arXiv subjects

Dong-Hui Yang

Publications and source records attributed to Dong-Hui Yang.

13 recordsLinked to original sources

arXiv2026

Application of the Shape Design Method to Hidden Regularity of Degenerate Hyperbolic Equations with Degenerate Boundary

This paper investigates the well-posedness and hidden regularity of boundary-degenerate hyperbolic equations in a two-dimensional setting. The shape design method is employed, which approximates the original degenerate problem by a family of uniformly elliptic problems on truncated subdomains and passes to the limit through uniform estimates. Within this framework, the existence and uniqueness of weak solutions are established in suitable weighted Sobolev spaces. A hidden regularity estimate for the conormal derivative on the nondegenerate portion of the boundary is obtained, providing a uniform bound in terms of the natural weighted energy norms of the initial data and source term. This estimate yields the boundary trace information essential for observability and controllability of degenerate hyperbolic systems.

math.AP↗

arXiv2026

Shape Design for Degenerate Hyperbolic Equation with Degenerate Boundary and Its Application to Observability

In this paper, we study the observability and controllability of a class of degenerate hyperbolic equations with a control region intersecting the degenerate set. Unlike the existing results that mainly deal with control regions separated from the degeneracy, we consider the case where the control region reaches the degenerate part. To handle the difficulty caused by the degeneracy, we introduce a shape-design-based approximation method based on shape design by approximating the degenerate equation with a family of uniformly hyperbolic equations. The proof relies on the spectral approximation of the associated operators, precise estimates for weak solutions, and the multiplier method. We first establish observability inequalities for the approximating equations with constants independent of the approximation parameter. Then, by passing to the degenerate limit, we obtain the observability inequality for the original degenerate equation.

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arXiv2026

Shape Design for Degenerate Parabolic Equations with Degenerate Boundaries and Its Application to Boundary Observability

In this study, we firstly establish the well-posedness of a degenerate parabolic equation under Dirichlet boundary conditions. Following this, we introduce a shape design problem, which acts as a framework for approximating the degenerate parabolic equation through a series of uniformly parabolic equations. Finally, as a tangible application of this shape design approach, we deduce a boundary observability inequality associated with the degenerate parabolic equation.

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arXiv2026

Some Key Properties of Eigenfunctions Linked to Degenerate Elliptic Differential Operators

In this study, we address the eigenvalue problem given by: \begin{equation*} \begin{cases} -\Div (w\nabla u_i)=\la_iu_i &\text{in } \Om\subset \mathbb{R}^n,\\ u_i=0 &\text{on } \pt \Om, \end{cases} \end{equation*} where $w > 0$ within $\Om$ and $w = 0$ on part of $\partial \Omega$. We establish Courant's nodal domain theorem for the corresponding degenerate elliptic differential operator $\mathcal{A}$. Unlike uniformly elliptic operators, degenerate cases often result in the loss of many advantageous properties. Despite this, we show that the essential property that the set $\{\rho \in L^\infty(\Omega) \colon \mathcal{A} + \rho \text{ has simple eigenvalues}\}$ forms a residual subset within $(L^\infty(\Omega), |\cdot|_\infty)$ still holds for the degenerate elliptic differential operator $\mathcal{A}$.

math.AP↗

arXiv2026

Approximation of Degenerate Hyperbolic Equations with Interior Degeneracy and Applications to Controllability

In this paper, we establish the existence of solutions for a particular class of degenerate hyperbolic equations. Following this, we approximate these degenerate equations by employing a sequence of uniformly hyperbolic equations. Notably, this specific approximation result has remained unexplored in the existing body of literature. Ultimately, we utilize this approximation framework to derive controllability results for the original degenerate hyperbolic equations, marking what could potentially be the inaugural investigation into higher-dimensional degenerate hyperbolic equations.

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arXiv2026

A Shape Design Approximation for Degenerate Partial Differential Equations and Its Application

In this paper, we focus on two types of degenerate partial differential equations: a degenerate elliptic equation and a degenerate parabolic equation. Significantly, both categories are characterized by the same principal operator. To obtain solutions for these equations, we introduce a novel approximation approach, termed the shape design approximation. As a practical application of this method, we derive a Carleman estimate for the backward degenerate parabolic equation. This estimate plays a pivotal role in establishing the null controllability of the degenerate parabolic equation. A notable advantage of employing the shape design approximation in deriving the Carleman estimate is that it enables us to bypass the requirement for second order derivatives in the degenerate equation. Usually, this has been a significant obstacle in the derivation of Carleman estimates for degenerate parabolic equations.

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arXiv2026

Quantitative Weak Unique Continuation on Annular Domains for Backward Degenerate Parabolic Equations with Degenerate Interior Points

In this paper, we establish a quantitative weak unique continuation theorem on an annular domain for a backward degenerate parabolic equation with a degenerate interior point. Our methodology hinges on approximating the solution of the degenerate parabolic equation through solutions of non-degenerate parabolic counterparts. Subsequently, we establish Carleman estimates for the non-degenerate parabolic equation across two separate domains. By virtue of these estimates, we deduce a quantitative weak unique continuation property for the degenerate parabolic equation, thereby substantiating the weak unique continuation result for the original degenerate parabolic equation.

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arXiv2026

Null Controllability for a Multi-Dimensional Degenerate Parabolic Equation with Degenerated Interior Point

In this study, we study the null controllability of a multi-dimensional degenerate parabolic equation characterized by a degenerate interior point. The control domain, which is an arbitrary inner region, does not encompass the degenerate point. To tackle this problem, we adopt a new approximation methodology. Specifically, we approximate the degenerate partial differential equations (PDEs) with a series of uniformly elliptic PDEs, notwithstanding their limited regularity. We then derive the Carleman estimate for these approximate uniformly parabolic equations and establish the observability inequality, which ultimately paves the way for demonstrating the null controllability of the system.

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arXiv2026

Hidden Boundary Trace Regularity and an Observability Estimate with Interior Remainder for Boundary-Degenerate Hyperbolic Equations

We study hidden boundary trace regularity for two-dimensional hyperbolic equations with boundary degeneracy governed by $\mcA\vp=-\Div(A\nabla \vp)$, where $A=\diag(1,r^\al)$ and $\al\in(0,1)$. We establish well-posedness in weighted Sobolev spaces and prove an $L^2$ trace estimate for the normal derivative on the nondegenerate side $r=1$. Using truncated geometries and Carleman weights adapted to the anisotropic degeneracy, we derive a large-time observability estimate with a lower-order interior remainder. We also identify a framework-level obstruction at the critical threshold $\al=1$: the weighted Dirichlet coercivity underlying the subcritical analysis loses uniformity and exhibits a logarithmic loss on truncated domains.

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arXiv2026

A Class of Degenerate Hyperbolic Equations with Neumann Boundary Conditions and Its Application to Observability

We establish a mixed observability inequality for a class of degenerate hyperbolic equations on the cylindrical domain $\Omega = \mathbb{T} \times (0,1)$ with mixed Neumann Dirichlet boundary conditions. The degeneracy acts only in the radial variable, whereas the periodic angular variable allows propagation with a strong tangential component, making a direct top boundary observation delicate. For $\alpha \in [1,2)$, we prove that the solution can be controlled by a boundary observation on the top boundary together with an interior observation on a narrow strip. The proof combines a weighted functional framework, improved regularity, a cutoff decomposition in the angular variable, a multiplier argument for the localized component, and an energy estimate for the remainder.

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arXiv2026

Shape-Design Approximation for a Class of Degenerate Hyperbolic Equations with a Degenerate Boundary Point and Its Application to Observability

We study a class of degenerate hyperbolic equations in a bounded domain whose degeneracy occurs at a boundary point. We first develop the weighted functional framework, prove well-posedness of the degenerate problem, and establish regularity away from the degenerate point. We then introduce a shape-design approximation obtained by removing a small neighborhood of the degenerate boundary point, which yields uniformly non-degenerate hyperbolic problems on regularized domains. We prove that the regularized solutions converge to the solution of the original degenerate equation, including the convergence of the boundary normal derivatives away from the degenerate point. Finally, under a geometric condition on the observation boundary, we derive an observability inequality for the degenerate equation by combining the uniform observability of the regularized problems with the limit passage.

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arXiv2020

Optimal Actuator Location of the Minimum Norm Controls for Heat Equation with General Controlled Domain

In this paper, we study optimal actuator location of the minimum norm controls for a multi-dimensional heat equation with control defined in the space $L^p(0,T;L^2(Ω))$. The actuator domain $ω$ is quite general in the sense that it is required only to have a prescribed Lebesgue measure. A relaxation problem is formulated and is transformed into a two-person zero-sum game problem. By the game theory, we develop a necessary and sufficient condition and the existence of relaxed optimal actuator location for $p\in[2,+\infty]$, which is characterized by the Nash equilibrium of the associated game problem. An interesting case is for the case of $p=2$, for which it is shown that the classical optimal actuator location can be obtained from the relaxed optimal actuator location without additional condition. Finally for $p=2$, a sufficient and necessary condition for classical optimal actuator location is presented.

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arXiv2017

Time-varying Bang-bang Property of Minimal Controls for Approximately Null-controllable Heat Equations

In this paper, optimal time control problems and optimal target control problems are studied for the approximately null-controllable heat equations. Compared with the existed results on these problems, the boundary of control variables are not constants but time varying functions. The time-varying bang-bang property for optimal time control problem, and an equivalence theorem for optimal control problem and optimal target problem are obtained.

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