Li–Yau Type Lower Bound

• The Li–Yau Type Lower Bound is a framework that provides explicit lower bounds for eigenvalue sums, incorporating domain geometry and correction terms to match Weyl asymptotics.
• The methodology employs Fourier analysis, phase-space localization, and curvature-dimension techniques to derive sharp estimates for Dirichlet, fractional, and mixed local-nonlocal operators.
• Recent extensions include applications to poly-Laplacians and combinatorial settings, with additional corrective measures like the moment of inertia enhancing spectral precision.

The Li–Yau Type Lower Bound is a general term encompassing a hierarchy of influential inequalities concerning the spectra and regularity of elliptic and parabolic operators, most notably the Dirichlet Laplacian and its nonlocal and higher order variants, as well as certain geometric and combinatorial structures. Originating from Li and Yau’s seminal 1983 work, these bounds precisely quantify the distribution of eigenvalues or the gradient behavior of heat evolution, with explicit constants and domain–dependent correction terms. The archetypal inequalities assert that, for a wide class of operators, the sum or certain functionals of eigenvalues are bounded below by explicit Weyl-type terms reflecting the ambient dimension, with possible sharp refinements via moment of inertia or boundary contributions. The methodology ranges from Fourier rearrangement and phase-space localization to maximum principle, Bochner–Weitzenböck identities, curvature-dimension calculus, and combinatorial Cheeger-type arguments. Below, the principal frameworks, refinements, and generalizations are organized and analyzed.

1. Classical Li–Yau Inequality and Berezin–Li–Yau Lower Bounds

The original Li–Yau lower bound applies to the Dirichlet Laplacian $-\Delta$ on a bounded domain $\Omega\subset\mathbb{R}^d$. For the ordered eigenvalues $0<\lambda_1\le\lambda_2\le\cdots$, Li–Yau proved

$$\sum_{j=1}^k\lambda_j \ge \frac{d}{d+2}\,(2\pi)^2\,|\Omega|^{-2/d}\,k^{1+2/d}$$

for all $k\in\mathbb{N}$. This is sometimes called the Berezin–Li–Yau inequality, as it can be viewed as a consequence of Berezin’s phase-space minimum principle via Legendre duality. The sharp constant matches the leading Weyl law asymptotics for eigenvalues and partial sums (Steinerberger, 2024). The underlying technique relies on the Fourier transform—Plancherel’s theorem combined with Bessel’s inequality yields a minimization problem for the $L^2$-mass distribution, solved by rearrangement methods.

2. Lower Bounds for Nonlocal and Fractional Operators

The Li–Yau paradigm extends to nonlocal settings, particularly the fractional Laplacian $(-\Delta)^{\alpha/2}$ and the pseudodifferential Klein–Gordon operator $H_{0,\Omega}=|p|$ with Dirichlet (killing) boundary condition. For $\beta_j$ the eigenvalues of $H_{0,\Omega}$ and for general bounded $\Omega\subset\mathbb{R}^d$, Harrell & Yolcu proved

$$\sum_{j=1}^k \beta_j \ge \frac{d}{d+1}\,C_d\,|\Omega|^{-1/d}\,k^{1+1/d}$$

with $C_d=(4\pi)^{-1}(1+d/2)^{1/d}$. Yolcu subsequently established a two–term refinement: $$\sum_{j=1}^k \beta_j \ge \frac{d}{d+1}\,C_d\,|\Omega|^{-1/d}\,k^{1+1/d} +\; M_d\,\frac{|\Omega|^{1+1/d}}{I(\Omega)}\,k^{1-1/d}$$ where $I(\Omega)$ is the moment of inertia and $M_d$ a dimension-dependent constant. The proof blends Fourier-side rearrangement, coarea formula–based derivative bounds, and a sharp one-dimensional integral inequality for radially decreasing rearrangements (Yolcu, 2009). The second term, of lower order $k^{1-1/d}$, replicates Melas’s correction for the Laplacian and is asymptotically precise.

For the fractional Laplacian, analogous results hold: $$\sum_{j=1}^N\lambda_j \ge \frac{d}{d+\alpha}\left(\frac{(2\pi)^dN}{\omega_d|\Omega|}\right)^{\alpha/d}N -\; \widetilde{C}_{d,\alpha}|\partial\Omega|N^{1+\alpha/d-1/d} + o(N^{1+\alpha/d-1/d})$$ with explicit constants $L_{d,\alpha}^{\mathrm{cl}}$ and $S_{d,\alpha}^{\mathrm{cl}}$ representing the interior and boundary phase-space volumes (Yolcu et al., 2010).

The spectral theory of mixed local-nonlocal operators $\mathcal{L}^{a,b}_\Omega=-a\Delta+b(-\Delta)^s$, $a>0$, $b\in\mathbb{R}$, $s\in(0,1)$, admits Berezin–Li–Yau-type lower bounds given by the maximum of the classical and fractional variants, with explicit dependence on the fractional-Sobolev embedding constant in the slightly negative $b$ regime (Kassymov et al., 21 Jun 2025).

3. Correction Terms and Domain Geometry: Moment of Inertia

A distinctive refinement of Li–Yau-type lower bounds is the appearance of geometric correction terms, notably involving the moment of inertia $I(\Omega)$. For the Laplacian, Melas demonstrated

$$\sum_{j=1}^k\lambda_j \ge \frac{d}{d+2}(2\pi)^2|\Omega|^{-2/d}k^{1+2/d} + M_d\,\frac{|\Omega|}{I(\Omega)}\,k$$

For the Klein–Gordon operator and other pseudodifferential analogues, the analogous correction term features $|\Omega|^{1+1/d}/I(\Omega)\,k^{1-1/d}$. The sharpness is tied to the domain’s geometry; extremal cases saturate the bound only if the mass distribution and inertia profiles fit the rearrangement minimizing scenario (Yolcu, 2009).

4. Generalizations: Poly-Laplacian, Fractional Orders, and Combinatorial Settings

Recent research has extended Li–Yau-type inequalities to poly-Laplacian eigenvalues on arbitrary bounded domains, yielding multi-term lower bounds for sums and individual eigenvalues, formally improving the classical constants and identifying up to five explicit lower order corrections in all dimensions $n\ge2$ and for all orders $l\ge1$ (Ji et al., 6 Aug 2025). These bounds are produced via refined control of the rearrangement critical polynomial equations originating from Fourier analysis and the Ilyin method.

Combinatorial versions have been proven for the gonality of curves over nonarchimedean fields by Cornelissen–Kato–Kool, where the minimal degree of certain harmonic graph morphisms is bounded below by explicit combinations of the first Laplacian eigenvalue and vertex count: $$\mathrm{sgon}(G) \ge \left\lceil \frac{\lambda_G}{\lambda_G+4(\Delta_G+1)}|V(G)| \right\rceil$$ with $\lambda_G$ the first nonzero Laplacian eigenvalue and $\Delta_G$ the maximal vertex degree (Cornelissen et al., 2012).

5. Li–Yau Inequalities for Heat Evolution and Higher Order Derivatives

For positive solutions $u(x,t)$ of the heat equation on manifolds (and in Euclidean space),

$\Delta\log u \ge -\frac{n}{2t}$

and equivalently,

$$\frac{\Delta u}{u} - \frac{|\nabla u|^2}{u^2} \ge -\frac{n}{2t}$$

Li–Chang Hung established a hierarchy of generalizations for second and fourth derivatives via explicit representation formulas, yielding parameterized lower bounds for mixed and gradient-cross terms (Hung, 2023). For instance, for suitable nonnegative parameters $\alpha,\beta,\gamma$,

$$\frac{\Delta u}{u} - \alpha\sum_{i\neq j}\frac{u_{x_ix_j}}{u} - \beta\sum_{i\neq j}\frac{u_{x_i}}{u}\frac{u_{x_j}}{u} - \gamma\frac{|\nabla u|^2}{u^2} \ge -\frac{n}{2t}$$

Higher order Li–Yau-type bounds are also derived for poly-Laplacian eigenvalues, and for stochastic evolution equations under hybrid curvature-dimension conditions (Kräss et al., 2023), as well as for curves in arbitrary codimension relating normalized bending energy to point multiplicity (Miura, 2021).

6. Connections to Curvature-Dimension Frameworks and Probability

In geometric analysis, curvature-dimension conditions (Bakry–Émery CD$(\rho,n)$) are both necessary and sufficient for Li–Yau-type inequalities for Markov generators, leading to improved and unified bounds for gradient and parabolic Harnack estimates, as in the Bakry–Bolley–Gentil formulation: $$\frac{\Gamma(P_tf)}{(P_tf)^2} < \frac{n}{2}\Phi_t\left(\frac{4}{n\rho}\frac{L\,P_tf}{P_tf}\right)$$ with $\Phi_t$ a convex profile controlling sharpness and ultracontractivity (Bakry et al., 2014). The approach generalizes to Finsler and Alexandrov spaces, where analogous maximal principle arguments and Bochner-type identities yield matching constants (Ohta et al., 2011, Ohta et al., 2011, Qian et al., 2011).

Probabilistic versions, exploiting stochastic calculus and semigroup representation, give Li–Yau-type bounds for diffusion processes on manifolds possibly with boundary, incorporating both curvature and boundary second fundamental form lower bounds with explicit stochastic integrals and weight functions (Wang et al., 2024).

7. Optimality, Extensions, and Open Problems

The leading term of all Li–Yau-type lower bounds is asymptotically sharp, consistent with Weyl’s law. Correction terms are not generally improvable in exponent absent additional geometric or analytic constraints. The methodology extends to generalized pseudodifferential operators, reaction-diffusion systems on mixed discrete-continuous spaces (Kräss et al., 2023), to combinatorial graph analogues, and to technical settings like Ricci flow (Song et al., 2023). However, simultaneous sharpness in multi-point (e.g., two-point or more indices) inequalities is obstructed by rigidity of mass distribution; thus, strict improvements over Berezin–Li–Yau are possible in averaged or multi-index settings (Steinerberger, 2024).

Further directions include explicit sharp bounds for fractional Laplacians with rough domains, lower bounds for boundary value problems under geometric flows, and multi-parameter extensions in combinatorial, probabilistic, or geometric settings.

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Key References

• "An Improvement to a Berezin-Li-Yau type inequality for the Klein-Gordon Operator" (Yolcu, 2009)
• "A Berezin-Li-Yau type inequality for the fractional Laplacian on a bounded domain" (Yolcu et al., 2010)
• "Berezin-Li-Yau inequality for mixed local-nonlocal Dirichlet-Laplacian" (Kassymov et al., 21 Jun 2025)
• "On generalized Li-Yau inequalities" (Hung, 2023)
• "Deep estimates for higher eigenvalues of the poly-Laplacian" (Ji et al., 6 Aug 2025)
• "Universal lower bounds for Dirichlet eigenvalues" (Steinerberger, 2024)
• "The Li-Yau inequality and applications under a curvature-dimension condition" (Bakry et al., 2014)
• "Bochner-Weitzenboeck formula and Li-Yau estimates on Finsler manifolds" (Ohta et al., 2011, Ohta et al., 2011)
• "Sharp Spectral Gap and Li-Yau's Estimate on Alexandrov Spaces" (Qian et al., 2011)
• "Probability Versions of Li-Yau Type Inequalities and Applications" (Wang et al., 2024)
• "A combinatorial Li-Yau inequality and rational points on curves" (Cornelissen et al., 2012)
• "Li-Yau type inequality for curves in any codimension" (Miura, 2021)
• "A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound" (Song et al., 2023)