Searcharxiv⌕ Search

arXiv subjects

Zhenqian Li

Publications and source records attributed to Zhenqian Li.

13 recordsLinked to original sources
arXiv2026

An inverse random source problem for the fractional Helmholtz equation

This paper investigates an inverse random source problem for the stochastic fractional Helmholtz equation. The source is modeled as a centered, complex-valued, microlocally isotropic generalized Gaussian random field whose covariance and relation operators are described by classical pseudo-differential operators. For sufficiently large wavenumbers, we first establish the well-posedness of the direct problem in the distributional sense by analyzing the corresponding Lippmann--Schwinger integral equation. For the inverse problem, we show that the principal symbols of both the covariance and relation operators can be uniquely determined, with probability one, from the far-field patterns generated by a single realization of the random source. The approach employs a combination of the Born linearization, asymptotic expansions of the fractional Helmholtz Green kernel at high wavenumbers, and microlocal analysis of associated Fourier integral operators.

math.AP↗
arXiv2024

Stability for inverse random source problems of the polyharmonic wave equation

This paper investigates stability estimates for inverse source problems in the stochastic polyharmonic wave equation, where the source is represented by white noise. The study examines the well-posedness of the direct problem and derives stability estimates for identifying the strength of the random source. Assuming a priori information of the regularity and support of the source strength, the Hölder stability is established in the absence of a potential. In the more challenging case where a potential is present, the logarithmic stability estimate is obtained by constructing specialized solutions to the polyharmonic wave equation.

math.AP↗
arXiv2022

On Skoda's theorem for Nadel-Lebesgue multiplier ideal sheaves on singular complex spaces and regularity of weak Kähler-Einstein metrics

In this article, we will characterize regular points respectively by the local vanishing, positivity of the Ricci curvature and $L^2$-solvability of the $\overline\partial$-equation together with Skoda's theorem for Nadel-Lebesgue multiplier ideal sheaves associated to plurisubharmonic (psh) functions on any (reduced) complex space of pure dimension. As a by-product, we show that any weak Kähler-Einstein metric on \emph{singular} $\mathbb{Q}$-Fano/Calabi-Yau/general type varieties cannot be smooth, and that in general there exists no \emph{singular} normal Kähler complex space such that the Kähler metric is Kähler-Einstein on the regular locus.

math.CV↗
arXiv2022

On a question of Kollár

In this note, we establish a generalized analytic inversion of adjunction via the Nadel-Ohsawa multiplier/adjoint ideal sheaves associated to plurisubharmonic (psh) functions for log pairs, by which we answer a question of Kollár in full generality.

math.CV↗
arXiv2021

Grauert-Riemenschneider multiplier ideal sheaves and the (optimal) Briançon-Skoda number

The goal of this note is to survey some recent results on the Grauert-Riemenschneider multiplier ideal sheaves on any (reduced) complex space of pure dimension. In particular, we obtain the Briançon-Skoda number for any Noetherian ring of weakly holomorphic functions with weakly rational singularities (\emph{not} essentially of finite type over $\mathbb{C}$ and Cohen-Macaulay local rings), which will partially answer a question of Huneke.

math.CV↗
arXiv2020

A note on multiplier ideal sheaves on complex spaces with singularities

The goal of this note is to present some recent results of our research concerning multiplier ideal sheaves on complex spaces and singularities of plurisubharmonic functions. We firstly introduce multiplier ideal sheaves on complex spaces (\emph{not} necessarily normal) via Ohsawa's extension measure, as a special case of which, it turns out to be the so-called Mather-Jacobian multiplier ideals in the algebro-geometric setting. As applications, we obtain a reasonable generalization of (algebraic) adjoint ideal sheaves to the analytic setting and establish some extension theorems on Kähler manifolds from \emph{singular} hypersurfaces. Relying on our multiplier and adjoint ideals, we also give characterizations for several important classes of singularities of pairs associated to plurisubharmonic functions. Moreover, we also investigate the local structure of singularities of log canonical locus of plurisubharmonic functions. Especially, in the three-dimensional case, we show that for any plurisubharmonic function with log canonical singularities, its associated multiplier ideal subscheme is weakly normal, by which we give a complete classification of multiplier ideal subschemes with log canonical singularities.

math.CV↗
arXiv2016

A Characterization of regular points by $L^2$ Extension Theorem

In this article, we present that the germ of a complex analytic set at the origin in $\mathbb{C}^n$ is regular if and only if the related $L^2$ extension theorem holds. We also obtain a necessary condition of the $L^2$ extension of bounded holomorphic sections from singular analytic sets.

math.CV↗