arXiv subjects

Yi-Sheng Lim

Publications and source records attributed to Yi-Sheng Lim.

5 recordsLinked to original sources

arXiv2026

Revisiting Picard's proof of the Picard--Weber--Weck selection theorem

Revisiting the rationale provided by Picard in the seminal paper [13], we provide an independent, self-contained proof of the Maxwell compactness property for weak Lipschitz domains in the Euclidean setting, detouring technical complications like the differential forms setting, Gaffney's inequality for smooth domains, Calderon's extension theorem or regularity theory for PDEs, or systems thereof, on smooth domains that are used in this context. As a by-product we provide an independent proof of the classical Gaffney estimate for cubes using only $L^2$-completeness of the Fourier bases.

math.AP↗

arXiv2025

Higher-order homogenization for equations of linearized elasticity using the operator-asymptotic approach

The operator-asymptotic approach was introduced by Lim-Žubrinić in [Asymptotic Analysis. 141(4), p. 211-256 (2025)] for the homogenization of an $\varepsilon\mathbb{Z}^d$-periodic composite media. In this article, we consider the setting of three-dimensional linearized elasticity, and extend the approach to obtain higher-order convergence rates. In particular, we consider the so-called ``Bloch approximation'' for vector-valued functions with compact Fourier support, and demonstrate that under such data, the approach provides an expansion that yields an error of order $\varepsilon^{n+1}$ in $L^2$ and $\varepsilon^n$ in $H^1$, for any $n$.

math.AP↗

arXiv2025

Operator aspects of wave propagation through periodic media

Recent results in quantitative homogenisation of the wave equation with rapidly oscillating coefficients are discussed from the operator-theoretic perspective, which views the solution as the result of applying the operator of hyperbolic dynamics, i.e. the unitary group of a self-adjoint operator on a suitable Hilbert space. A prototype one-dimensional example of utilising the framework of Ryzhov boundary triples is analysed, where operator-norm resolvent estimates for the problem of classical moderate-contrast homogenisation are obtained. By an appropriate "dilation" procedure, these are shown to upgrade to second-order (and more generally, higher-order) estimates for the resolvent and the unitary group describing the evolution for the related wave equation.

math.AP↗

arXiv2024

An operator-asymptotic approach to periodic homogenization for equations of linearized elasticity

We present an operator-asymptotic approach to the problem of homogenization of periodic composite media in the setting of three-dimensional linearized elasticity. This is based on a uniform approximation with respect to the inverse wavelength $|χ|$ for the solution to the resolvent problem when written as a superposition of elementary plane waves with wave vector (``quasimomentum") $χ$. We develop an asymptotic procedure in powers of $|χ|$, combined with a new uniform version of the classical Korn inequality. As a consequence, we obtain $L^2\to L^2$, $L^2\to H^1$, and higher-order $L^2\to L^2$ norm-resolvent estimates in $\mathbb{R}^3$. The $L^2 \to H^1$ and higher-order $L^2 \to L^2$ correctors emerge naturally from the asymptotic procedure, and the former is shown to coincide with the classical formulae.

math.AP↗

arXiv2023

A high-contrast composite with annular inclusions: Norm-resolvent asymptotics

We investigate the operator-norm resolvent asymptotics of a high-contrast composite, consisting of a "stiff" material, with annular "soft" inclusions (a "stiff-soft-stiff" setup). This setup is derived from two models with very different effective wave propagation behaviors. Our analysis is based on an operator-framework proposed by Cherednichenko, Ershova, and Kiselev in [Effective Behaviour of Critical-Contrast PDEs: Micro-resonances, Frequency Conversion, and Time Dispersive Properties. I. Commun. Math. Phys. 375, p. 1833-1884]. Then, as a first step towards studying wave propagation on the stiff-soft-stiff composite, we use the effective description to derive analogous "dispersion functions".

math.AP↗