A capacitary approach to Lyapunov-type inequalities for elliptic problems on weighted graphs

Technical Summary: A Capacitary Approach to Lyapunov-type Inequalities for Elliptic Problems on Weighted Graphs

Problem Statement
The paper addresses the lack of investigation into Lyapunov-type inequalities within the context of weighted graphs. While such inequalities are well-established for ordinary and partial differential equations (providing necessary conditions for the existence of nontrivial solutions to boundary value problems and relating to stability and eigenvalue estimates), their discrete counterparts for the discrete $p$-Laplacian on weighted graphs had not been systematically studied. The authors focus on the Dirichlet problem driven by the discrete $p$-Laplacian on a connected, locally finite weighted graph $(V, \omega, \mu)$:
$$\begin{cases} -\Delta^\omega_p u(x) = w(x)|u(x)|^{p-2}u(x), & x \in \Omega, \\ u(x) = 0, & x \in V \setminus \Omega, \end{cases}$$
where $\Omega \subset V$ is a nonempty finite subset, $w$ is a potential, and $1 < p < \infty$. The goal is to derive lower bounds for the potential $w$ (specifically its positive part $w_+$) in terms of geometric and capacitary properties of the domain $\Omega$, ensuring that if a nontrivial solution exists, the potential cannot be arbitrarily small.

Methodology
The authors adopt a capacitary approach, which is the central methodological innovation of the work. Rather than relying solely on metric or volume assumptions, the study is based on:

1. Point $p$-capacities: Defined for a vertex $x \in \Omega$ relative to the domain $\Omega$ as $C_{p}(x, \Omega) = \inf \{ E_p(\phi) : \phi \in X_V(\Omega), \phi(x)=1 \}$, where $E_p$ is the discrete $p$-energy.
2. Capacitary Radii: Two key quantities are introduced:
   • $R_p(\Omega) = \max_{x \in \Omega} C_p(x, \Omega)^{-1}$.
   • $R_{p,s}(\Omega) = \max_{u \in X_V(\Omega), u \not\equiv 0} \frac{\|u\|_{\ell^{ps'}(\Omega, \mu)}^p}{E_p(u)}$, where $s'$ is the conjugate exponent of $s$.
3. Variational Arguments: The proofs utilize variational principles (specifically Lemma 2.1 regarding localized eigenvalue problems) and discrete integration-by-parts identities to relate the energy of a solution to the potential.
4. Geometric Estimates: The general capacitary inequalities are specialized by estimating the capacitary radii under specific geometric assumptions, including bounded geometry, polynomial volume growth, relative isoperimetric inequalities, and specific graph structures (radial path graphs and regular trees).

Key Contributions and Results

1. General Intrinsic Inequalities:
   The paper establishes general Lyapunov-type inequalities valid on any connected locally finite weighted graph, independent of specific metric or growth assumptions.

   • If the Dirichlet problem admits a nontrivial solution, then $\|w_+\|_{\ell^1(\Omega, \mu)} \geq 1/R_p(\Omega)$.
   • More generally, for $1 < s < \infty$, $\|w_+\|_{\ell^s(\Omega, \mu)} \geq 1/R_{p,s}(\Omega)$.
     These results reduce the problem of finding Lyapunov inequalities to the estimation of capacitary radii.

2. Supercritical Regime on Polynomial Growth Graphs:
   Under assumptions of bounded geometry, polynomial volume growth with exponent $D$, and a relative isoperimetric inequality, the authors derive an "inner-radius" Lyapunov inequality for the supercritical case $p > D$:
   $$\|w_+\|_{\ell^1(\Omega, \mu)} \geq c r_\Omega^{-(p-D)}$$
   where $r_\Omega$ is the inner radius of $\Omega$. The paper proves that the exponent $p-D$ is sharp by constructing counterexamples on the integer lattice $\mathbb{Z}^D$ where the inequality fails for any exponent $\beta < p-D$.

3. Radial Path Graphs (Three Regimes):
   For weighted radial path graphs with effective dimension $D$, the authors derive explicit inequalities covering three distinct regimes:

   • $p > D$: Power scale: $\|w_+\|_{\ell^1} \geq c \tau_\Omega^{-(p-D)}$.
   • $p = D$: Logarithmic scale: $\|w_+\|_{\ell^1} \geq c (\log \tau_\Omega)^{1-p}$.
   • $1 < p < D$: $L^s$ scale (for $s > D/p$): $\|w_+\|_{\ell^s} \geq c \tau_\Omega^{-(p-D)/s}$.
     Here, $\tau_\Omega$ is the radial outer radius. The paper demonstrates the sharpness of these scales, noting that the relevant dimension is the "capacitary dimension" encoded by the weights and measures, not merely the combinatorial dimension.

4. Regular Trees:
   For $q$-regular trees ($q \geq 3$), the authors obtain a uniform Lyapunov-type inequality with an explicit sharp constant $c_{p,q}$:
   $$\|w_+\|_{\ell^1(\Omega, \mu)} \geq c_{p,q} = q \left( 1 - (q-1)^{-\frac{1}{p-1}} \right)^{p-1}.$$
   This constant is shown to be optimal.

5. Applications to Eigenvalues:
   As a direct application, the paper derives lower bounds for the first weighted Dirichlet eigenvalue $\lambda_{1,p}(\Omega, w)$ of the discrete $p$-Laplacian. These bounds are expressed in terms of the capacitary radii and the norms of the weight $w$, providing discrete analogues to continuous eigenvalue estimates derived from Lyapunov inequalities.

Significance and Claims
The paper claims to initiate the study of Lyapunov-type inequalities for discrete $p$-Laplacians on weighted graphs from a capacitary perspective. Its primary significance lies in:

• Unification: Providing a framework that works for arbitrary graphs via capacitary radii, which can then be specialized to various geometric settings.
• Sharpness: Unlike many previous works that provide existence of constants, this paper rigorously proves the sharpness of the exponents and constants in specific geometric settings (lattices, radial paths, trees).
• Dimensional Insight: Highlighting that for graphs, the "effective dimension" governing the inequalities is determined by the interplay of vertex measures and edge weights (capacitary dimension), which can differ from the combinatorial dimension.
• Eigenvalue Estimates: Offering new, explicit lower bounds for the first Dirichlet eigenvalue that complement existing Cheeger-type and Faber-Krahn-type estimates.

The authors do not claim to solve the general case for all possible graph geometries without assumptions but rather provide a robust method to derive sharp inequalities once the specific capacitary properties of a graph are understood.