New global stability estimates for the Gel'fand-Calderon inverse problem

Instability in the Gel'fand inverse problem at high energies

We give an instability estimate for the Gel'fand inverse boundary value problem at high energies. Our instability estimate shows an optimality of several important preceeding stability results on inverse problems of such a type.

Partial regularity of stable solutions to the fractional Gel'fand-Liouville equation

We analyze stable weak solutions to the fractional Gel'fand problem \begin{equation*} (-Δ)^su=e^u\quad\mathrm{in}\quad Ω\subset\mathbb{R}^n. \end{equation*} We prove that the dimension of the singular set is at most $n-10s.$

The Gel'fand problem for the biharmonic operator

We study stable solutions of a fourth order nonlinear elliptic equation, both in entire space and in bounded domains.

On the number of peaks of the eigenfunctions of the linearized Gel'fand problem

We derive a second order estimate for the first m eigenvalues and eigenfunctions of the linearized Gel'fand problem associated to solutions which blow-up at m points. This allows us to determine, in some suitable situations, some qualitative properties of the first m eigenfunctions as the number of points of concentration or the multiplicity of the eigenvalue .

Morse indices of multiple blow-up solutions to the two-dimensional Gel'fand problem

Blow-up solutions to the two-dimensional Gel'fand problem are studied. It is known that the location of the blow-up points of these solutions is related to a Hamiltonian function involving the Green function of the domain. We show that this implies an equivalence between the Morse indices of the solutions and the associated critical points of the Hamiltonian.

On stable determination of potential by boundary measurements

We give new stability estimates for the Gel'fand-Calderon inverse boundary value problem

Local uniqueness of $m$-bubbling sequences for the Gel'fand equation

We consider the Gel'fand problem, $$ \begin{cases} Δw_{\varepsilon}+\varepsilon^2 h e^{w_{\varepsilon}}=0\quad&\mbox{in}\quadΩ, w_{\varepsilon}=0\quad&\mbox{on}\quad\partialΩ, \end{cases} $$ where $h$ is a nonnegative function in ${Ω\subset\mathbb{R}^2}$. Under suitable assumptions on $h$ and $Ω$, we prove the local uniqueness of $m-$bubbling solutions for any $\varepsilon>0$ small enough.

Action-Angle variables for the Gel'fand-Dikii flows

Using the scattering transform for $n^{th}$ order linear scalar operators, the Poisson bracket found by Gel'fand and Dikii, which generalizes the Gardner Poisson bracket for the KdV hierarchy, is computed on the scattering side. Action-angle variables are then constructed. Using this, complete integrability is demonstrated in the strong sense. Real action-angle variables are constructed in the self-adjoint case.

Exponential instability in the Gel'fand inverse problem on the energy intervals

We consider the Gel'fand inverse problem and continue studies of [Mandache,2001]. We show that the Mandache-type instability remains valid even in the case of Dirichlet-to-Neumann map given on the energy intervals. These instability results show, in particular, that the logarithmic stability estimates of [Alessandrini,1988], [Novikov,Santacesaria,2010] and especially of [Novikov,2010] are optimal (up to the value of the exponent).

线性化Liouville-Gel'fand问题特征值渐近行为的精细化

Refinement of asymptotic behavior of the eigenvalues for the linearized Liouville-Gel'fand Problem

本文研究了二维区域中带非齐次系数的线性化Liouville-Gel'fand问题，确定了在m个点爆破的解对应的前m个特征值与特征函数渐近展开的第二项，并分析了系数所需的正则性条件。

Stability and Reconstruction in Gel'fand Inverse Boundary Spectral Problem

We consider stability and approximate reconstruction of Riemannian manifold when the finite number of eigenvalues of the Laplace-Beltrami operator and the boundary values of the corresponding eigenfunctions are given. The reconstruction can be done in stable way when manifold is a priori known to satisfy natural geometrical conditions related to curvature and other invariant quantities.

Exact eigenvalues and eigenfunctinos for a one-dimensional Gel'fand problem

It is known that every positive solution of a one-dimensional Gel'fand problem can be written explicitly. In this paper we obtain exact expressions of all the eigenvalues and eigenfunctions of the linearized eigenvalue problem at each solution. We also study asymptotic behaviors of eigenvalues and eigenfunctions as the $L^{\infty}$-norm of the solution goes to the infinity.

New global stability estimates for the Calderón problem in two dimensions

We prove a new global stability estimate for the Gel'fand-Calderón inverse problem on a two-dimensional bounded domain or, more precisely, the inverse boundary value problem for the equation $-Δψ+ v\, ψ= 0$ on $D$, where $v$ is a smooth real-valued potential of conductivity type defined on a bounded planar domain $D$. The principal feature of this estimate is that it shows that the more a potential is smooth, the more its reconstruction is stable, and the stability varies exponentially with respect to the smoothness (in a sense to be made precise). As a corollary we obtain a similar estimate f…

On the global bifurcation diagram of the Gel'fand problem

For domains of first kind [7,13] we describe the qualitative behavior of the global bifurcation diagram of the unbounded branch of solutions of the Gel'fand problem crossing the origin. At least to our knowledge this is the first result about the exact monotonicity of the branch of non-minimal solutions which is not just concerned with radial solutions [28] and/or with symmetric domains [23]. Toward our goal we parametrize the branch not by the $L^{\infty}(Ω)$-norm of the solutions but by the energy of the associated mean field problem. The proof relies on a carefully modified spectral analysi…

Gel'fand的逆问题在Ricci曲率界下

Gel'fand's inverse problem under Ricci curvature bounds

本文在RCD(K,N)空间框架下，研究了Gel'fand逆问题的唯一可解性及在闭合黎曼流形上的稳定性。

Boundary regularity for the Ricci equation, geometric convergence, and Gel'fand's inverse boundary problem

This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional st…

Reconstruction and stability in Gel'fand's inverse interior spectral problem

Assume that $M$ is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian $Δ_g$ on $M$ as well as the corresponding eigenfunctions restricted on an open set in $M$. We then construct a stable approximation to the manifold $(M,g)$. Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from $M$ when the above data are given with a small error. We give an explicit $\log\log$-type stability esti…

On the Gel'fand-Calderón inverse problem in two dimensions

We prove uniqueness and stability for the inverse boundary value problem of the two dimensional Schrödinger equation. We do not assume the potentials to be continuous or even bounded. Instead, we assume that some of their positive fractional derivatives are in a specific Lorentz space. These spaces are a natural generalization to the usual fractional Sobolev spaces. The thesis consists of two parts. In the first part, we define the generalized fractional Sobolev spaces and prove some of their properties including embeddings and interpolation identities. In particular we sharpen the usual Sobol…

Gel'fand反边界问题的定量稳定性

Quantitative stability of Gel'fand's inverse boundary problem

本文研究Gel'fand反边界问题，证明在有界几何流形类中，有限个含小误差的特征值及特征函数边界值可确定接近原流形的度量空间，并给出构造算法，基于波动算子唯一连续性的显式稳定性估计。

Schrödinger算子的Gel'fand逆内部谱问题的稳定性

Stability of Gel'fand's inverse interior spectral problem for Schrödinger operators

本文研究Schrödinger算子的Gel'fand逆内部谱问题，证明有限谱数据近似可在Gromov-Hausdorff拓扑下恢复流形并一致逼近势函数，从而得到一般情形下的定量稳定性估计。

Calderón's Inverse Problem with a Finite Number of Measurements II: Independent Data

We prove a local Lipschitz stability estimate for Gel'fand-Calderón's inverse problem for the Schrödinger equation. The main novelty is that only a finite number of boundary input data is available, and those are independent of the unknown potential, provided it belongs to a known finite-dimensional subspace of $L^\infty$. A similar result for Calderón's problem is obtained as a corollary. This improves upon two previous results of the authors on several aspects, namely the number of measurements and the stability with respect to mismodeling errors. A new iterative reconstruction scheme based …

The Calderon problem in transversally anisotropic geometries

We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two conditions: (1) the metric is conformally transversally anisotropic (CTA), and (2) the transversal manifold is simple. In this paper we will consider geometries satisfying (1) but not (2). The first main result states that the boundary measurements uniquely determine a mixed Four…

闭黎曼流形上的分数阶 Calderón 问题

Fractional Calderón problem on a closed Riemannian manifold

本文证明闭黎曼流形上分数阶 Laplace 方程的源到解映射可在任意小观测区恢复流形等距类，并借助 Carlson 定理变体将其归约为 Gel'fand 反谱问题。