Lipschitz stability estimate for the simultaneous recovery of two coefficients in the anisotropic Schrödinger type equation via local Cauchy data

We consider the inverse problem of the simultaneous identification of the coefficients σ and q of the equation from the knowledge of the Cauchy data set. We assume that , where A is a given matrix function and γ and q are unknown piecewise affine scalar functions. No sign, nor spectrum condition on q is assumed. We derive a result of global Lipschitz stability in dimension . The proof relies on the method of singular solutions and on the quantitative estimates of unique continuation

Stability estimates for some anisotropic inverse problems

This doctoral thesis is devoted to the study of the stability of some inverse problems concerning media with anisotropic properties. Specifically, we have studied the conditional stability, which refers to the dependence of the unknown parameters or defects on known measurement data when the unknowns satisfy certain constraints. Inverse problems are known to be ill-posed in the Hadamard sense, making it difficult to obtain reliable numerical reconstructions of the parameters of interest. In this thesis, we discuss the stability issue for two types of inverse problems: the coefficient identification problem and the determination of inclusions. The underlying mathematical model is given by a second-order linear elliptic equation of the form: \begin{equation}\label{eqnAb: elliptic} \text{div}(\sigma\nabla u) + q\:u=0,\qquad \text{in }\Omega, \end{equation} with prescribed Cauchy data $\{u|_{\partial\Omega},\sigma\nabla u\cdot \nu|_{\partial\Omega}\}$ or the Dirichlet-to-Neumann (DtoN) map, when the latter is defined. The leading-order term exhibits an anisotropic behaviour, which is described by a coefficient of the form \begin{equation}\label{eqnAb: conductivity} \sigma(x)=\gamma(x)A(x), \end{equation} where $\gamma$ is a suitably regular scalar function and $A$ is a matrix-valued function. We address the stability issue for the inverse conductivity problem, when $q=0$ in \eqref{eqnAb: elliptic}, of determining the conductivity $\sigma$ of the form \eqref{eqnAb: conductivity}, where $\gamma$ is a piecewise affine function, from the knowledge of the boundary data. We construct an ad-hoc misfit functional to encode the error arising from the comparison with two different known boundary data. We prove a H\"older stability estimate in terms of the misfit functional and then we derive a Lipschitz stability estimate in terms of the local Dirichlet to Neumann map. In the general case, when $q\neq 0$ and no sign or spectrum condition on $q$ is assumed in the generalised Schr\"odinger equation \eqref{eqnAb: elliptic}, we address the inverse problem of determining an anisotropic inclusion in a medium $\Omega$ from the knowledge of the local Cauchy data and the problem of the simultaneous determination of the two coefficients in \eqref{eqnAb: elliptic}. For the first problem, we provide a log-type stability estimate which holds under mild constraints on the unknown inclusion. For the latter, we derive a global Lipschitz stability estimate which holds for the coefficients $\sigma$ and $q$ in \eqref{eqnAb: elliptic} with a piecewise affine scalar part on a known partition of the domain.This doctoral thesis is devoted to the study of the stability of some inverse problems concerning media with anisotropic properties. Specifically, we have studied the conditional stability, which refers to the dependence of the unknown parameters or defects on known measurement data when the unknowns satisfy certain constraints. Inverse problems are known to be ill-posed in the Hadamard sense, making it difficult to obtain reliable numerical reconstructions of the parameters of interest. In this thesis, we discuss the stability issue for two types of inverse problems: the coefficient identification problem and the determination of inclusions. The underlying mathematical model is given by a second-order linear elliptic equation of the form: \begin{equation}\label{eqnAb: elliptic} \text{div}(\sigma\nabla u) + q\:u=0,\qquad \text{in }\Omega, \end{equation} with prescribed Cauchy data $\{u|_{\partial\Omega},\sigma\nabla u\cdot \nu|_{\partial\Omega}\}$ or the Dirichlet-to-Neumann (DtoN) map, when the latter is defined. The leading-order term exhibits an anisotropic behaviour, which is described by a coefficient of the form \begin{equation}\label{eqnAb: conductivity} \sigma(x)=\gamma(x)A(x), \end{equation} where $\gamma$ is a suitably regular scalar function and $A$ is a matrix-valued function. We address the stability issue for the inverse conductivity problem, when $q=0$ in \eqref{eqnAb: elliptic}, of determining the conductivity $\sigma$ of the form \eqref{eqnAb: conductivity}, where $\gamma$ is a piecewise affine function, from the knowledge of the boundary data. We construct an ad-hoc misfit functional to encode the error arising from the comparison with two different known boundary data. We prove a H\"older stability estimate in terms of the misfit functional and then we derive a Lipschitz stability estimate in terms of the local Dirichlet to Neumann map. In the general case, when $q\neq 0$ and no sign or spectrum condition on $q$ is assumed in the generalised Schr\"odinger equation \eqref{eqnAb: elliptic}, we address the inverse problem of determining an anisotropic inclusion in a medium $\Omega$ from the knowledge of the local Cauchy data and the problem of the simultaneous determination of the two coefficients in \eqref{eqnAb: elliptic}. For the first problem, we provide a log-type stability estimate which holds under mild constraints on the unknown inclusion. For the latter, we derive a global Lipschitz stability estimate which holds for the coefficients $\sigma$ and $q$ in \eqref{eqnAb: elliptic} with a piecewise affine scalar part on a known partition of the domain

Stable determination of polyhedral interfaces from boundary data for the Helmholtz equation

We study an inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map as the data. We consider piecewise constant wavespeeds on an unknown tetrahedral partition and prove a Lipschitz stability estimate in terms of the Hausdorff distance between partitions

[Book of abstracts]

USPFAPESPCAPESICMC Summer Meeting on Differential Equations (2015 São Carlos

Evolution systems for paraxial wave equations of Schrödinger-type with non-smooth coefficients

AbstractWe prove existence of strongly continuous evolution systems in L2 for Schrödinger-type equations with non-Lipschitz coefficients in the principal part. The underlying operator structure is motivated from models of paraxial approximations of wave propagation in geophysics. Thus, the evolution direction is a spatial coordinate (depth) with additional pseudodifferential terms in time and low regularity in the lateral space variables. We formulate and analyze the Cauchy problem in distribution spaces with mixed regularity. The key point in the evolution system construction is an elliptic regularity result, which enables us to precisely determine the common domain of the generators. The construction of a solution with low regularity in the coefficients is the basis for an inverse analysis which allows to infer the lack of lateral regularity in the medium from measured data

On the global existence for the modified Camassa-Holm equation via the inverse scattering method

In this paper, we address the existence of global solutions to the Cauchy problem of the modified Camassa-Holm (mCH) equation, which is known as a model for the unidirectional propagation of shallow water waves. Based on the spectral analysis of the Lax pair, we apply the inverse scattering transform to rigorously analyze the mCH equation with zero background. By connecting the Cauchy problem to the Riemann-Hilbert (RH) problem, we establish a bijective map between potential and reflection coefficients within the $L^2$-Sobolev space framework. Utilizing a reconstruction formula and estimates on the time-dependent RH problem, we obtain a unique global solution to the Cauchy problem for the mCH equation.Comment: 29 page

Minimizing travelling waves for the one-dimensional nonlinear Schr\"odinger equation with non-zero condition at infinity

This paper deals with the existence of travelling wave solutions for a general one-dimensional nonlinear Schr\"odinger equation. We construct these solutions by minimizing the energy under the constraint of fixed momentum. We also prove that the family of minimizers is stable. Our method is based on recent articles about the orbital stability for the classical and non-local Gross-Pitaevskii equations [3, 10]. It relies on a concentration-compactness theorem, which provides some compactness for the minimizing sequences and thus the convergence (up to a subsequence) towards a travelling wave solution