arXiv subjects

Sze-Man Ngai

Publications and source records attributed to Sze-Man Ngai.

10 recordsLinked to original sources

arXiv2026

Krein-Feller operators on Riemannian manifolds: compactness of embedding and Hodge's theorem

For a bounded open set Omega in a complete oriented Riemannian n-manifold and a positive finite Borel measure mu with support contained in Omega, we define an associated Krein-Feller operators (or Laplacian) Delta_mu by assuming the Poincar'e inequalities for the measure mu. We obtain sufficient conditions for the operator to have compact resolvent and in this case, we prove the Hodge theorem for functions, which states that there exists an orthonormal basis of L^2(Omega,mu) consisting of eigenfunctions of Delta_mu, the eigenspaces are finite-dimensional, and the eigenvalues of -Delta_mu are real, countable, and increasing to infinity. One of these sufficient conditions is that the lower L^infty-dimension dim_infty(mu) of mu is greater than n-2. We prove that the compactness of embedding for functions also hold for measures without compact support, provided the manifold is of bounded geometry. The main idea of our proof is to use Toponogov's and Rauch's comparison theorems to extend a classical compact embedding theorem of Maz'ja to Riemannian manifolds. For a compact Riemannian manifold, using the above results, we also obtain sufficient conditions for Hodge Laplacian on k-forms, to have compact resolvent. Our result extends the classical Hodge theorem to Krein-Feller operators. We study the condition dim_infty(mu)>n-2 for self-similar and self-conformal measures. Results in this paper extend analogous ones by Hu et al. in J. Funct. Anal., which are established for measures on R^n.

math.FA↗

arXiv2025

Hodge-de Rham Theory on Higher-Dimensional Level-L Sierpinski Gaskets

This paper extends the Hodge-de Rham theory of Aaron \textit{et al.} [Commun. Pure Appl. Anal. {\bf 13} (2014)] to higher-dimensional level-$l$ Sierpinski gaskets $SG_{\ell}^{n},$ providing a framework for analyzing differential forms and Laplacians on these fractal structures. We construct a sequence of graphs approximating $SG_{\ell}^{n}$ and define $k$-forms, de Rham derivatives, and their duals on these graphs. We prove that the extension of a $1$-form on a generation-$m$ graph to a $1$-form on a generation-$(m+1)$ graph is harmonic. We obtain a basis for the space of harmonic $1$-forms. We also explore the properties of $2$-forms on the level-$3$ Sierpinski gasket, under the assumptions that the $2$-forms are absolutely continuous with respect to the Kusuoka measure or the standard self-similar measure and that the Radon-Nikodym derivatives are continuous.

math.DG↗

arXiv2024

Iterated relation systems on Riemannian manifolds

For fractals on Riemannian manifolds, the theory of iterated function systems often does not apply well directly, as fractal sets are often defined by relations that are multivalued or non-contractive. To overcome this difficulty, we introduce the notion of iterated relation systems. We study the attractor of an iterated relation system and formulate a condition under which such an attractor can be identified with that of an associated graph-directed iterated function system. Using this method, we obtain dimension formulas for the attractor of an iterated relation system under the graph open set condition or the graph finite type condition. This method improves the one in [Ngai-Xu, J. Geom. Anal. {\bf 33} (2023), 262], which relies on knowing the specific structure of the attractor.

math.DS↗

arXiv2024

Nodal sets and continuity of eigenfunctions of Krein-Feller operators on Riemannian manifolds

Let $d\geq1$, $Ω$ be a bounded domain of a smooth complete Riemannian d-manifold M, and $μ$ be a positive finite Borel measure with compact support in $\overlineΩ$. We prove the Courant nodal domain theorem for the eigenfunctions of Kreĭn-Feller operator $Δ_μ$ under the assumption that such eigenfunctions are continuous on $\overlineΩ$. For $d\geq2$, We prove that on a bounded domain $Ω\subset M$ with smooth boundary and on which the Green's function of the Laplace-Beltrami operator exists, the eigenfunctions of $Δ_μ$ are continuous on $Ω$. We also prove that if M is compact and $\partial M=\emptyset$, then the eigenfuctions of $Δ_μ$ are continuous on M.

math.FA↗

arXiv2024

Nodal sets and continuity of eigenfunctions of Kre\uı-Feller operators

Let $μ$ be a compactly supported positive finite Borel measure on $\R^{d}$. Let $0<λ_{1}\leqλ_{2}\leq\ldots$ be eigenvalues of the Kre$\breve{ı}$n-Feller operator $Δ_μ$. We prove that, on a bounded domain, the nodal set of a continuous $λ_{n}$-eigenfunction of a Kre$\breve{ı}$n-Feller operator divides the domain into at least 2 and at most $n+r-1$ subdomains, where $r$ is the multiplicity of $λ_{n}$. This work generalizes the nodal set theorem of the classical Laplace operator to Kre$\breve{ı}$n-Feller operators on bounded domains. We also prove that on bounded domains on which the classical Green function exists, the eigenfunctions of a Kre$\breve{ı}$n-Feller operator are continuous.

math.AP↗

arXiv2024

Differential equations defined by Kre\uın-Feller operators on Riemannian manifolds

We study linear and semi-linear wave, heat, and Schrödinger equations defined by Kre\uın-Feller operator $-Δ_μ$ on a complete Riemannian $n$-manifolds $M$, where $μ$ is a finite positive Borel measure on a bounded open subset $Ω$ of $M$ with support contained in $\overlineΩ$. Under the assumption that $\underline{\operatorname{dim}}_{\infty}(μ)>n-2$, we prove that for a linear or semi-linear equation of each of the above three types, there exists a unique weak solution. We study the crucial condition $\dim_(μ)>n-2$ and provide examples of measures on $\mathbb{S}^2$ and $\mathbb{T}^2$ that satisfy the condition. We also study weak solutions of linear equations of the above three classes by using examples on $\mathbb{S}^1$

math.FA↗

arXiv2024

Hodge Theorem for Krein-Feller operators on compact Riemannian manifolds

For an open set in a compact smooth oriented Riemannian n-manifold and a positive finite Borel measure with support contained in the closure of the open set, we define an associated Krein-Feller operator on k-forms by assuming the Poincare inequality. Krein-Feller operators on Euclidean space have been studied extensively in fractal geometry. Using results established by the authors, we obtain sufficient conditions for Krein-Feller operator to have compact resolvent. Under these conditions, we prove the Hodge theorem for forms, which states that there exists an orthonormal basis of L2 consisting of eigenforms of Krein-Feller operator, the eigenspaces are finite-dimensional, and the eigenvalues of Krein-Feller operator are real, countable, and increasing to infinity. One of these sufficient conditions is that the dimension of the measure is greater than n-2. Our result extends the classical Hodge theorem to Krein-Feller operators.

math.FA↗

arXiv2022

Separation conditions for iterated function systems with overlaps on Riemannian manifolds

We formulate the weak separation condition and the finite type condition for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, and generalize the main theorems by Lau \textit{et al.} in [Monatsch. Math. 156 (2009), 325-355]. We also obtain a formula for the Hausdorff dimension of a self-similar set defined by an iterated function system satisfying the finite type condition, generalizing a corresponding result by Jin-Yau [Comm. Anal. Geom. 13 (2005), 821--843] and Lau-Ngai [Adv. Math. 208 (2007), 647-671] on Euclidean spaces. Moreover, we obtain a formula for the Hausdorff dimension of a graph self-similar set generated by a graph-directed iterated function system satisfying the graph finite type condition, extending a result by Ngai \textit{et al.} in [Nonlinearity 23 (2010), 2333--2350].

math.FA↗

arXiv2022

Existence of $L^q$-dimension and entropy dimension of self-conformal measures on Riemannian manifolds

Peres and Solomyak proved that on $\mathbb R^n$, the limits defining the $L^q$-dimension for any $q\in(0,\infty)\setminus\{1\}$, and the entropy dimension of a self-conformal measure exist, without assuming any separation condition. By introducing the notions of heavy maximal packings and partitions, we prove that on a doubling metric space the $L^q$-dimension, $q\in(0,\infty)\setminus\{1\}$, is equivalent to the generalized dimension. We also generalize the result on the existence of the $L^q$-dimension to self-conformal measures on complete Riemannian manifolds with the doubling property. In particular, these results hold for complete Riemannian manifolds with nonnegative Ricci curvature. Moreover, by assuming that the measure is doubling, we extend the result on the existence of the entropy dimension to self-conformal measures on complete Riemannian manifolds.

math.FA↗

arXiv2014

One-dimensional wave equations defined by fractal Laplacians

We study one-dimensional wave equations defined by a class of fractal Laplacians. These Laplacians are defined by fractal measures generated by iterated function systems with overlaps, such as the well-known infinite Bernoulli convolution associated with the golden ratio and the 3-fold convolution of the Cantor measure. The iterated function systems defining these measures do not satisfy the post-critically finite condition or the open set condition. By using second-order self-similar identities introduced by Strichartz et al., we discretize the equations and use the finite element and central difference methods to obtain numerical approximations to the weak solutions. We prove that the numerical solutions converge to the weak solution, and obtain estimates for the rate of convergence.

math-ph↗