arXiv subjects

Hua-Yang Wang

Publications and source records attributed to Hua-Yang Wang.

7 recordsLinked to original sources

arXiv2026

The Critical Semilinear Elliptic Equation with Isolated Boundary Singularities II

Continuing the work of the second author (2017), we study the Sobolev critical semilinear elliptic equation in the half-space with an isolated boundary singularity and zero Dirichlet boundary condition. This paper addresses two open questions in this setting: the existence of Delaunay-type log-periodic solutions posed by del~Pino--Musso--Pacard (2007), and the asymptotic classification of singular solutions posed by Bidaut-Véron--Ponce--Véron (2007). We establish a global branch of positive log-periodic solutions along which blow-up occurs at a uniquely determined period. We also construct the corresponding concentrating family and prove its local uniqueness. Consequently, the expected stationary asymptotic classification fails, and no universal critical scale-invariant upper bound can hold throughout the half-space. This behavior contrasts sharply with the classical interior singularity theory of Caffarelli--Gidas--Spruck (1989).

math.AP↗

arXiv2026

Optimal Weighted Smoothing and Asymptotics of Ancient Solutions for Fast Diffusion Equations

The Cauchy--Dirichlet problem for the fast diffusion equation on a smooth bounded domain admits a natural bound in the weighted space $L^p_{Φ_1}$, where $Φ_1$ is the first Dirichlet eigenfunction. A key regularization question is whether this implies the stronger $L^\infty$ bound. We provide a complete resolution, showing that the critical exponent coincides with the classical Brezis--Turner exponent from semilinear elliptic theory. As a primary application, we derive improved global Harnack inequalities and describe asymptotic behavior of positive ancient solutions.

math.AP↗

arXiv2026

A duality approach to gradient Hölder estimates for linear divergence form elliptic equations

We prove a sparse bound in the context of Schauder theory for divergence form elliptic partial differential equations. In addition, we show how an iteration argument inspired by sparse domination bounds can be used to deduce gradient reverse Hölder inequalities for equations with non-constant coefficients from the theory for constant coefficient equations. We deal with coefficient matrices whose entries are either Hölder continuous or just uniformly continuous, leading to different results. The purpose of the approach is to highlight the connection between Schauder theory and duality of local Hardy spaces and local Hölder spaces.

math.AP↗

arXiv2025

Solutions to Sobolev Supercritical Nonlinear Schrodinger Equations on an Annulus via a Hopf Reduction Method

This paper investigates the existence of positive solutions with a prescribed mass for nonlinear Schrodinger equations on an annulus, possibly in the Sobolev supercritical regime. A reduction method based on the Hopf fibration is used to transform the problem into a lower-dimensional one. We obtain a new mass critical threshold and we show that in the new mass subcritical or critical regimes there exists a positive solution which corresponds to a global minimizer, while in the mass supercritical regime, there exists two positive solutions which correspond to a local minimizer and a mountain pass solution respectively. Some other problems are also discussed in this paper.

math.AP↗

arXiv2025

Enhancing the fidelity of stimulated Raman transitions with simple phase shifts

We demonstrate that in stimulated Raman transitions, introducing one or two simple phase shifts to the control fields significantly enhances the fidelity of state manipulation while simultaneously reducing leakage to the intermediate excited state. Our approach achieves high-fidelity quantum gate operations between the two target states under arbitrary detuning conditions. Notably, the average population in the intermediate excited state is approximately halved, without extending the overall evolution time. Additionally, our method exhibits greater robustness to static amplitude and detuning errors compared to conventional adiabatic elimination techniques, and maintains higher fidelity even in the presence of dissipation.

quant-ph↗

arXiv2024

Sparse gradient bounds for divergence form elliptic equations

We provide sparse estimates for gradients of solutions to divergence form elliptic partial differential equations in terms of the source data. We give a general result of Meyers (or Gehring) type, a result for linear equations with VMO coefficients and a result for linear equations with Dini continuous coefficients. In addition, we provide an abstract theorem conditional on PDE estimates available. The linear results have the full range of weighted estimates with Muckenhoupt weights as a consequence.

math.AP↗

arXiv2022

Normalized Solutions to Schrödinger Equations with Critical Exponent and Mixed Nonlocal Nonlinearities

We study the existence and nonexistence of normalized solutions $(u_a, λ_a)\in H^{1}(\mathbb{R}^N)\times \mathbb{R}$ to the nonlinear Schrödinger equation with mixed nonlocal nonlinearities. This study can be viewed as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions to the nonlocal Schrödiger equation with a fixed $L^2$-norm $\|u\|_2=a>0$. The leading term is $L^2$-supercritical, that is, $p\in (\frac{N+α+2}{N},\frac{N+α}{N-2}]$, where the Hardy-Littlewood-Sobolev critical exponent $p=\frac{N+α}{N-2}$ appears. We first prove that there exist two normalized solutions if $q\in (\frac{N+α}{N},\frac{N+α+2}{N})$ with $μ>0$ small, that is, one is at the negative energy level while the other one is at the positive energy level. For $q=\frac{N+α+2}{N}$, we show that there is a normalized ground state for $0<μ< \tildeμ $ and there exist no ground states for $μ>\tildeμ$, where $\tildeμ$ is a sharp positive constant. If $q\in (\frac{N+α+2}{N},\frac{N+α}{N-2})$, we deduce that there exists a normalized ground state for any $μ>0$. We also obtain some existence and nonexistence results for the case $μ<0$ and $q\in (\frac{N+α}{N},\frac{N+α+2}{N}]$. Besides, we analyze the asymptotic behavior of normalized ground states as $μ\rightarrow 0^{+}$.

math.AP↗