arXiv subjects

Huaian Diao

Publications and source records attributed to Huaian Diao.

At least 37 records · Page 2Linked to original sources

arXiv2025

Inverse elastic obstacle scattering problems by monotonicity method

We consider the elastic wave scattering problem involving rigid obstacles. This work addresses the inverse problem of reconstructing the position and shape of such obstacles using far-field measurements. A novel monotonicity-based approach is developed for this purpose. By factorizing the far-field operator and utilizing the existence of localized wave functions, we derive a shape characterization criterion for the obstacle boundary. The proposed method employs monotonicity tests to determine the geometric relationship between any given test domain and the actual scatterer. As a result, the shape and location of rigid elastic obstacles can be uniquely identified without requiring any initial guesses or prior knowledge of the physical parameters of the homogeneous background medium.

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arXiv2025

Dislocations in a multi-layered elastic solid with applications to fault and interface identifications

This paper investigates an elastic dislocation problem within a bounded and multi-layered solid governed by the Lamé system. We address the simultaneous reconstruction of the faults, the jumps in displacement and traction fields across the faults, and the interfaces of layers using a single passive boundary measurement. This inverse problem is particularly challenging due to the discontinuities in both the displacement and traction fields across the faults and the inherent difficulty of establishing uniqueness results with limited measurement data. Under the assumptions that the Lamé parameters are piecewise constants within each layer, satisfying strong convexity conditions, and that the faults exhibit corner singularities, we establish local uniqueness identifiability results for both the interfaces and the faults, as well as the jumps across the faults. Furthermore, we derive global uniqueness results for reconstructing the interfaces, the faults, and the corresponding displacement and traction jumps in generic scenarios under a priori geometric information, where the faults are geometrically general and may be either open or closed.

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arXiv2025

Quasi-Minnaert Resonances in 2D Elastic Wave Scattering with Applications

In our earlier work [13], we introduced a novel quasi-Minnaert resonance for three-dimensional elastic wave scattering in the sub-wavelength regime. Therein, we provided a rigorous analysis of the boundary localization and surface resonance phenomena for both the total and scattered waves, achieved through carefully selected incident waves and tailored physical parameters of the elastic medium. In the present study, we focus on quasi-Minnaert resonances in the context of two-dimensional elastic wave scattering. Unlike the 3D case [13], the 2D setting introduces fundamental theoretical challenges stemming from (i) the intrinsic coupling between shear and compressional waves, and (ii) the complex spectral properties of the associated layer potential operators. By combining layer potential techniques with refined asymptotic analysis and strategically designed incident waves, we rigorously establish quasi-Minnaert resonances in both the internal and scattered fields. In addition, the associated stress concentration effects are quantitatively characterized. Notably, the boundary-localized nature of the scattered field reveals potential applications in near-cloaking (via wave manipulation around boundaries). Our results contribute to a more comprehensive framework for studying resonance behaviors in high-contrast elastic systems.

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arXiv2025

On quasi-Minnaert resonances in elasticity and their applications to stress concentrations

This paper unveils and investigates a novel quasi-Minnaert resonance for an elastic hard inclusion embedded in a soft homogeneous medium in the sub-wavelength regime. The quasi-Minnaert resonance consists of boundary localization and surface resonance for the generated internal total and external scattered wave fields associated with the hard inclusion. It possesses similar quantitative behaviours as those for the classical Minnaert resonance due to high-contrast material structures, but occurs for a continuous spectrum of frequencies instead of certain discrete Minnaert resonant frequencies. We present a comprehensive analysis to uncover the physical origin and the mechanism of this new physical phenomenon. It is shown that the delicate high-contrast material structures and the properly tailored incident waves which are coupled together in a subtle manner play a crucial role in ensuring such phenomena. The stress concentration phenomena in both the internal total field and the scattered field components are also rigorously established. The analysis in this paper is deeply rooted in layer potential theory and intricate asymptotic analysis. We believe that our findings can have a significant impact on the theory of composite materials and metamaterials.

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arXiv2025

On a novel UCP result and its application to inverse conductive scattering

In this paper, we derive a novel Unique Continuation Principle (UCP) for a system of second-order elliptic PDEs system and apply it to investigate inverse problems in conductive scattering. The UCP relaxes the typical assumptions imposed on the domain or boundary with certain interior transmission conditions. This is motivated by the study of the associated inverse scattering problem and enables us to establish several novel unique identifiability results for the determination of generalized conductive scatterers using a single far-field pattern, significantly extending the results in [15,23]. A key technical advancement in our work is the combination of Complex Geometric Optics (CGO) techniques from [15,23] with the Fourier expansion method to microlocally analyze corner singularities and their implications for inverse problems. We believe that the methods developed can have broader applications in other contexts.

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arXiv2024

On a coupled-physics transmission eigenvalue problem and its spectral properties with applications

In this paper, we investigate a transmission eigenvalue problem that couples the principles of acoustics and elasticity. This problem naturally arises when studying fluid-solid interactions and constructing bubbly-elastic structures to create metamaterials. We uncover intriguing local geometric structures of the transmission eigenfunctions near the corners of the domains, under typical regularity conditions. As applications, we present novel unique identifiability and visibility results for an inverse problem associated with an acoustoelastic system, which hold practical significance.

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arXiv2024

Effective medium theory for embedded obstacles in electromagnetic scattering with applications

This paper focuses on the time-harmonic electromagnetic (EM) scattering problem in a general medium which may possess a nontrivial topological structure. We model this by an inhomogeneous and possibly anisotropic medium with embedded obstacles and the EM waves cannot penetrate inside the obstacles. Such a situation naturally arises in studying inverse EM scattering problems from complex mediums with partial boundary measurements, or inverse problems from EM mediums with metal inclusions. We develop a novel theoretical framework by showing that the embedded obstacles can be effectively approximated by a certain isotropic medium with a specific choice of material parameters. We derive sharp estimates to verify this effective approximation and also discuss the practical implications of our results to the inverse problems mentioned above, which are longstanding topics in inverse scattering theory.

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arXiv2023

Unique determination by a single far-field measurement for an inverse elastic problem

This paper is concerned with the unique identification of the shape of a scatterer through a single far-field pattern in an inverse elastic medium scattering problem with a generalized transmission boundary condition. The uniqueness issue by a single far-field measurement is a challenging problem in inverse scattering theory, which has a long and colorful history. In this paper, we demonstrate the well-posedness of the direct problem by the variational approach. We establish the uniqueness results by a single far-field measurement under a generic scenario when dealing with underlying elastic scatterers exhibiting polygonal-nest or polygonal-cell structures. Furthermore, for a polygonal-nest or polygonal-cell structure scatterer associated with density and boundary impedance parameters as piecewise constants, we show that these physical quantities can be uniquely determined simultaneously by a single far-field measurement. The corresponding proof relies heavily on examining the singular behaviour of a coupled PDE system near a corner in a microlocal manner.

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arXiv2023

Stable determination of an impedance obstacle by a single far-field measurement

We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in $\mathbb{R}^2$. The obstacle is of general polygonal shape and the impedance parameter can be variable. We establish the stability results by using a single far-field pattern, which constitutes a longstanding problem in the inverse scattering theory. This is the first stability result in the literature in determining an impedance obstacle by a single far-field measurement. If the obstacle is of a generally polygonal shape, the stability in determining the obstacle is established in terms of a modified Hausdorff distance and is independent of the boundary impedance parameter. If the obstacle is further known to be convex, the stability in simultaneously determining the obstacle and the boundary impedance is established in terms of the classical Hausdorff distance. There are several technical novelties and development in the mathematical strategy developed for establishing the aforementioned stability results. First, the stability analysis is conducted around a corner point in a micro-local manner. Second, our stability estimates establish explicit relationships among the geometric configurations of the obstacle and the vanishing order of the wave field at the corner point. Third, we develop novel error propagation techniques to tackle singularities of the wave field at a corner as well as to tackle the impedance boundary condition.

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arXiv2023

Spectral properties of an acoustic-elastic transmission eigenvalue problem with applications

We are concerned with a coupled-physics spectral problem arising in the coupled propagation of acoustic and elastic waves, which is referred to as the acoustic-elastic transmission eigenvalue problem. There are two major contributions in this work which are new to the literature. First, under a mild condition on the medium parameters, we prove the existence of an acoustic-elastic transmission eigenvalue. Second, we establish a geometric rigidity result of the transmission eigenfunctions by showing that they tend to localize on the boundary of the underlying domain. Moreover, we also consider interesting implications of the obtained results to the effective construction of metamaterials by using bubbly elastic structures and to the inverse problem associated with the fluid-structure interaction.

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arXiv2022

Determining anomalies in a semilinear elliptic equation by a minimal number of measurements

We are concerned with the inverse boundary problem of determining anomalies associated with a semilinear elliptic equation of the form $-Δu+a(\mathbf x, u)=0$, where $a(\mathbf x, u)$ is a general nonlinear term that belongs to a Hölder class. It is assumed that the inhomogeneity of $f(\mathbf x, u)$ is contained in a bounded domain $D$ in the sense that outside $D$, $a(\mathbf x, u)=λu$ with $λ\in\mathbb{C}$. We establish novel unique identifiability results in several general scenarios of practical interest. These include determining the support of the inclusion (i.e. $D$) independent of its content (i.e. $a(\mathbf{x}, u)$ in $D$) by a single boundary measurement; and determining both $D$ and $a(\mathbf{x}, u)|_D$ by $M$ boundary measurements, where $M\in\mathbb{N}$ signifies the number of unknown coefficients in $a(\mathbf x, u)$. The mathematical argument is based on microlocally characterising the singularities in the solution $u$ induced by the geometric singularities of $D$, and does not rely on any linearisation technique.

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arXiv2022

Local geometric properties of conductive transmission eigenfunctions and applications

The purpose of the paper is twofold. First, we show that partial-data transmission eigenfunctions associated with a conductive boundary condition vanish locally around a polyhedral or conic corner in $\mathbb{R}^n$, $n=2,3$. Second, we apply the spectral property to the geometrical inverse scattering problem of determining the shape as well as its boundary impedance parameter of a conductive scatterer, independent of its medium content, by a single far-field measurement. We establish several new unique recovery results. The results extend the relevant ones in [30] in two directions: first, we consider a more general geometric setup where both polyhedral and conic corners are investigated, whereas in [30] only polyhedral corners are concerned; second, we significantly relax the regularity assumptions in [30] which is particularly useful for the geometrical inverse problem mentioned above. We develop novel technical strategies to achieve these new results.

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arXiv2022

Visibility, invisibility and unique recovery of inverse electromagnetic problems with conical singularities

In this paper, we study time-harmonic electromagnetic scattering in two scenarios, where the anomalous scatterer is either a pair of electromagnetic sources or an inhomogeneous medium, both with compact supports. We are mainly concerned with the geometrical inverse scattering problem of recovering the support of the scatterer, independent of its physical contents, by a single far-field measurement. It is assumed that the support of the scatterer (locally) possesses a conical singularity. We establish a local characterisation of the scatterer when invisibility/transparency occurs, showing that its characteristic parameters must vanish locally around the conical point. Using this characterisation, we establish several local and global uniqueness results for the aforementioned inverse scattering problems, showing that visibility must imply unique recovery. In the process, we also establish the local vanishing property of the electromagnetic transmission eigenfunctions around a conical point under the Hölder regularity or a regularity condition in terms of Herglotz approximation.

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arXiv2021

Two unique Identifiability results for inverse scattering problems within polyhedral geometries

We consider the unique determinations of impenetrable obstacles or diffraction grating profiles in $\mathbb{R}^3$ by a single far-field measurement within polyhedral geometries. We are particularly interested in the case that the scattering objects are of impedance type. We derive two new unique identifiability results for the inverse scattering problem in the aforementioned two challenging setups. The main technical idea is to exploit certain quantitative geometric properties of the Laplacian eigenfunctions which were initiated in our recent works [8,9]. In this paper, we derive novel geometric properties that generalize and extend the related results in [9], which further enable us to establish the new unique identifiability results. It is pointed out that in addition to the shape of the obstacle or the grating profile, we can simultaneously recover the boundary impedance parameters.

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arXiv2021

Stable determination of an elastic medium scatterer by a single far-field measurement and beyond

We are concerned with the time-harmonic elastic scattering due to an inhomogeneous elastic material inclusion located inside a uniformly homogeneous isotropic medium. We establish a sharp stability estimate of logarithmic type in determining the support of the elastic scatterer, independent of its material content, by a single far-field measurement when the support is a convex polyhedral domain in $\mathbb{R}^n$, $n=2,3$. Our argument in establishing the stability result is localized around a corner of the medium scatterer. This enables us to further establish a byproduct result by proving that if a generic medium scatterer, not necessary to be a polyhedral shape, possesses a corner, then there exists a positive lower bound of the scattered far-field patterns. The latter result indicates that if an elastic material object possesses a corner on its support, then it scatters every incident wave stably and invisibility phenomenon does not occur.

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arXiv2021

On a local geometric property of the generalized elastic transmission eigenfunctions and application

Consider the nonlinear and completely continuous scattering map \[ \mathcal{S}\big((Ω; λ, μ, V), \mathbf{u}^i\big)=\mathbf{u}_t^\infty(\hat{\mathbf{x}}), \quad \hat{\mathbf{x}}\in\mathbb{S}^{n-1}, \] which sends an inhomogeneous elastic scatterer $(Ω; λ, μ, V)$ to its far-field pattern $\mathbf{u}_t^\infty$ due to an incident wave field $\mathbf{u}^i$ via the Lamé system. Here, $(λ, μ, V)$ signifies the medium configuration of an elastic scatterer that is compactly supported in $Ω$. In this paper, we are concerned with the intrinsic geometric structure of the kernel space of $\mathcal{S}$, which is of fundamental importance to the theory of inverse scattering and invisibility cloaking for elastic waves and has received considerable attention recently. It turns out that the study is contained in analysing the geometric properties of a certain non-selfadjoint and non-elliptic transmission eigenvalue problem. We propose a generalized elastic transmission eigenvalue problem and prove that the transmission eigenfunctions vanish locally around a corner of $\partialΩ$ under generic regularity criteria. The regularity criteria are characerized by the Hölder continuity or a certain Fourier extension property of the transmission eigenfunctions. As an interesting and significant application, we apply the local geometric property to derive several novel unique identifiability results for a longstanding inverse elastic problem by a single far-field measurement.

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arXiv2021

Effective medium theory for embedded obstacles in elasticity with applications to inverse problems

We consider the time-harmonic elastic wave scattering from a general (possibly anisotropic) inhomogeneous medium with an embedded impenetrable obstacle. We show that the impenetrable obstacle can be effectively approximated by an isotropic elastic medium with a particular choice of material parameters. We derive sharp estimates to rigorously verify such an effective approximation. Our study is strongly motivated by the related studies of two challenging inverse elastic problems including the inverse boundary problem with partial data and the inverse scattering problem of recovering mediums with buried obstacles. The proposed effective medium theory readily yields some interesting applications of practical significance to these inverse problems.

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arXiv2021

Further results on generalized Holmgren's principle to the Lamé operator and applications

In our earlier paper [9], it is proved that a homogeneous rigid, traction or impedance condition on one or two intersecting line segments together with a certain zero point-value condition implies that the solution to the Lamé system must be identically zero, which is referred to as the generalized Holmgren principle (GHP). The GHP enables us to solve a longstanding inverse scattering problem of determining a polygonal elastic obstacle of general impedance type by at most a few far-field measurements. In this paper, we include all the possible physical boundary conditions from linear elasticity into the GHP study with additionally the soft-clamped, simply-supported as well as the associated impedance-type conditions. We derive a comprehensive and complete characterisation of the GHP associated with all of the aforementioned physical conditions. As significant applications, we establish novel unique identifiability results by at most a few scattering measurements not only for the inverse elastic obstacle problem but also for the inverse elastic diffraction grating problem within polygonal geometry in the most general physical scenario. We follow the general strategy from [9] in establishing the results. However, we develop technically new ingredients to tackle the more general and challenging physical and mathematical setups. It is particularly worth noting that in [9], the impedance parameters were assumed to be constant whereas in this work they can be variable functions.

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