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Donghui Yang

Publications and source records attributed to Donghui Yang.

At least 19 recordsLinked to original sources
arXiv2026

Null Controllability for Degenerate Parabolic Equations with Internal Control Applied on a Measurable Subset

This work serves as a continuation of our preceding paper [28]. In that study, we presented a separable variable method to derive the Lebeau-Robbiano spectral inequality for a specific degenerate parabolic equation and subsequently employed it to demonstrate the null controllability of said equation when internal control is applied to an open subset. In the current paper, we reapply the separable variable method to attain the Lebeau-Robbiano spectral inequality for a different degenerate parabolic equation, and we substantiate the null controllability of this equation with internal control acting on a measurable subset. This approach may offer an alternative means of proving controllability results for degenerate parabolic equations.

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arXiv2026

On Controllability of a Class of N -dimensional Hyperbolic Equations with Internal Single-point Degeneracy

This paper explores the controllability of a class of N-dimensional hyperbolic equations featuring a single interior degenerate point. Firstly, we establish the well-posedness of the equation through the application of the Hardy inequality. Following this, we primarily utilize the Carleman estimate method to derive the observability inequality. By leveraging the equivalence between observability and controllability, we deduce the exact controllability of the equation. It is worth noting that our selected control region includes the degenerate null point. In the Carleman estimate, we adopt a unique approach to construct the weight function, effectively negating the influence of the degenerate region.

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arXiv2026

Shape Design for a Class of Degenerate Parabolic Equations with Boundary Point Degeneracy and Its Application to Boundary Observability

We study a class of degenerate parabolic equations with boundary point degeneracy in dimensions N>=2 and investigate the associated boundary observability problem by means of shape design. While one-dimensional degenerate models have been treated in the literature, the genuinely higher-dimensional case remains much more delicate because the degeneracy occurs at a boundary point and the boundary normal trace cannot be extracted directly near the singularity. We approximate the degenerate equation by a family of uniformly parabolic problems on truncated domains obtained by removing a small neighborhood of the degenerate point. Under a geometric condition on the boundary, we establish uniform estimates for the approximate problems, prove convergence to the solution of the original degenerate equation, and identify the convergence of the boundary normal derivatives under additional regularity. We then combine this approximation scheme with a parabolic Carleman estimate for the approximate backward equations and derive a boundary observability inequality for the limiting degenerate equation. In this way, we obtain a higher-dimensional parabolic counterpart of the shape-design program previously developed for degenerate hyperbolic equations.

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arXiv2026

Non-homogeneous boundary value problems for second-order degenerate hyperbolic equations and their application

We study second-order hyperbolic equations with degenerate elliptic operators and non-homogeneous Dirichlet boundary inputs. We establish existence and regularity of weak solutions in weighted Sobolev spaces under mild assumptions on the degenerate weight. A Dirichlet map is constructed for the degenerate elliptic operator, leading to a solution theory that extends classical approaches to the degenerate setting. In particular, we derive energy estimates and well-posedness for boundary inputs of low regularity (in appropriate trace spaces), even though the classical Dirichlet-to-Neumann framework is not directly applicable in the degenerate setting. As an application, we prove an approximate controllability criterion, which generalizes the Hilbert Uniqueness Method to degenerate wave equations. Our framework accommodates higher-dimensional degenerate waves, non-homogeneous boundary conditions, and weighted functional analysis. We also illustrate how our criterion connects to higher-dimensional Grushin equations and waves with single-point degeneracy, and we highlight the remaining unique continuation/observability issue as an open problem.

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arXiv2025

Benchmarking Endoscopic Surgical Image Restoration and Beyond

In endoscopic surgery, a clear and high-quality visual field is critical for surgeons to make accurate intraoperative decisions. However, persistent visual degradation, including smoke generated by energy devices, lens fogging from thermal gradients, and lens contamination due to blood or tissue fluid splashes during surgical procedures, severely impairs visual clarity. These degenerations can seriously hinder surgical workflow and pose risks to patient safety. To systematically investigate and address various forms of surgical scene degradation, we introduce a real- world open-source surgical image restoration dataset covering endoscopic environments, called SurgClean, which involves multi-type image restoration tasks from two medical sites, i.e., desmoking, defogging, and desplashing. SurgClean comprises 3,113 images with diverse degradation types and corresponding paired reference labels. Based on SurgClean, we establish a standardized evaluation benchmark and provide performance for 22 representative generic task-specific image restoration approaches, including 12 generic and 10 task-specific image restoration approaches. Experimental results reveal substantial performance gaps relative to clinical requirements, highlighting a critical opportunity for algorithm advancements in intelligent surgical restoration. Furthermore, we explore the degradation discrepancies between surgical and natural scenes from structural perception and semantic under- standing perspectives, providing fundamental insights for domain-specific image restoration research. Our work aims to empower restoration algorithms and improve the efficiency of clinical procedures.

cs.CV↗
arXiv2025

Approximation of Elliptic Equations with Interior Single-Point Degeneracy and Its Application to Weak Unique Continuation Property

This paper investigates the quantitative weak unique continuation property (QWUCP) for a class of high-dimensional elliptic equations with interior point degeneracy. First, we establish well-posedness results in weighted function spaces. Then, using an innovative approximation method, we derive the three-ball theorem at the degenerate point. Finally, we apply the three-ball theorem to prove QWUCP for two different cases.

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arXiv2024

Quantitative uniqueness estimates for stochastic parabolic equations on the whole Euclidean space

In this paper, a quantitative estimate of unique continuation for the stochastic heat equation with bounded potentials on the whole Euclidean space is established. This paper generalizes the earlier results in [29] and [17] from a bounded domain to an unbounded one. The proof is based on the locally parabolic-type frequency function method. An observability estimate from measurable sets in time for the same equation is also derived.

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arXiv2023

The solvability and a Stackelberg-Nash game problem for degenerate elliptic equations

This study aims to investigate the functional properties of weak solution spaces and their compact embedding properties in relation to the Dirichlet problem associated with a specific class of degenerate elliptic equations. To expand the scope of analysis for degenerate elliptic problems, we employ weighted Sobolev inequalities and compact embedding techniques within the framework of weighted Sobolev spaces. Additionally, we apply these findings to examine a Stackelberg-Nash game problem and obtain the existence of the Stackelberg-Nash equilibrium.

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arXiv2023

Observability inequalities for the backward stochastic evolution equations and their applications

The present article delves into the investigation of observability inequalities pertaining to backward stochastic evolution equations. We employ a combination of spectral inequalities, interpolation inequalities, and the telegraph series method as our primary tools to directly establish observability inequalities. Furthermore, we explore three specific equations as application examples: a stochastic degenerate equation, a stochastic fourth order parabolic equation and a stochastic heat equation. It is noteworthy that these equations can be rendered null controllability with only one control in the drift term to each system.

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arXiv2023

Observability inequality, the interpolation inequality and the spectral inequality for the degenerate parabolic equation in R

This paper investigates the interrelationships between the observability inequality, the Hölder-type interpolation inequality, and the spectral inequality for the degenerate parabolic equation in $\mathbb{R}$. We elucidate the distinctive properties of observable sets pertaining to the degenerate parabolic equation. Specifically, we establish that a measurable set in $\mathbb{R}$ fulfills the observability inequality when it exhibits $γ$-thickness at a scale $L$, where $γ>0$ and $L>0$.

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arXiv2023

Carleman estimates for degenerate parabolic equations with single interior point degeneracy and its applications

We study the controllability of a class of $N$-dimensional degenerate parabolic equations with single interior point degeneracy. We employ the Galerkin method to prove the existence of solutions for the equations. The analysis is then divided into two cases based on whether the degenerate point $x=0$ lies within the control region $\omega_0$ or not. For each case, we establish specific Carleman estimates. As a result, we achieve null controllability in the first case $0\in\omega_0$ and unique continuation and approximate controllability in the second case $0\notin\omega_0$.

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arXiv2023

Optimal Actuator Location of the Norm Optimal Controls for Degenerate Parabolic Equations

This paper focuses on investigating the optimal actuator location for achieving minimum norm controls in the context of approximate controllability for degenerate parabolic equations. We propose a formulation of the optimization problem that encompasses both the actuator location and its associated minimum norm control. Specifically, we transform the problem into a two-person zero-sum game problem, resulting in the development of four equivalent formulations. Finally, we establish the crucial result that the solution to the relaxed optimization problem serves as an optimal actuator location for the classical problem.

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arXiv2023

Null controllability of two kinds of coupled parabolic systems with switching control

The focus of this paper is on the null controllability of two kinds of coupled systems including both degenerate and non-degenerate equations with switching control. We first establish the observability inequality for measurable subsets in time for such coupled system, and then by the HUM method to obtain the null controllability. Next, we investigate the null controllability of such coupled system for segmented time intervals. Notably, these results are obtained through spectral inequalities rather than using the method of Carleman estimates. Such coupled systems with switching control, to the best of our knowledge, are among the first to discuss.

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arXiv2023

Norm and time optimal control problems of stochastic heat equations

This paper investigates the norm and time optimal control problems for stochastic heat equations. We begin by presenting a characterization of the norm optimal control, followed by a discussion of its properties. We then explore the equivalence between the norm optimal control and time optimal control, and subsequently establish the bang-bang property of the time optimal control. These problems, to the best of our knowledge, are among the first to discuss in the stochastic case.

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arXiv2023

The fundamental gap of a kind of two dimensional sub-elliptic operator

This paper is concerned at the minimization fundamental gap problem for a class of two-dimensional degenerate sub-elliptic operators. We establish existence results for weak solutions, Sobolev embedding theorem and spectral theory of sub-elliptic operators. We provide the existence and characterization theorems for extremizing potentials $V(x)$ when $V(x)$ is subject to $L^\infty$ norm constraint.

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arXiv2023

Null controllability of n-dimensional parabolic equations degenerated on partial boundary

This paper extends the Carleman estimates to high dimensional parabolic equations with highly degenerate symmetric coefficients on a bounded domain of Lipschitz boundary and use these estimates to study the controlla?bility the corresponding equations. Due to the nonsmoothness and degeneracy of boundary, the partial integration by parts in Carleman estimates have no meaning on the degenerate and nonsmooth parts of the boundary. To get around of this difficulty, we construct special weight function, and transform some integral terms in degenerate regions into a non-degenerate ones carefully so that the obtained Carleman estimates can still be used to the controllability problem. Our results includes some well-known works as some special cases as well as some interesting new examples.

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arXiv2023

Observability Inequality from Measurable Sets and the Stackelberg-Nash Game Problem for Degenerate Parabolic Equations

In this study, we employ the established Carleman estimates and propagation estimates of smallness from measurable sets for real analytic functions, along with the telescoping series method, to establish an observability inequality for the degenerate parabolic equation over measurable subsets in the time-space domain. As a direct application, we formulate a captivating Stackelberg-Nash game problem and provide a proof of the existence of its equilibrium. Additionally, we characterize the set of Stackelberg-Nash equilibria and delve into the analysis of a norm optimal control problem.

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arXiv2023

Extremal properties of the first eigenvalue and the fundamental gap of a sub-elliptic operator

We consider the problems of extreming the first eigenvalue and the fundamental gap of a sub-elliptic operator with Dirichlet boundary condition, when the potential $V$ is subjected to a $p$-norm constraint. The existence results for weak solutions, compact embedding theorem and spectral theory for sub-elliptic equation are given. Moreover, we provide the specific characteristics of the corresponding optimal potential function.

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