Singular limit of lattice graphs

Technical Summary: Singular Limit of Lattice Graphs

Problem Statement
This paper investigates the asymptotic behavior of solutions to the nonlinear Schrödinger (NLS) equation on $d$-dimensional lattice graphs ($\mathbb{Z}^d_\varepsilon$) as the edge length $\varepsilon$ tends to zero. The central question is whether the ground states (minimizers of action and energy functionals) on these discrete structures converge to the corresponding ground states on the continuous Euclidean space $\mathbb{R}^d$. While analogous results for metric grids were established in previous work (specifically [11]), this study addresses the lattice graph setting, which presents distinct analytical challenges due to the absence of direct restriction and extension estimates between lattice functions and Euclidean functions. Furthermore, the paper extends the analysis to Sobolev supercritical cases ($p > 2^*$ for action, $p > 2 + 4/d$ for energy) and the critical case ($p = 2^*$), areas previously unexplored for lattice graphs.

Methodology
The authors develop a unified framework connecting lattice graphs, metric grids, and Euclidean spaces through the following methodological steps:

1. Establishment of Restriction and Extension Estimates: A primary technical hurdle is the lack of direct estimates between functions on lattice graphs $V^d_\varepsilon$ and $\mathbb{R}^d$. To overcome this, the authors introduce an intermediate space: the metric grid $G^d_\varepsilon$. They construct linear extension operators (mapping lattice functions to metric grid functions via piecewise-affine interpolation) and restriction operators (mapping metric grid functions to lattice values).
2. Norm Equivalence and Inequalities: Using these operators, the authors prove that the Sobolev norms on lattice graphs and metric grids are equivalent up to scaling factors involving $\varepsilon$. They derive new Gagliardo-Nirenberg type inequalities for concentrated nonlinearities on metric grids, which are crucial for controlling the interaction between scales.
3. Variational Analysis: The study employs variational methods to analyze the action functional $\hat{J}_{\omega, V^d_\varepsilon}$ and the energy functional $\hat{E}_{V^d_\varepsilon}$. By establishing that sequences of ground states on lattice graphs form minimizing sequences for the corresponding functionals on $\mathbb{R}^d$ (after appropriate scaling and extension), the authors utilize compactness arguments to prove convergence.
4. Handling Supercritical and Critical Cases: For supercritical exponents ($p > 2^*$), the authors utilize scaling arguments to show that the infimum of the action functional on $\mathbb{R}^d$ is zero, leading to the convergence of lattice ground states to zero. For energy functionals in the supercritical mass-constrained regime, they demonstrate divergence.

Key Contributions and Results

• Convergence in the Subcritical Regime:

  • Action Ground States: For $2 < p < 2^*$, the paper proves that the extensions of action ground states on lattice graphs converge strongly in $H^1(\mathbb{R}^d)$ to the unique positive ground state of the NLS equation on $\mathbb{R}^d$. The authors provide precise error estimates for the convergence rate, distinguishing between dimensions $d=1,2$ and $d \ge 3$, and between sub-subcritical and super-subcritical ranges of $p$.
  • Energy Ground States: Similarly, for $2 < p < 2 + 4/d$, energy ground states with fixed mass converge strongly to the corresponding energy ground states on $\mathbb{R}^d$.
  • Refinement of Previous Work: The authors improve upon the convergence rate results of [11] for metric grids by establishing a new Gagliardo-Nirenberg type inequality, answering an open question posed by Dovetta regarding the rate $O(\varepsilon^{\frac{d-2}{2}(2^*-p)})$.

• Supercritical and Divergence Results:

  • Action Supercritical ($d \ge 3, p > 2^*$): The authors show that the scaled action functional value tends to zero, and the ground states converge strongly to zero in $H^1(\mathbb{R}^d)$.
  • Energy Supercritical ($d \ge 2, p > 2 + 4/d$): In contrast to the subcritical case, the paper demonstrates that for large masses, the energy ground states on lattice graphs diverge in $H^1(\mathbb{R}^d)$ as $\varepsilon \to 0$, with the energy tending to $-\infty$.

• Multiplicity of Critical Points:

  • As a by-product of the singular limit analysis, the paper establishes a multiplicity result for fixed-mass critical points of the energy functional on lattice graphs. Specifically, for $d \ge 2$ and $p \in (2 + 4/d, 2^*)$, there exist large masses $\mu$ for which the energy functional possesses at least two distinct positive critical points: one being the energy ground state and the other being an action ground state (which is not an energy ground state). This highlights a failure of equivalence between action and energy ground states on lattice graphs, a phenomenon not observed in the continuous setting or on certain metric graphs.

• Optimal Constants in Gagliardo-Nirenberg Inequalities:

  • The paper investigates the sharp constants $K_{q, V^d_1}$ for Gagliardo-Nirenberg inequalities on lattice graphs. It proves that $K_{q, V^d_1} \ge K_{q, \mathbb{R}^d}$.
  • Crucially, for the range $q \in [2 + 4/d, 2^*)$ (and $q=2^*$ for $d \ge 3$), the authors prove that $K_{q, V^d_1} = K_{q, \mathbb{R}^d}$.
  • They further show that for $q \in (2 + 4/d, 2^*)$, the optimal constant on the lattice is not attained.
  • The paper settles an open problem regarding the case $d \ge 3$ and $q = 2^*$, establishing that the optimal constant on the lattice equals that on $\mathbb{R}^d$ (specifically $K_{2^*, V^d_1} = K_{2^*, \mathbb{R}^d}$), a result that contrasts with the behavior of $L^1$-based Sobolev inequalities on metric grids.

Significance and Claims
The paper claims to provide a unified framework for studying singular limit problems on lattice graphs, bridging the gap between discrete microscale structures and continuous macroscale models. Its significance lies in:

1. Extending the Literature: It substantially extends existing results by covering the Sobolev supercritical and critical cases, which were previously unstudied for lattice graphs.
2. Resolving Open Problems: It positively answers a question posed by Dovetta regarding the convergence rate of ground states and establishes new Gagliardo-Nirenberg inequalities for the critical exponent case.
3. Revealing Discrete Phenomena: The multiplicity result for fixed-mass critical points reveals a structural difference between lattice graphs and continuous domains (or certain metric graphs), where action and energy ground states are not always equivalent.
4. Numerical Implications: The authors suggest that the result showing ground states on finite graphs can approximate those on $\mathbb{R}^d$ (Remark 1.5) holds significance for numerical analysis.

The paper maintains a rigorous variational approach, relying on the construction of specific extension/restriction operators and the derivation of new discrete inequalities to handle the lack of smoothness inherent in lattice structures.