We consider the inverse conductivity problem of identifying embedded objects in unbounded domains. The main tool is a set of special solutions to the Schroedinger equation, the complex spherical waves, which are constructed by a Carleman estimate. Using these solutions we can treat the inverse problem in the unbounded domain like in the bounded case, and it is possible to localize the boundary measurements to a finite part of the boundary.
The present paper considers a homogeneous bubble inside an unbounded polytropic compressible liquid with viscosity. The system is governed by the Navier-Stokes equation with free boundary which is determined by the kinematic and dynamic boundary conditions on the bubble-liquid interface. The global existence of solution is proved, and the $\dot{H}^1$ asymptotic stability of the spherical equilibrium in terms of viscous damping together with a explicit decay rate is given in bare energy methods.
Global stability of the spherically symmetric nonisentropic compressible Euler equations with positive density around global-in-time background affine solutions is shown in the presence of free vacuum boundaries. Vacuum is achieved despite a non-vanishing density by considering a negatively unbounded entropy and we use a novel weighted energy method whereby the exponential of the entropy will act as a changing weight to handle the degeneracy of the vacuum boundary. Spherical symmetry introduces a coordinate singularity near the origin for which we adapt a method developed for the Euler-Poisson…
This paper deals with the spherically symmetric flow of compressible viscous and polytropic ideal fluid in unbounded domain exterior to a ball in $\mathbb{R}^n$ with $n\ge2$. We show that the global solutions are convergent as time goes to infinity. The critical step is obtaining the point-wise bound of the specific volume $v(x,t)$ and the absolute temperature $θ(x,t)$ from up and below both for $x$ and $t$. Note that the initial data can be arbitrarily large and, compared with \cite{nn}, our method applies to the spatial dimension $n=2.$ The proof is based on the elementary energy methods.
We consider the operator $H:= i \partial_t + \nabla \cdot (c \nabla)$ in an unbounded strip $Ω$ in $\mathbb{R}^2$, where $c(x,y) \in \mathcal{C}^3(\barΩ)$. We prove adapted a global Carleman estimate and an energy estimate for this operator. Using these estimates, we give a stability result for the diffusion coefficient $c(x,y)$.
We use sphericalization to study the Dirichlet problem, Perron solutions and boundary regularity for p-harmonic functions on unbounded sets in Ahlfors regular metric spaces. Boundary regularity for the point at infinity is given special attention. In particular, we allow for several "approach directions" towards infinity and take into account the massiveness of their complements. In 2005, Llorente-Manfredi-Wu showed that the p-harmonic measure on the upper half space $R^n_+, n \ge 2$, is not subadditive on null sets when $p \neq 2$. Using their result and spherical inversion, we create similar…
This paper is concerned with analysis of electromagnetic wave scattering by an obstacle which is embedded in a two-layered lossy medium separated by an unbounded rough surface. Given a dipole point source, the direct problem is to determine the electromagnetic wave field for the given obstacle and unbounded rough surface; the inverse problem is to reconstruct simultaneously the obstacle and unbounded rough surface from the electromagnetic field measured on a plane surface above the obstacle. For the direct problem, a new boundary integral equation is proposed and its well-posedness is establis…
The monotonicity method for the inverse elastic scattering on unbounded domains
Institute for Mathematics, Goethe-University Frankfurt（法兰克福歌德大学数学研究所）；Three Gorges Mathematical Research Center, College of Mathematics and Physics, China Three Gorges University（三峡大学数学与物理学院三峡数学研究中心）
We summarize in these notes the course given at the Summer School of AIP 2019 held in Grenoble from July 1st to July 5th. This course was mainly devoted to the determination of the unbounded potential in a Schrödinger equation from the associated Dirichlet-to-Neumann map (abbreviated to DN map in this text). We establish a stability inequality for potentials belonging to $L^n$, where $n\ge 3$ is the dimension of the space. Next, we prove a uniqueness result for potentials in $L^{n/2}$, $n\ge 3$, and apply this uniqueness result to demonstrate a Borg-Levinson type theorem. We use a classical ap…
In this paper, we consider the direct and inverse problem of scattering of time-harmonic waves by an unbounded rough interface with a buried impenetrable obstacle. We first study the well-posedness of the direct problem with a local source by the variational method; the well-posedness result is then extended to scattering problems induced by point source waves (PSWs) and hyper-singular point source waves (HSPSWs). For PSW or HSPSW incident waves, the induced total field admits a uniformly bounded estimate in any compact subset far from the source position. Moreover, we show that the scattered …
This paper is concerned with inverse acoustic source problems in an unbounded domain with dynamical boundary surface data of Dirichlet kind. The measurement data are taken at a surface far away from the source support. We prove uniqueness in recovering source terms of the form $f(x)g(t)$ and $f(x_1,x_2,t) h(x_3)$, where $g(t)$ and $h(x_3)$ are given and $x=(x_1, x_2, x_3)$ is the spatial variable in three dimensions. Without these a priori information, we prove that the boundary data of a family of solutions can be used to recover general source terms depending on both time and spatial variabl…
We discuss a time-harmonic inverse scattering problem for the Helmholtz equation with compactly supported penetrable and possibly inhomogeneous scattering objects in an unbounded homogeneous background medium, and we develop a monotonicity relation for the far field operator that maps superpositions of incident plane waves to the far field patterns of the corresponding scattered waves. We utilize this monotonicity relation to establish novel characterizations of the support of the scattering objects in terms of the far field operator. These are related to and extend corresponding results known…
We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across $Σ\subseteqΓ=\partialΩ$, where $Ω$ is a bounded open subset of $\mathbb{R}^{3}$ with a Lipschitz boundary. This follows from a uniqueness result in inverse scattering for Schrödinger operators with singular $δ$-type potential supported on the surface $Γ$ and of strength $α\in L^{p}(Γ)$, $p>2$.
We consider the inverse problem of determining the time independent scalar potential of the dynamic Schrödinger equation in an infinite cylindrical domain from one boundary Neumann observation of the solution. We prove Hölder stability by choosing the Dirichlet boundary condition suitably.
In this paper we give sufficient conditions on $α\geq 0$ and $c\in \mathbb{R}$ ensuring that the space of test functions $C_c^\infty(\mathbb{R}^N)$ is a core for the operator $$L_0u=(1+|x|^α)Δu+\frac{c}{|x|^2}u=:Lu+\frac{c}{|x|^2}u,$$ and $L_0$ with suitable domain generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p(\mathbb{R}^N),\,1<p<\infty$. The proofs are based on some $L^p$-weighted Hardy's inequality and perturbation techniques.
The endpoint Strichartz estimates for two-dimensional Schrodinger equations were recovered by averaging the solutions in L^2 in the angular variable by Tao. For Schrodinger equations with defocusing inverse square potential, we proved that the homogeneous endpoint estimates hold under this setting. In particular, the original versions of endpoint estimates hold for radial data.
For a second order formally symmetric elliptic differential expression we show that the knowledge of the Dirichlet-to-Neumann map or Robin-to-Dirichlet map for suitably many energies on an arbitrarily small open subset of the boundary determines the self-adjoint operator with a Dirichlet boundary condition or with a (possibly non-self-adjoint) Robin boundary condition uniquely up to unitary equivalence. These results hold for general Lipschitz domains, which can be unbounded and may have a non-compact boundary, and under weak regularity assumptions on the coefficients of the differential expre…
We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded open set in $R^n$ for the perturbed polyharmonic operator $(-Δ)^m +q$ with $q\in L^{n/2m}$, $n>2m$, determines the potential $q$ in the set uniquely. In the course of the proof, we construct a special Green function for the polyharmonic operator and establish its mapping properties in suitable weighted $L^2$ and $L^p$ spaces. The $L^p$ estimates for the special Green function are derived from $L^p$ Carleman estimates with linear weights for the polyharmonic operator.
We study inverse conductivity problem for an anisotropic conductivity in $L^\infty$ in bounded and unbounded domains. Also, we give applications of the results in the case when Dirichlet-to-Neumann and Neumann-to-Dirichlet maps are given only on a part of the boundary.
We consider the inverse problem of determining an electromagnetic potential appearing in an infinite cylindrical domain from boundary measurements. More precisely, we prove the stable recovery of some general class of magnetic field and electric potential from boundary measurements. Assuming some knowledge of the unknown coefficients close to the boundary, we obtain also some results of stable recovery with measurements restricted to some portion of the boundary. Our approach combines construction of complex geometric optics solutions and Carleman estimates suitably designed for our stability …
In this article we focus on inverse problems for a semilinear elliptic equation. We show that a potential $q$ in $L^{n/2+\varepsilon}$, $\varepsilon>0$, can be determined from the full and partial Dirichlet-to-Neumann map. This extends the results from [M. Lassas, T. Liimatainen, Y.-H. Lin, and M. Salo, Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations, Rev. Mat. Iberoam. (2021)] where this is shown for Hölder continuous potentials. Also we show that when the Dirichlet-to-Neumann map is restricted to one point on the boundary…
In this paper, we study the global existence, uniqueness and large-time behavior of spherically symmetric solution of a viscous radiative and reactive gas in an unbounded domain exterior to the unit sphere in $\mathbb{R}^{n}$ for $n\geq 2$. The key point in the analysis is to deduce certain uniform estimates on the solutions, especially on the uniform positive lower and upper bounds on the specific volume and the temperature.
We prove uniqueness in the inverse conductivity problem for uniformly elliptic conductivities in $W^{s,p}(Ω)$, where $Ω\subset \mathbb R^n$ is Lipschitz, $3\leq n \leq 6$, and $s$ and $p$ are such that $ W^{s,p}(Ω)\not \subset W^{1,\infty}(Ω)$. In particular, we obtain uniqueness for conductivities in $W^{1,n}(Ω)$ ($n=3,4$). This improves on the result of the author and Tataru, who assumed that the conductivity is Lipschitz.
We consider the Dirichlet Laplacian $A_q=-Δ+q$ in a bounded domain $Ω\subset \mathbb{R}^d$, $d \ge 3$, with real-valued perturbation $q \in L^{\max(2 , 3 d / 5)}(Ω)$. We examine the stability issue in the inverse problem of determining the electric potential $q$ from the asymptotic behavior of the eigenvalues of $A_q$. Assuming that the boundary measurement of the normal derivative of the eigenfunctions is a square summable sequence in $L^2(\partial Ω)$, we prove that $q$ can be Hölder stably retrieved through knowledge of the asymptotics of the eigenvalues
We study the inverse problem of determining a magnetic Schrödinger operator in an unbounded closed waveguide from boundary measurements. We consider this problem with a general closed waveguide in the sense that we only require our unbounded domain to be contained into an infinite cylinder. In this context we prove the unique recovery of the magnetic field and the electric potential associated with general bounded and non-compactly supported electromagnetic potentials. By assuming that the electromagnetic potentials are known on the neighborhood of the boundary outside a compact set, we even p…
We discuss a time-harmonic inverse scattering problem for a nonlinear Helmholtz equation with compactly supported inhomogeneous scattering objects that are described by a nonlinear refractive index in unbounded free space. Assuming the knowledge of a nonlinear far field operator, which maps Herglotz incident waves to the far field patterns of corresponding solutions of the nonlinear scattering problem, we show that the nonlinear index of refraction is uniquely determined. We also generalize two reconstruction methods, a factorization method and a monotonicity method, to recover the support of …
In this paper, we represent the exact solution of a two phase inverse spherical Stefan problem, where along with unknown temperature functions heat flux function has to be determined. Suggested solution is obtained from new form of integral error function and its properties which are represented in the form of series whose coefficients have to be determined. The model problem can be used for mathematical modeling and investigation of arc phenomena in electrical contacts.
We consider the inverse problem of determining the time independent scalar potential of the dynamic Schrödinger equation in an infinite cylindrical domain, from one Neumann boundary observation of the solution. Assuming that this potential is known outside some fixed compact subset of the waveguide, we prove that it may be Lipschitz stably retrieved by choosing the Dirichlet boundary condition of the system suitably. Since the proof is by means of a global Carleman estimate designed specifically for the Schrödinger operator acting in an unbounded cylindrical domain, the Neumann data is measure…
We consider an inverse problem arising in thermo-/photo- acoustic tomography that amounts to reconstructing a function $f$ from its circular or spherical means with the centers lying on a given measurement surface. (Equivalently, these means can be expressed through the solution $u(t,x)$ of the wave equation with the initial pressure equal to $f$.) An explicit solution of this inverse problem is obtained in 3D for the surface that is the boundary of an open octet, and in 2D for the case when the centers of integration circles lie on two rays starting at the origin and intersecting at the angle…
We consider the inverse problem of recovering the spherically symmetric sound speed, density and attenuation in the Sun from the observations of the acoustic field randomly excited by turbulent convection. We show that observations at two heights above the photosphere and at two frequencies above the acoustic cutoff frequency uniquely determine the solar parameters. We also present numerical simulations which confirm this theoretical result.
General perturbations of a spherical gas bubble in a compressible and inviscid fluid with surface tension were proved in Shapiro and Weinstein (2011), in the linearized approximation, to decay exponentially, $\sim e^{-Γt}, Γ>0$, as time advances. Formal asymptotic and numerical evidence led to the conjecture that $Γ\approx \frac{A}ε \frac{We}{ε^{2}} \exp(-B \frac{We}{ε^2})$, where $0<ε\ll1$ is the Mach number, We is the Weber number, and $A$ and $B$ are positive constants. In this paper, we prove this conjecture and calculate $A$ and $B$ to leading order in $ε$.
We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in $W^{2,p}$. The result is improved for $p=2$ avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications…
In this paper we fix $1\le p<\infty$ and consider $(\Om,d,μ)$ be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure $μ$ supporting a $p$-Poincaré inequality such that $\Om$ is a uniform domain in its completion $\bar\Om$. We realize the trace of functions in the Dirichlet-Sobolev space $D^{1,p}(\Om)$ on the boundary $\partial\Om$ as functions in the homogeneous Besov space $HB^α_{p,p}(\partial\Om)$ for suitable $α$; here, $\partial\Om$ is equipped with a non-atomic Borel regular measure $ν$. We show that if $ν$ satisfies a $θ$-codimensional condit…
