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Yifei Pan

Publications and source records attributed to Yifei Pan.

At least 19 recordsLinked to original sources
arXiv2026

Unique continuation for $\bar\partial u = Vu$ at infinity

Motivated by Landis's conjecture on unique continuation at infinity for the Laplacian, we study the corresponding property for the Cauchy-Riemann operator. We prove that every weak solution of $\bar\partial u=Vu$ on a neighborhood of infinity, with $V\in L^\infty$, vanishes identically if it decays exponentially at a rate greater than $ 2\|V\|_{L^\infty}$. This conclusion is sharp both in the constant $2\|V\|_{L^\infty}$ and in the order of exponential decay required. More generally, we establish unique continuation at infinity for a broad class of radially decaying bounded potentials, with optimal decay rates determined by the decay of the potential. We also obtain related unique continuation results for $L^2$ potentials and for compactly supported potentials under weaker assumptions at infinity.

math.CV↗
arXiv2026

On a Local Radial Rigidity Phenomenon of Elliptic Systems on Riemannian Manifolds

In this paper, we study local radial rigidity phenomena for a broad class of general-order elliptic systems on Riemannian manifolds via a reduction to singular ordinary differential equations of Euler type. These rigidity properties have natural applications to several geometric problems, including prescribed curvature problems of Yamabe and Paneitz type in conformal geometry, higher-order analogues arising from harmonic maps, and Calabi-type problems on K\"ahler manifolds. The method applies also to higher-order nonlinear Poisson systems and, more generally, to weighted divergence-form elliptic systems, yielding local uniqueness and existence results for solutions with prescribed initial jets at the singularity.

math.AP↗
arXiv2026

Energy estimates for level sets of holomorphic functions and universal counterexamples to Calder\'on-Zygmund theory

We demonstrate that the failure of $L^1$ regularity in Calder\'on-Zygmund theory is a universal phenomenon: every non-constant holomorphic function in $\C^n$ generates a counterexample to the Poisson equation. In order to achieve this goal, we shall establish sharp level-set estimates that link harmonic analysis to the geometry of complex structure through Hironaka's resolution of singularities and the \L{}ojasiewicz gradient inequality.

math.CV↗
arXiv2025

Non-quadratic solutions to the Monge-Amp\`ere equation

We construct ample smooth strictly plurisubharmonic non-quadratic solutions to the Monge-Amp\`ere equation on either cylindrical type domains or the whole complex Euclidean space $\mathbb C^2$. Among these, the entire solutions defined on $\mathbb C^2$ induce flat Kahler metrics, as expected by a question of Calabi. In contrast, those on cylindrical domains produce a family of nowhere flat Kahler metrics. Beyond these smooth solutions, we also classify solutions that are radially symmetric in one variable, which exhibit various types of singularities. Finally, we explore analogous solutions to Donaldson's equation motivated by a result of He.

math.CV↗
arXiv2024

Unique continuation of Schr\"odinger-type equations for $\bar\partial$ II

In this paper, we extend our earlier unique continuation results \cite{PZ2} for the Schr\"odinger-type inequality $ |\bar\partial u| \le V|u|$ on a domain in $\mathbb C^n$ by removing the smoothness assumption on solutions $u = (u_1, \ldots, u_N)$. More specifically, we establish the unique continuation property for $W_{loc}^{1,1}$ solutions when the potential $V\in L_{loc}^p $, $ p>2n$; and for $W_{loc}^{1,2n+\epsilon}$ solutions when $V\in L_{loc}^{2n}$ with $N=1$ or $n = 2$. Although the unique continuation property fails in general if $V\in L_{loc}^{p}, p<2n$, we show that the property still holds for $W_{loc}^{1,1}$ solutions when $V $ is a small constant multiple of $ \frac{1}{|z|}$.

math.CV↗
arXiv2024

Unique continuation of Schr\"odinger-type equations for $\bar\partial$

The purpose of this paper is to study the unique continuation property for a Schr\"odinger-type equation $ \bar\partial u = Vu$ on a domain in $\mathbb C^n$, where the solution $u$ may be a scalar function, or a vector-valued function. While simple examples show that the unique continuation property fails in general if the potential $V\in L^{p}, p<2n$, we first prove that, in the case when $u $ is a scalar function, the unique continuation property holds when $V\in L_{loc}^{2n}$ and is $\bar\partial$-closed. For vector-valued smooth solutions, we establish the unique continuation property either when $V\in L_{loc}^p $, $ p>2n$ for $n\ge 3$, or when $V\in L_{loc}^{2n}$ for $n = 2$. Finally, we discuss the unique continuation property for some special cases where $V\notin L_{loc}^{2n}$, for instance, $V $ is a constant multiple of $ \frac{1}{|z|}$.

math.CV↗
arXiv2024

Unique continuation for a gradient inequality with $L^n$ potential

We establish a unique continuation property for solutions of the differential inequality $|\nabla u|\leq V|u|$, where $V$ is locally $L^n$ integrable on a domain in $\mathbb R^n$. A stronger uniqueness result is obtained if in addition the solutions are locally Lipschitz. One application is a finite order vanishing property in the $L^2$ sense for the exponential of $W^{1,n}$ functions. We further discuss related results for the Cauchy-Riemann operator $\bar\partial$ and characterize the vanishing order for smooth extension of holomorphic functions across the boundary.

math.AP↗
arXiv2024

On solutions to $-\Delta u = V u$ near infinity

We investigate the unique continuation property and the sign changing behavior of weak solutions to $-\Delta u =Vu$ near infinity under certain conditions on the blow-up rate of the potential $V$ near infinity.

math.AP↗
arXiv2022

Optimal Sobolev regularity of $\bar\partial$ on the Hartogs triangle

In this paper, we show that for each $k\in \mathbb Z^+, p>4$, there exists a solution operator $\mathcal T_k$ to the $\bar\partial$ problem on the Hartogs triangle that maintains the same $W^{k, p}$ regularity as that of the data. According to a Kerzman-type example, this operator provides solutions with the optimal Sobolev regularity.

math.CV↗
arXiv2022

Weighted Sobolev estimates of the truncated Beurling operator

Given a bounded planar domain $D$ with $W^{k+1, \infty}$ boundary, $ k\in \mathbb Z^+$, and a weight $μ\in A_p, 1<p<\infty$, we show that the corresponding truncated Beurling transform is a bounded operator sending $W^{k, p}(D, μ)$ into itself. Weighted Sobolev estimates for other Cauchy-type integrals are also obtained.

math.CV↗