Nader Masmoudi, Fabrice Planchon
We prove the uniqueness of weak solutions to the critical defocusing wave equation in 3D under a local energy inequality condition. More precisely, we prove the uniqueness of $ u \in L^\infty\_t(\dot{H}^{1})\cap \dot{W}^{1,\infty}\_t(L^2)$, under the condition that $u$ verifies some local energy inequalities.
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We show the existence and uniqueness of solutions (either local or global for small data) for an equation arising in different aspects of surface growth. Following the work of Koch and Tataru we consider spaces critical with respect to scaling and we prove our results in the largest possible critical space such that weak solutions are defined. The uniqueness of global weak solutions remains unfortunately open, unless the initial conditions are sufficiently small.
On the uniqueness of the mild solution of the critical quasi-geostrophic equation
本文在尺度临界齐次 Besov 空间中证明了带临界耗散的二维准地转方程温和解的唯一性，且无需正则性假设。
In this paper, we investigate the critical points of solutions to the prescribed constant mean curvature equation with Neumann and Robin boundary conditions respectively in a bounded smooth convex domain $Ω$ of $\mathbb{R}^{n}(n\geq2)$. Firstly, we show the non-degeneracy and uniqueness of the critical points of solutions in a planar domain by using the local Chen & Huang's comparison technique and the geometric properties of approximate surfaces at the non-degenerate critical points. Secondly, we deduce the uniqueness and non-degeneracy of the critical points of solutions in a rotationally sy…
We give a description of a magnetohydrodynamical system in $n$ dimension using the exterior derivative. We then prove existence of global solutions for small initial data and local existence for arbitrary large data in two classes of critical spaces -- $L^q_tL^p_x$ and $\mathcal{C}_tL^p_x$, as well as uniqueness for solutions in $\mathcal{C}_tL^p_x$.
In this paper we deal with the Cauchy problem for the incompressible Euler equations in the three-dimensional periodic setting. We prove non-uniqueness for an $L^2$-dense set of Hölder continuous initial data in the class of Hölder continuous admissible weak solutions for all exponents below the Onsager-critical $1/3$. This improves previous results on non-uniqueness obtained by Daneri in arXiv:1302.0988 and by Daneri and Szekelyhidi Jr. in arXiv:1603.09714 and generalizes the result obtained by Buckmaster, De Lellis, Szekelyhidi Jr. and Vicol in arXiv:1701.08678.
In this paper we show the uniqueness of the critical point for \emph{semi-stable} solutions of the problem $$\begin{cases} -Δu=f(u)&\text{in }Ω\\ u>0&\text{in }Ω\\ u=0&\text{on } \partialΩ,\end{cases}$$ where $Ω\subset\mathbb{R}^2$ is a smooth bounded domain whose boundary has \emph{nonnegative} curvature and $f(0)\ge0$. It extends a result by Cabré-Chanillo to the case where the curvature of $\partialΩ$ vanishes.
We construct a singular solution of a stationary nonlinear Schrödinger equation on $\mathbb{R}^2$ with square-exponential nonlinearity having linear behavior around zero. In view of Trudinger-Moser inequality, this type of nonlinearity has an energy-critical growth. We use this singular solution to prove non-uniqueness of strong solutions for the Cauchy problem of the corresponding semilinear heat equation. The proof relies on explicit computation showing a regularizing effect of the heat equation in an appropriate functional space.
We show that 1-D rarefaction wave solutions are unique in the class of bounded entropy solutions to the multidimensional compressible Euler system. Such a result may be viewed as a counterpart of the recent examples of non-uniqueness of the shock wave solutions to the Riemann problem, where infinitely many solutions are constructed by the method of convex integration.
We establish the uniqueness of semi-wavefront solution for a non-local delayed reaction-diffusion equation. This result is obtained by using a generalization of the Diekman-Kaper theory for a nonlinear convolution equation. Several applications to the systems of non-local reaction-diffusion equations with distributed time delay are also considered.
We prove a unified and general criterion for the uniqueness of critical points of a functional in the presence of constraints such as positivity, boundedness, or fixed mass. Our method relies on convexity properties along suitable paths and significantly generalizes well-known uniqueness theorems. Due to the flexibility in the construction of the paths, our approach does not depend on the convexity of the domain and can be used to prove uniqueness in subsets, even if it does not hold globally. The results apply to all critical points and not only to minimizers, thus they provide uniqueness of …
Uniqueness and Nondegeneracy of Ground State for the $L^2$-Critical Boson Star Equation
本文证明了$L^2$临界玻色星方程基态在相位旋转和平移下的唯一性，并确定了线性化算子的核，进而验证了质量接近钱德拉塞卡质量时Lieb-Yau唯一性猜想。
We prove that quasi-concave positive solutions to a class of quasi-linear elliptic equations driven by the $p$-Laplacian in convex bounded domains of the plane have only one critical point. As a consequence, we obtain strict concavity results for suitable transformations of these solutions.
In this paper, we study the existence and uniqueness of traveling wave solution for the accelerated Frenkel-Kontorova model. This model consists in a system of ODE that describes the motion particles in interaction. The most important applications we have in mind is the motion of crystal defects called dislocations. For this model, we prove the existence of traveling wave solutions under very weak assumptions. The uniqueness of the velocity is also studied as well as the uniqueness of the profile which used different types of strong maximum principle. As far as we know, this is the first resul…
In this paper, we consider the following critical fractional Kirchhoff equation \begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su=|u|^{2^*_s-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} where $a,b>0$, $\frac{N}{4}<s<1$, $2^*_s=\frac{2N}{N-2s}$ and $(-Δ)^s$ is the fractional Laplacian. We prove the uniqueness and nondegeneracy of positive solutions to the problem, which can be used to study the singular perturbation problems concerning fractional Kirchhoff equations.
We prove uniqueness for the vortex-wave system with a single point vortex introduced by Marchioro and Pulvirenti in the case where the vorticity is initially constant near the point vortex. Our method relies on the Eulerian approach for this problem and in particular on the formulation in terms of the velocity.
There is a few results about the global stability of nontrivial solutions to quasilinear wave equations. In this paper we are concerned with the uniqueness and stability of traveling waves to the time-like extremal hypersurface in Minkowski space. Firstly, we can get the existence and uniqueness of traveling wave solutions to the time-like extremal hypersurface in R1+(n+1), which can be considered as the generalized Bernstein theorem in Minkowski space. Furthermore, we also get the stability of traveling wave solutions with speed of light to time-like extremal hypersurface in 1 + (2 + 1) dimen…
In this short note, based on Carleman estimates and Holmgren's type theorems, we provide a converse theorem of the classical Huygens principle for free wave equations. Possible generalizations to other underlying space-times or other wave type equations are also discussed.
Unconditional Uniqueness for the Energy-critical and Energy-supercritical Quadratic NLS
研究能量临界和超临界二次非线性薛定谔方程，通过引入新的无限二次层级，利用其与双线性\(U\)-\(V\)估计兼容的组合结构，以二次层级方法在临界正则性下建立无条件唯一性，回答布尔甘的唯一性问题。
We study the uniqueness of solutions of Helmholtz equation for a problem that concerns wave propagation in waveguides. The classical radiation condition does not apply to our problem because the inhomogeneity of the index of refraction extends to infinity in one direction. Also, because of the presence of a waveguide, some waves propagate in one direction with different propagation constants and without decaying in amplitude. Our main result provides an explicit condition for uniqueness which takes into account the physically significant components, corresponding to guided and non-guided waves…
Non-Uniqueness of Smooth Solutions of the Navier-Stokes Equations from Critical Data
研究在临界正则性下构造了纳维-斯托克斯方程的非唯一光滑解，证明了小数据全球适定性结果的尖锐性。
We prove the uniqueness for the positive solutions of the following elliptic systems: \begin{eqnarray*} \left\{\begin{array}{ll} - \lap (u(x)) = u(x)^αv(x)^β - \lap (v(x)) = u(x)^β v(x)^α \end{array} \right. \end{eqnarray*} Here $x\in R^n$, $n\geq 3$, and $1\leq α, β\leq \frac{n+2}{n-2}$ with $α+β=\frac{n+2}{n-2}$. In the special case when $n=3$ and $α=2, β=3$, the systems come from the stationary Schrodinger system with critical exponents for Bose-Einstein condensate. As a key step, we prove the radial symmetry of the positive solutions to the elliptic system above with critical exponents.
This article deals with the uniqueness in identifying multiple parameters simultaneously in the one-dimensional time-fractional diffusion-wave equation of fractional time-derivative order $\in (0,2)$ with the zero Robin boundary condition. Using the Laplace transform and a transformation formula, we prove the uniqueness in determining an order of the fractional derivative, a spatially varying potential, initial values and Robin coefficients simultaneously by boundary measurement data, provided that all the eigenmodes of an initial value do not vanish. Furthermore, for another formulation of in…
We prove the uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent and a large frequency parameter. This study is motivated by the paper [2] and aims to remove the restriction on dimension imposed there. In this paper, we employ the fixed-point argument developed in [7] to prove the uniquness. Hence, the linearization around the Aubin-Talenti function plays a key role. Furthermore, we need some estimates for the associated perturbed resolvents (see Proposition 3.1).
We are concerned with the problem of determining the damping boundary coefficient appearing in a dissipative wave equation from a single boundary measurement. We prove that the uniqueness holds at the origin provided that the initial condition is appropriately chosen. We show that the choice of the initial condition leading to uniqueness is related to a fine version of unique continuation property for elliptic operators. We also establish a Lipschitz directional stability estimate at the origin, which is obtained by a linearization process.
Uniqueness of conservative solutions to a one-dimensional general quasilinear wave equation through variational principle
本文利用特征分析证明一般拟线性波动方程能量守恒 Hölder 连续弱解的唯一性，且结论适用于任意大小数据。
Sharp non-uniqueness for the Navier-Stokes equations in scaling critical spaces
本文研究Navier-Stokes方程在临界空间中的解的唯一性，证明在某些稍大的临界空间中唯一性失效，并构造了从零初始状态出发的无限多个全局解。
We prove existence, uniqueness, and stability of transition fronts (generalized traveling waves) for reaction-diffusion equations in cylindrical domains with general inhomogeneous ignition reactions. We also show uniform convergence of solutions with exponentially decaying initial data to time translates of the front. In the case of stationary ergodic reactions the fronts are proved to propagate with a deterministic positive speed. Our results extend to reaction-advection-diffusion equations with periodic advection and diffusion.
This paper is devoted to the inverse problem of determining the spatially dependent source in a time fractional diffusion-wave equation, with the aid of extra measurement data at subboundary. Uniqueness result is obtained by using the analyticity and the new established unique continuation principle provided that the coefficients are all temporally independent. We also derive a Lipschitz stability of our inverse source problem under a suitable topology whose norm is given via the adjoint system of the fractional diffusion-wave equation.
Non-uniqueness of smooth solutions of the 5D magnetohydrodynamic equations from critical data
本文基于Coiculescu和Palasek的工作，研究5维不可压缩磁流体动力学方程的光滑解非唯一性，构造了两个不同的全局解，初始数据在BMO^{-1}空间中非零。
\begin{abstract} We consider the following poly-harmonic equations with critical exponents: \begin{equation}\label{P} (-Δ)^m u =K(y)u^{\frac{N+2m}{N-2m}},\;\;\; u>0\;\;\;\hbox{in} \mathbb{R}^N, \end{equation} where $N> 2m+2,m\in\mathbb{N}_{+}, K(y)$ is positive and periodic in its first $k$ variables $(y_1,\cdots, y_k)$, $1\leq k<\frac{N-2m}{2}$. Under some conditions on $K(y)$ near its critical point, we prove not only that problem~\eqref{P} admits solutions with infinitely many bubbles, but also that the bubbling solutions obtained in our existence result are locally unique. This local uniqu…
The existence, uniqueness and stability of periodic traveling waves for the fractional Benjamin-Bona-Mahony equation is considered. In our approach, we give sufficient conditions to prove a uniqueness result for the single-lobe solution obtained by a constrained minimization problem. The spectral stability is then shown by determining that the associated linearized operator around the wave restricted to the orthogonal of the tangent space related to the momentum and mass at the periodic wave has no negative eigenvalues. We propose the Petviashvili's method to investigate the spectral stability…
Uniqueness and multiplicity of positive radial solutions to the super-critical Brezis-Nirenberg problem in an annulus
本文研究环形区域中超临界Brezis-Nirenberg问题，建立三维情形正径向解的唯一性结果，证明内半径足够小且外半径在某范围时至少有三个正径向解，指数在临界Sobolev指数与Joseph-Lundgren指数间时对任意正整数$k$至少有$k$个正径向解。
We are concerned with the focusing $L^2$-critical nonlinear Schrödinger equations in $\mathbb{R}^d$ for $d=1,2$. The uniqueness is proved for a large energy class of multi-bubble blow-up solutions, which converge to a sum of $K$ pseudo-conformal blow-up solutions particularly with low rate $(T-t)^{0+}$, as $t\to T$, $1\leq K<\infty$. Moreover, we also prove the uniqueness in the energy class of multi-solitons which converge to a sum of $K$ solitary waves with convergence rate $(1/t)^{2+}$, as $t\to \infty$. The uniqueness class is further enlarged to contain the multi-solitons with even lower …
In a recent paper, Struwe considered the Cauchy problem for a class of nonlinear wave and Scrödinger equations. Under some assumptions on the nonlinearities, it was shown that uniqueness of classical solutions can be obtained in the much larger class of distribution solutions satisfying the energy inequality. As pointed out in [38], the conditions on the nonlinearities are satisfied for any polynomial growth but they fail to hold for higher growth (for example ${\rm e}^{u^2}$). Our aim here is to improve Struwe's result by showing that uniqueness holds for more general nonlinearities including…
Non-uniqueness and failure of Calderón-Zygmund estimates below the critical exponent for non-monotone PDE with linear growth
该研究通过凸积分方法，展示了非单调PDE在低于临界指数下解的非唯一性及Calderón-Zygmund估计的失效
We consider fractional diffusion-wave equations with source term which is represented in a form of a product of a temporal function and a spatial function. We prove the uniqueness for inveres source problem of determining spatially varying factor by decay of data as the time tends to $\infty$, provided that the source does not work during the observations. Our main result asserts the uniqueness if data decay more rapidly than $\left(\frac{1}{t^p}\right)$ with any $p\in \N$ as $t\to\infty$. Date taken not from the initial time are realistic but the uniqueness was not known in general. The proof…
We consider the inverse problem of determining the Lamé parameters and the density of a three-dimensional elastic body from the local time-harmonic Dirichlet-to-Neumann map. We prove uniqueness and Lipschitz stability of this inverse problem when the Lamé parameters and the density are assumed to be piecewise constant on a given domain partition.
In this paper we are interested in the following critical Hartree equation \begin{equation*} \begin{cases} -Δu =\displaystyle{\Big(\int_Ω\frac{u^{2_μ^\ast} (ξ)}{|x-ξ|^μ}dξ\Big)u^{2_μ^\ast-1}}+\varepsilon u ,~~~\text{in}~Ω,\\ u=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}~\partialΩ, \end{cases} \end{equation*} where $N\geq4$, $0<μ\leq4$, $\varepsilon>0$ is a small parameter, $Ω$ is a bounded domain in $\mathbb{R}^N$, and $2_μ^\ast=\frac{2N-μ}{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. By establishing various versions of local Pohozaev ident…
A $σ$-homothetic uniqueness of the critical catenoid
本文证明了单位欧氏三维球中与临界悬链面σ-位似的自由边界极小环面具有唯一性。
Uniqueness for the spatially homogeneous Boltzmann equation in critical Sobolev spaces
本文研究极软势下无角截断的空间均匀玻尔兹曼方程，在带对数修正的临界索伯列夫空间中建立解的存在性、唯一性等，结合费舍尔信息单调性实现解的全局时间延拓，采用相空间与频率二进局域化及精确交换子估计作为核心工具。
Uniqueness in inverse elastic scattering from traction-free polygonal obstacles with a single incoming wave
本文针对二维有界无牵引力障碍物的时谐弹性散射逆问题，利用Navier方程反射原理与弹性场唯一延拓性，证明单个入射平面波下连通多边形障碍物可由远场模式唯一确定。
The first part of this paper is concerned with the uniqueness to inverse time-harmonic elastic scattering from bounded rigid obstacles in two dimensions. It is proved that a connected polygonal obstacle can be uniquely identified by the far-field pattern over all observation directions corresponding to a single incident plane wave. Our approach is based on a new reflection principle for the first boundary value problem of the Navier equation. In the second part, we propose a revisited factorization method to recover a rigid elastic body with a single far-field pattern.
The aim of this paper is to classify the positive solutions of the nonlocal critical equation: $$ -Δu=\left(I_μ\ast u^{2^{\ast}_μ}\right)u^{{2}^{\ast}_μ-1},~~x\in\mathbb{R}^{N}, $$ where $0<μ<N$, if $N=3\ \hbox{or} \ 4$ and $0<μ\leq4$ if $N\geq5$, $I_μ$ is the Riesz potential defined by $$ I_μ(x)=\frac{Γ(\fracμ{2})}{Γ(\frac{N-μ}{2})π^{\frac{N}{2}}2^{N-μ}|x|^μ} $$ with $Γ(s)=\int^{+\infty}_{0}x^{s-1}e^{-x}dx$, $s>0$ and $2^{\ast}_μ=\frac{2N-μ}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We apply the moving plane method in integral forms to prove…
In this paper, we prove that there exists at most one positive radial weak solution to the following quasilinear elliptic equation with singular critical growth \[ \begin{cases} -Δ_{p}u-{\displaystyle \fracμ{|x|^{p}}|u|^{p-2}u}{\displaystyle =\frac{|u|^{\frac{(N-s)p}{N-p}-2}u}{|x|^{s}}}+λ|u|^{p-2}u & \text{in }B,\\ u=0 & \text{on }\partial B, \end{cases} \] where $B$ is an open finite ball in $\mathbb{R}^{N}$ centered at the origin, $1<p<N$, $-\infty<μ<((N-p)/p)^{p}$, $0\le s<p$ and $λ\in\mathbb{R}$. A related limiting problem is also considered.
A uniqueness result for the two vortex travelling wave in the Nonlinear Schrodinger equation
本文证明了2维非线性薛定谔方程在大动量极限下双涡旋行波极小值子的唯一性，从而确立了该行波的轨道稳定性。
Nonlinear heat equations in two dimensions with singular initial data are studied. In recent works nonlinearities with exponential growth of Trudinger-Moser type have been shown to manifest critical behavior: well-posedness in the subcritical case and non-existence for certain supercritical data. In this article we propose a specific model nonlinearity with Trudinger-Moser growth for which we obtain surprisingly complete results: a) for initial data strictly below a certain singular threshold function $\widetilde u$ the problem is well-posed, b) for initial data above this threshold function $…
On bilinear estimates and critical uniqueness classes for Navier-Stokes equations
本文研究临界空间中 Navier-Stokes 方程温和解的双线性估计与唯一性，构建无需辅助范数的一般框架，并在 Besov、Morrey 等空间及 Besov-weak-Herz 空间中取得新成果。
We consider Dirichlet problems for linear elliptic equations of second order in divergence form on a bounded or exterior smooth domain $Ω$ in $\mathbb{R}^n$, $n \ge 3$, with drifts $\mathbf{b}$ in the critical weak $L^n$-space $L^{n,\infty}(Ω; \mathbb{R}^n )$. First, assuming that the drift $\mathbf{b}$ has nonnegative weak divergence in $L^{n/2, \infty }(Ω)$, we establish existence and uniqueness of weak solutions in $W^{1,p}(Ω)$ or $D^{1,p}(Ω)$ for any $p$ with $n' = n/(n-1)< p < n$. By duality, a similar result also holds for the dual problem. Next, we prove $W^{1,n+\varepsilon}$ or $W^{2, …
Uniqueness and stability of entropy waves among vanishing dissipation limits from the Navier--Stokes--Fourier System
该研究在纳维-斯托克斯-傅里叶系统的消失耗散极限类中，基于a-压缩理论，证明了可压缩欧拉方程熵波解的唯一性与稳定性，且当初值收敛时解会收敛到熵波。
Uniqueness in inverse scattering with phaseless near-field data generated by superpositions of two incident plane waves
本文研究固定频率下两入射平面波叠加产生的无相位近场数据的逆散射问题，证明未知散射体可被该数据唯一确定，证明结合了相位分析及相关方程的经典引理与公式。
We study the equation \begin{equation*}\label{P0} (-Δ)^s u = |x|^α u^{\frac{N+2s+2α}{N-2s}}\mbox{ in }\mathbb{R}^N,\tag{P} \end{equation*} where $(-Δ)^s$ is the fractional Laplacian operator with $0 < s < 1$, $α>-2s$ and $N>2s$. We prove the linear non-degeneracy of positive radially symmetric solutions of the equation (\ref{P0}) and, as a consequence, a uniqueness result of those solutions with Morse index equal to one. In particular, the ground state solution is unique. Our non-degeneracy result extends in the radial setting some known theorems done by Dávila, Del Pino and Sire (see \cite[Th…
On uniqueness for hyperbolic half-wave maps in dimension $d \geq 3$
本文研究了维度 $d \geq 3$ 下 Half-Wave Maps 方程的唯一性问题，通过时间求导、Nash 嵌入定理及几何性质，结合分数阶 Leibniz 规则与交换子估计，利用 Grönwall 不等式在自然能量类中证明了唯一性。
A uniqueness theorem for 3D semilinear wave equations satisfying the null condition
该研究针对三维闵可夫斯基时空中满足零条件的半线性波动方程组，通过给定球外初值相等、光锥外辐射场相等的前提，证明了解的唯一性结论，即解在双曲面外或整个时空相等，具体取决于球与光锥的尺寸。
Small mass uniqueness in the anisotropic liquid drop model and the critical mass problem
本文获得了所有参数下各向异性原子液滴模型的小质量唯一性，并探讨了临界质量猜想。
We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i\partial_tu+Δu+k(x)|u|^{2}u=0$. From standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\|u\|_{L^2}<M_k$ are global in time while a finite time blow up singularity formation may occur for $\|u\|_{L^2}>M_k$. In this paper, we consider the dynamics at threshold $\|u_0\|_{L^2}=M_k$ and give a necessary and sufficient condition on $k$ to ensure the existence of critical mass finite time blow up elements. Moreover, we give a complete classification in the energy class of t…
We are revisiting the topic of travelling fronts for the food-limited (FL) model with spatio-temporal nonlocal reaction. These solutions are crucial for understanding the whole model dynamics. Firstly, we prove the existence of monotone wavefronts. In difference with all previous results formulated in terms of `sufficiently small parameters', our existence theorem indicates a reasonably broad and explicit range of the model key parameters allowing the existence of monotone waves. Secondly, numerical simulations realized on the base of our analysis show appearance of non-oscillating and non-mon…
We study a non-linear and non-local evolution equation for curves obtained as the sharp interface limit of a phase-field model for crawling motion of eukaryotic cells on a substrate. We establish uniqueness of solutions to the sharp interface limit equation in the so-called subcritical parameter regime. The proof relies on a Grönwall estimate for a specially chosen weighted $L^2$ norm. Next, as persistent motion of crawling cells is of central interest to biologists we study the existence of traveling wave solutions. We prove that traveling wave solutions exist in the supercritical parameter r…
For a class of Kirchhoff functional, we first give a complete classification with respect to the exponent $p$ for its $L^2$-normalized critical points, and show that the minimizer of the functional, if exists, is unique up to translations. Secondly, we search for the mountain pass type critical point for the functional on the $L^2$-normalized manifold, and also prove that this type critical point is unique up to translations. Our proof relies only on some simple energy estimates and avoids using the concentration-compactness principles. These conclusions extend some known results in previous p…
Sharp non-uniqueness of weak solutions to the semilinear wave equation
本文针对半线性波动方程，在低正则性下构造无穷多个零初值弱解，证明非唯一性，并指出该结果与经典唯一性理论互补，达到尖锐界限。
Uniqueness of an inverse cavity scattering problem for the biharmonic wave equation
针对二维双调和波方程的腔体反散射问题，论文利用 Green 表示、混合互易关系及对称性，证明由远场模式或无相位近场数据可唯一确定腔体形状。
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