Inverse spectral problem for analytic $(Z/2Z)^n$-symmetric domains in $R^n$

完备黎曼流形上对称双曲偏微分算子的反问题

An Inverse Problem for symmetric hyperbolic Partial Differential Operators on Complete Riemannian Manifolds

本文研究完备黎曼流形上对称双曲偏微分算子的反问题，通过改进边界控制方法（BC-method），从局部源到解映射唯一恢复流形及低阶项。

The factorization method for the inverse acoustic scattering problems with limited aperture data

We study the factorization method for the inverse acoustic scattering problems in the case of limited aperture data. In this case, the factorization of the far field operator is not symmetric. So, we can not apply the original factorization method which requires the symmetricity of the factorization. In this paper, we provide a new functional analytic theorem corresponding to the case of limited aperture data, and by applying it, we give the characterizations of the unknown domain.

Self-Expanders to Inverse Curvature Flows by Homogeneous Functions

In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-expanders to any of these flows are round spheres. Secondly, we show that complete non-compact self-expanders to any of these flows with asymptotically cylindrical ends must be rotationally symmetric. Thirdly, we show that when such a flow is uniformly parabolic, there exist comp…

Navier-Stokes equations, symmetric and uniform analytic solutions in phase space

For incompressible Navier-Stokes equations, Necas-Ruzicka-Sverak proved that self-similar solution has to be zero in 1996. Further, Yang-Yang-Wu find symmetry property plays an important role in the study of ill-posedness. In this paper, we consider two types of symmetry property. We search special symmetric and uniform analytic functions to approach the solution and establish global uniform analytic and symmetric solution with initial value in general symmetric Fourier-Herz space. For two kinds of symmetry of initial data, we prove that the solution has also the same symmetric structure. Furt…

Uniqueness for fractional nonsymmetric diffusion equations and an application to an inverse source problem

In this paper, we discuss the uniqueness for solution to time-fractional diffusion equation $\partial_t^α(u-u_0) + Au=0$ with the homogeneous Dirichlet boundary condition, where an elliptic operator $-A$ is not necessarily symmetric. We prove that the solution is identically zero if its normal derivative with respect to the operator $A$ vanishes on an arbitrary small part of the spatial domain over a time interval. The proof is based on the Laplace transform and the spectral decomposition, and is valid for more general time-fractional partial differential equations, including those involving n…

Inverse antiplane problem on $n$ uniformly stressed inclusions

The inverse problem of antiplane elasticity on determination of the profiles of $n$ uniformly stressed inclusions is studied. The inclusions are in ideal contact with the surrounding matrix, the stress field inside the inclusions is uniform, and at infinity the body is subjected to antiplane uniform shear. The exterior of the inclusions, an $n$-connected domain, is treated as the image by a conformal map of an $n$-connected slit domain with the slits lying in the same line. The inverse problem is solved by quadratures by reducing it to two Riemann-Hilbert problems on a Riemann surface of genus…

旋转对称超曲面的逆均值流

Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces

研究了旋转对称超曲面的逆均值流，证明其收敛性并扩展了Minkowski不等式，同时建立了与极小曲面理论的联系。

Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type

Let $(M,g)$ be a globally symmetric space of noncompact type, of arbitrary rank, and $Δ$ its Laplacian. We prove the existence of a meromorphic continuation of the resolvent $(Δ-\ev)^{-1}$ across the continuous spectrum to a Riemann surface multiply covering the plane. The methods are purely analytic and are adapted from quantum $N$-body scattering.

具有有限个随机测量的解析逆问题

Analytic inverse problems with finitely many random measurements

该研究针对严重不适定逆问题，证明随机采样可将所需测量数从随d指数增长降至2d+1，并将结果应用于Calderón问题和逆介质散射问题。