all $k$. Previous work, specifically Levine and Weinberger, established that for convex domains, $\mu_{k+d} \leq \lambda_k$. The central problem addressed is the behavior of the "shift" $\Psi(d, k, \Omega)$, defined such that $\mu_{k+\Psi} \leq \lambda_k$, particularly in the regime where the dimension $d$ tends to infinity. The paper seeks to determine how much the index shift $\Psi$ grows as $d$ increases, moving beyond the linear dependence on $d$ found in previous convex domain inequalities.

Methodology
The paper employs a combination of spectral counting functions, variational principles, and asymptotic analysis of special functions.

1. Counting Functions and Definitions: The author defines $\Phi(d, k, \Omega) = \#\{j : \mu_j \leq \lambda_k\}$ and $\Psi = \Phi - k$. The goal is to lower-bound $\Phi$ (and thus $\Psi$) for large $d$.
2. Variational Construction for Convex Domains: To prove the main result for convex domains, the author constructs a specific subspace $F$ of the Sobolev space $W^1_2(\Omega)$ that is orthogonal to the first $k$ Dirichlet eigenfunctions.
   • The construction utilizes the fact that a convex domain is contained within a layer of minimal width $w(\Omega)$.
   • Test functions are chosen in the form $u_\eta(x) = \cos(x_1 \delta) e^{i\langle y, \eta \rangle}$, where $x = (x_1, y)$ and $\eta$ lies on a sphere in $\mathbb{R}^{d-1}$.
   • A matrix $B(H)$ is formed from the inner products of these test functions and their gradients. The author analyzes the positive spectrum of this matrix using a probabilistic argument (averaging over the choice of points $\eta$ on the sphere) to show that a subspace of sufficiently large dimension exists where the quadratic form is positive. This dimension provides a lower bound for the difference $N_N(\Omega, \lambda) - N_D(\Omega, \lambda)$.
3. Asymptotic Analysis: The proofs rely heavily on the asymptotic behavior of Bessel function zeros ($j_\nu$), the Gamma function (Stirling's formula), and combinatorial estimates for high-dimensional volumes.
4. Specific Geometries: For the ball and cube cases, the author explicitly computes the asymptotics of the counting function by analyzing the roots of Bessel functions and the multiplicity of eigenvalues in high dimensions.

Key Contributions and Results

• General Domains ($k=1, 2$): The paper establishes lower bounds for $\Phi(d, 1, \Omega)$ and $\Phi(d, 2, \Omega)$ for any bounded domain.

  • Theorem 1.1: Provides explicit lower bounds involving the first positive root of the Bessel function $j_{d/2-1}$.
  • Corollary 1.2: Demonstrates that for any domain, $\Phi(d, 1, \Omega) \geq C_1 (e/2)^d$. This implies an exponential growth of the index shift with dimension.
  • Corollary 1.3: For sufficiently large dimensions ($d \geq 7$), $\mu_3 < \lambda_1$; for $d \geq 6$, $\mu_4 < \lambda_2$.

• Convex Domains (Main Result):

  • Theorem 1.4: For all natural $k$ and all bounded convex domains $\Omega$, there exists an absolute constant $C_{conv}$ such that $\Psi(d, k, \Omega) \geq C_{conv} (e/2)^d$. This confirms that the index shift grows exponentially with dimension for convex domains, significantly improving upon the linear bound $\mu_{k+d} \leq \lambda_k$.

• Specific Geometries:

  • Theorem 1.6 (Ball): For a $d$-dimensional ball $B$, the asymptotics of the logarithm of the counting function is derived: $\log \Phi(d, 1, B) = \alpha_{ball} d + O(d^{1/3})$, where $\alpha_{ball} = \sqrt{2}\log(\sqrt{2}+1) - \log 2 \approx 0.55$.
  • Theorem 1.8 (Cube): For a $d$-dimensional cube $Q$, $\log \Phi(d, 1, Q) = \alpha_{cube} d + O(d^{1/3})$, with $\alpha_{cube} \approx 1.04$. The proof of this specific theorem is noted to be in a separate paper.

Significance and Claims
The paper claims to resolve the question of how the inequality between Neumann and Dirichlet eigenvalues behaves in high dimensions. While previous results suggested a linear shift for convex domains, this work proves that the shift is actually exponential in $d$.

The author notes that the exponential growth rate $(e/2)^d$ derived in the general convex case is a lower bound. By analyzing the ball and cube, the paper shows that the growth rate is indeed exponential, with specific constants $\alpha_{ball}$ and $\alpha_{cube}$. The author posits Conjecture 1.5, suggesting that for all domains (not just convex ones), the shift grows exponentially as $C^* e^{\alpha^* d}$. The results for the ball and cube serve to refine the understanding of the constant $\alpha^*$, showing that $\alpha^* \leq \alpha_{ball}$.

The paper does not propose new applications or experimental setups but focuses on refining the theoretical bounds of spectral inequalities in high-dimensional geometric analysis. The significance lies in demonstrating that the spectral gap between Neumann and Dirichlet problems widens dramatically as the dimension of the space increases.