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Nam Q. Le

Publications and source records attributed to Nam Q. Le.

At least 19 recordsLinked to original sources
arXiv2026

Isolation of scalar Allen-Cahn local minimizers

We show that scalar local minimizers of the Allen-Cahn functional with the double-well potential and homogeneous Neumann boundary conditions on a bounded Lipschitz domain are isolated in the $L^1$-topology. This affirmatively answers a question raised by Kohn and Sternberg (Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989)). The proof implements an analytic Lyapunov-Schmidt reduction inspired by the theory of monotone dynamical systems in works of Jiang-Yu and Hirsch-Smith in an elliptic setting.

math.AP↗
arXiv2026

Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three

For each nonnegative integer $m$, we construct smooth symmetric $3\times 3$ coefficient matrices $A_m$ satisfying the fixed ellipticity bound \[ I\leq A_m\leq 2^{81}I \] for which the smooth solutions of uniformly elliptic equations in nondivergence form \[ \text{tr}(A_m(x)D^2 u_m)=A_m(x):D^2u_m=0\qquad\text{in }B_2\subset {\mathbb R}^3 \] have common Dirichlet data, satisfy $\|u_m\|_{L^\infty(B_2)}\leq1$, but \[ \lim_{m\to \infty}\|Du_m\|_{L^1(B_1)}=\infty. \] Thus, there is no interior $W^{1,1}$ estimate depending only on ellipticity in dimension three, and consequently no such $W^{1,p}$ estimate for any $p\geq1$. This resolves in the negative an open question raised by Nadirashvili, Tkachev, and Vlăduţ. The construction also gives a uniformly convergent limit $u\notin \text{BV}_{\rm loc}(B_1)$ for a measurable uniformly elliptic coefficient matrix obtained as an $L^1$ limit of the $A_m$.

math.AP↗
arXiv2026

A Green's function approach to linearized Monge-Ampère equations in divergence form and application to singular Abreu type equations

In this paper, we establish local and global regularity estimates for linearized Monge-Ampère equations in divergence form via critical Lorentz space estimates for the Green's function of the linearized Monge-Ampère operator and its gradient. These estimates hold under suitable conditions on the data and the convex Monge-Ampère potential is assumed to have Hessian determinant bounded between two positive constants. As an application, we obtain the solvability in all dimensions of the second boundary value problem for a class of singular fourth-order Abreu type equations that arise from the approximation analysis of variational problems subject to convexity constraints.

math.AP↗
arXiv2026

Coupling Language Models with Physics-based Simulation for Synthesis of Inorganic Materials

Modern generative machine learning (ML) models can propose novel inorganic crystalline materials with targeted properties; however, synthesis planning of these materials remains difficult due to the complexity of the associated physical processes and limited availability of computational tools. We introduce a novel hybrid framework to evaluate Large Language Models (LLMs) in inorganic synthesis planning by combining thermodynamic databases with simplified kinetics models to approximate realistic synthesis conditions. As a case study, we focus on the niobium-oxygen system, which features multiple industrially relevant oxide phases with well-characterized data. In computational simulations, we compare LLM-generated synthesis routes with classical path-planning algorithms, showing that the implicit priors in LLMs can yield more viable strategies. In our evaluation setting, classical search methods serve primarily as a foil rather than a direct competitor. This illustrates the relative complexity of the problem and highlights where the LLM's implicit priors add value.

cs.AI↗
arXiv2026

Global $C^{1,β}$ and $W^{2, p}$ regularity for some singular Monge-Ampère equations

We establish global $C^{1,β}$ and $W^{2, p}$ regularity for singular Monge-Ampère equations of the form \[\det D^2 u \sim \text{dist}^{-α}(\cdot,\partialΩ),\quad α\in (0, 1),\] under suitable conditions on the boundary data and domains. Our results imply that the convex Aleksandrov solution to the singular Monge-Ampère equation \[\det D^2 u=|u|^{-α}\quad \text{in}\quadΩ,\quad u=0\quad \text{in}\quad \partialΩ, \quad α\in (0, 1),\] where $Ω$ is a $C^3$, bounded, and uniformly convex domain, is globally $C^{1,β}$ and belongs to $W^{2, p}$ for all $p<1/α$.

math.AP↗
arXiv2026

A Variational Approach to Degenerate Monge--Ampère Equations with Mixed Measures and Monotonicity

We study the solvability and uniqueness for several degenerate Monge--Ampère equations including the Monge--Ampère eigenvalue problem in real Euclidean spaces that involve singular Borel measures. Our approach systematically analyzes the Monge--Ampère energy from the variational point of view and appropriately exploits monotonicity arguments. Our main tools consist of the mixed Monge--Ampère measure, Aleksandrov--Blocki--Jerison-type maximum principles, integration by parts, convex envelope, and comparison principles for subcritical equations. For the Monge--Ampère eigenvalue problem, we contrast the analysis within and without the energy class; even if it might not have solutions in the energy class, we show that the infimum of the Rayleigh quotient can be approximated from above by Monge--Ampère eigenvalues of the truncated measures, and by Rayleigh quotients of an inverse iterative scheme. We give examples showing that for very singular Borel measures, the Monge--Ampère eigenvalue problem has only solutions outside the energy class together with symmetry breaking and nonuniqueness.

math.AP↗
arXiv2025

Convergence of an iterative scheme for the Monge-Ampère eigenvalue problem with general initial data

In this note, we revisit an iterative scheme, due to Abedin and Kitagawa (Inverse Iteration for the Monge-Ampère Eigenvalue Problem, Proc. Amer. Math. Soc. 148 (2020), no. 11, 4875--4886), to solve the Monge-Ampère eigenvalue problem on a general bounded convex domain. Using a nonlinear integration by parts, we show that the scheme converges for all convex initial data having finite and nonzero Rayleigh quotient to a nonzero Monge-Ampère eigenfunction. As an application, we obtain an energy characterization of the Monge--Ampère eigenfunctions.

math.AP↗
arXiv2025

Large dimension behavior of the Hessian eigenvalues of the unit balls

We show that a sequence of $k$-Hessian eigenvalues of the unit ball in ${\mathbb R}^n$ stays bounded as long as the ratio $n/k$ stays bounded. Moreover, we identify their growth of order at least $(2-1/k)$ in $n/k$. In the case $k=n$, we show that the Monge--Ampère eigenvalues of the unit balls tend to $4$ in the large dimension limit.

math.AP↗
arXiv2025

Global Lipschitz and Sobolev estimates for the Monge-Ampère eigenfunctions of general bounded convex domains

We show that the Monge-Ampère eigenfunctions of general bounded convex domains are globally Lipschitz. The same result holds for convex solutions to degenerate Monge-Ampère equations of the form $\det D^2 u =M|u|^p$ with zero boundary condition on general bounded convex domains in ${\mathbb R}^n$ within the sharp threshold $p>n-2$. As a consequence, we obtain global $W^{2, 1}$ estimates for these solutions.

math.AP↗
arXiv2024

Self-supervised learning for crystal property prediction via denoising

Accurate prediction of the properties of crystalline materials is crucial for targeted discovery, and this prediction is increasingly done with data-driven models. However, for many properties of interest, the number of materials for which a specific property has been determined is much smaller than the number of known materials. To overcome this disparity, we propose a novel self-supervised learning (SSL) strategy for material property prediction. Our approach, crystal denoising self-supervised learning (CDSSL), pretrains predictive models (e.g., graph networks) with a pretext task based on recovering valid material structures when given perturbed versions of these structures. We demonstrate that CDSSL models out-perform models trained without SSL, across material types, properties, and dataset sizes.

cs.LG↗
arXiv2024

Singular Abreu equations and linearized Monge-Ampère equations with drifts

We study the solvability of singular Abreu equations which arise in the approximation of convex functionals subject to a convexity constraint. Previous works established the solvability of their second boundary value problems either in two dimensions, or in higher dimensions under either a smallness condition or a radial symmetry condition. Here, we solve the higher dimensional case by transforming singular Abreu equations into linearized Monge-Ampère equations with drifts. We establish global Hölder estimates for the linearized Monge-Ampère equation with drifts under suitable hypotheses, and then use them to the regularity and solvability of the second boundary value problem for singular Abreu equations in higher dimensions. Many cases with general right-hand side will also be discussed.

math.AP↗
arXiv2024

On global $W^{2,δ}$ estimates for the Monge-Ampère equation on general bounded convex domains

We establish global $W^{2,δ}$ estimates, for all $δ<\frac{1}{n-1}$, for convex solutions to the Monge-Ampère equation with positive $C^{2,β}$ right-hand side and zero boundary values on general bounded convex domains in ${\mathbb R}^n$ ($n\geq 2$). We exhibit examples showing that global $W^{2, \frac{n}{2(n-1)}}$ estimates fail in all dimensions, so the range of $δ$ is sharp in two dimensions.

math.AP↗
arXiv2023

Twisted Harnack inequality and approximation of variational problems with a convexity constraint by singular Abreu equations

We show in all dimensions that minimizers of variational problems with a convexity constraint, which arise from the Rochet-Choné model with a quadratic cost in the monopolist's problem in economics, can be approximated in the uniform norm by solutions of singular Abreu equations. The difficulty of our Abreu equations consists of having singularities that occur only in a proper subdomain and they cannot be completely removed by any transformations. To solve them, we rely on a new tool which we establish here: a Harnack inequality for singular linearized Monge-Ampère type equations that satisfy certain twisted conditions.

math.AP↗
arXiv2023

Evaluating the diversity and utility of materials proposed by generative models

Generative machine learning models can use data generated by scientific modeling to create large quantities of novel material structures. Here, we assess how one state-of-the-art generative model, the physics-guided crystal generation model (PGCGM), can be used as part of the inverse design process. We show that the default PGCGM's input space is not smooth with respect to parameter variation, making material optimization difficult and limited. We also demonstrate that most generated structures are predicted to be thermodynamically unstable by a separate property-prediction model, partially due to out-of-domain data challenges. Our findings suggest how generative models might be improved to enable better inverse design.

cond-mat.mtrl-sci↗
arXiv2022

Closed-loop machine learning for discovery of novel superconductors

The discovery of novel materials drives industrial innovation, although the pace of discovery tends to be slow due to the infrequency of "Eureka!" moments. These moments are typically tangential to the original target of the experimental work: "accidental discoveries". Here we demonstrate the acceleration of intentional materials discovery - targeting material properties of interest while generalizing the search to a large materials space with machine learning (ML) methods. We demonstrate a closed-loop ML discovery process targeting novel superconducting materials, which have industrial applications ranging from quantum computing to sensors to power delivery. By closing the loop, i.e. by experimentally testing the results of the ML-generated superconductivity predictions and feeding data back into the ML model to refine, we demonstrate that success rates for superconductor discovery can be more than doubled. In four closed-loop cycles, we discovered a new superconductor in the Zr-In-Ni system, re-discovered five superconductors unknown in the training datasets, and identified two additional phase diagrams of interest for new superconducting materials. Our work demonstrates the critical role experimental feedback provides in ML-driven discovery, and provides definite evidence that such technologies can accelerate discovery even in the absence of knowledge of the underlying physics.

cond-mat.supr-con↗
arXiv2022

Remarks on sharp boundary estimates for singular and degenerate Monge-Ampère equations

By constructing appropriate smooth, possibly non-convex supersolutions, we establish sharp lower bounds near the boundary for the modulus of nontrivial solutions to singular and degenerate Monge-Ampère equations of the form $\det D^2 u =|u|^q$ with zero boundary condition on a bounded domain in $\mathbb{R}^n$. These bounds imply that currently known global Hölder regularity results for these equations are optimal for all $q$ negative, and almost optimal for $0\leq q\leq n-2$. Our study also establishes the optimality of global $C^{\frac{1}{n}}$ regularity for convex solutions to the Monge-Ampère equation with finite total Monge-Ampère measure. Moreover, when $0\leq q<n-2$, the unique solution has its gradient blowing up near any flat part of the boundary. The case of $q$ being $0$ is related to surface tensions in dimer models. We also obtain new global log-Lipschitz estimates, and apply them to the Abreu's equation with degenerate boundary data.

math.AP↗
arXiv2022

Curvature-informed multi-task learning for graph networks

Properties of interest for crystals and molecules, such as band gap, elasticity, and solubility, are generally related to each other: they are governed by the same underlying laws of physics. However, when state-of-the-art graph neural networks attempt to predict multiple properties simultaneously (the multi-task learning (MTL) setting), they frequently underperform a suite of single property predictors. This suggests graph networks may not be fully leveraging these underlying similarities. Here we investigate a potential explanation for this phenomenon: the curvature of each property's loss surface significantly varies, leading to inefficient learning. This difference in curvature can be assessed by looking at spectral properties of the Hessians of each property's loss function, which is done in a matrix-free manner via randomized numerical linear algebra. We evaluate our hypothesis on two benchmark datasets (Materials Project (MP) and QM8) and consider how these findings can inform the training of novel multi-task learning models.

cs.LG↗