arXiv subjects

Zhonggan Huang

Publications and source records attributed to Zhonggan Huang.

8 recordsLinked to original sources

arXiv2026

An indefinite Coulomb interaction from the Steklov spectrum of perforated manifolds

We establish optimal convergence rates for Steklov eigenvalues and harmonically extended eigenfunctions toward their weighted Laplace--Beltrami counterparts on a closed manifold perforated by many small geodesic balls. The holes have radii that scale critically with respect to their spacing in the sense that the boundary area of each hole balances with the weighted volume of its Voronoi cell. We then derive a higher-order expansion of the Steklov eigenvalues; in dimensions two and three, we identify two correction scales. The expansion is governed by an indefinite Coulomb-type energy of the discrepancy between the boundary and bulk measures, mediated by the reduced Green function of the limiting operator. The proof relies on sharp estimates for certain auxiliary functions that we introduce in order to quantify the discrepancy measure between the surface measure on the holes and their background density. Our paper serves to bridge an emerging literature in spectral geometry with one on systems of points interacting via Coulomb-type energies.

math.AP↗

arXiv2026

On the continuum limit of t-SNE for data visualization

This work is concerned with the continuum limit of a graph-based data visualization technique called the t-Distributed Stochastic Neighbor Embedding (t-SNE), which is widely used for visualizing data in a variety of applications, but is still poorly understood from a theoretical standpoint. The t-SNE algorithm produces visualizations by minimizing the Kullback-Leibler divergence between similarity matrices representing the high dimensional data and its low dimensional representation. We prove that as the number of data points $n \to \infty$, after a natural rescaling and in applicable parameter regimes, the Kullback-Leibler divergence is consistent as the number of data points $n \to \infty$ and the similarity graph remains sparse with a continuum variational problem that involves a non-convex gradient regularization term and a penalty on the magnitude of the probability density function in the visualization space. These two terms represent the continuum limits of the attraction and repulsion forces in the t-SNE algorithm. Due to the lack of convexity in the continuum variational problem, the question of well-posedeness is only partially resolved. We show that when both dimensions are $1$, the problem admits a unique smooth minimizer, along with an infinite number of discontinuous minimizers (interpreted in a relaxed sense). This aligns well with the empirically observed ability of t-SNE to separate data in seemingly arbitrary ways in the visualization. The energy is also very closely related to the famously ill-posed Perona-Malik equation, which is used for denoising and simplifying images. We present numerical results validating the continuum limit, provide some preliminary results about the delicate nature of the limiting energetic problem in higher dimensions, and highlight several problems for future work.

stat.ML↗

arXiv2026

On the attainability of the singular Wiener bound

We characterize the lower and upper attainability of the Wiener bound (also known as the conductive analogue of the Voigt-Reuss-Hill bound in elasticity theory) for singularly distributed conductive material mixtures. For the lower attainability we consider mixtures in which high-conductance materials support on sets having finite one-dimensional Hausdorff measures. We show that, under a mild coercivity condition, the kernel of the effective tensor of the mixture is equal to the orthogonal complement of the homotopy classes of closed paths in the supporting set. This shows that a periodic planar network has positive definite effective tensor, i.e., it is resilient to fluctuations, if and only if the network is reticulate. We prove a geometric characterization of the upper attainability by applying a transformation from varifolds to matrix-valued measures. We show that this transformation leads to an equivalence between two distinct notions from material science and geometric measure theory respectively: conductance maximality and area criticality. Based on this relation we show a pointwise dimension bound for mixtures that attain the upper Wiener bound by applying a fractional version of the monotonicity formula for stationary varifolds. This dimension bound illustrates how the maximality condition constrains the local anisotropy and the local distribution of conductance magnitudes. Both the lower and upper attainability results have potential novel applications in modeling leaf venation patterns.

math.AP↗

arXiv2025

Homogenization of a vertical oscillating Neumann condition

We homogenize the Laplace and heat equations with the Neumann data oscillating in the ``vertical" $u$-variable. These are simplified models for interface motion in heterogeneous media, particularly capillary contact lines. The homogenization limit reveals a pinning effect at zero tangential slope, leading to a novel singularly anisotropic pinned Neumann condition. The singular pinning creates an unconstrained contact set, generalizing the contact set in the classical thin obstacle problem. We establish a comparison principle for the heat equation with this new type of boundary condition. The comparison principle enables a proof of homogenization via the method of half-relaxed limits from viscosity solution theory. Our work also demonstrates, for the first time in a PDE problem in multiple dimensions, the emergence of rate-independent pinning from gradient flows with wiggly energies. Prior limit theorems of this type, in rate-independent contexts, were limited to ODEs and PDEs in one dimension.

math.AP↗

arXiv2024

Regularity theory of a gradient degenerate Neumann problem

We study the regularity and comparison principle for a gradient degenerate Neumann problem. The problem is a generalization of the Signorini or thin obstacle problem which appears in the study of certain singular anisotropic free boundary problems arising from homogenization. In scaling terms, the problem is critical since the gradient degeneracy and the Neumann PDE operator are of the same order. We show the (optimal) $C^{1,\frac{1}{2}}$ regularity in dimension $d=2$ and we show the same regularity result in $d\geq 3$ conditional on the assumption that the degenerate values of the solution do not accumulate. We also prove a comparison principle characterizing minimal supersolutions, which we believe will have applications to homogenization and other related scaling limits.

math.AP↗

arXiv2022

Is Mean Curvature Flow a Gradient Flow?

It is well-known that the mean curvature flow is a formal gradient flow of the perimeter functional. However, by the work of Michor and Mumford [7,8], the formal Riemannian structure that is compatible with the gradient flow structure induces a degenerate metric on the space of hypersurfaces. It is then natural to ask whether there is a nondegenerate metric space of hypersurfaces, on which the mean curvature flow admits a gradient flow structure. In this paper we study the mean curvature flow on two nondegenerate metric spaces of simple closed plane curves: the uniformness-preserving metric structure proposed by Shi and Vorotnikov [11] and the curvature-weighted structure proposed by Michor and Mumford [8], and prove that the mean curvature flow is not a gradient flow in either of the spaces.

math.DG↗

arXiv2020

Homogenization of Enhancing Thin Layers

This paper derives an explicit formula for the effective diffusion tensor by using the solutions to some effective cell problems after homogenizing Road effective boundary conditions (EBCs). The concept of Road EBCs was proposed recently by H. Li and X. Wang, and in this paper, we extend the effective conditions on closed curves to those on patterns, especially on the included nodes. We also prove that homogenization process commutes with the derivation of Road EBCs. By analyzing the effective diffusion tensor, we obtain several rules for maximizing its trace with given Road-effective-diffusivity/scale and length/scale in each cell and define a notion of balanced patterns. Moreover, we give an estimate of the trace of effective diffusion tensor of general patterns as Road-effective-diffusivity/scale goes to infinity.

math.AP↗

arXiv2018

Aut($Γ$)=det($M_Γ$)

In this article, we will show that the automorphism group of any hypergraph is essentially equal to the determinant of some matrix over a ring generated from the set of ground points. With this, we are also able to determine whether two graphs are isomorphic.

math.GM↗