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A. G. Ramm

Publications and source records attributed to A. G. Ramm.

At least 19 recordsLinked to original sources
arXiv2020

Stability of solutions to some abstract evolution equations with delay

The global existence and stability of the solution to the delay differential equation (*)$\dot{u} = A(t)u + G(t,u(t-τ)) + f(t)$, $t\ge 0$, $u(t) = v(t)$, $-τ\le t\le 0$, are studied. Here $A(t):\mathcal{H}\to \mathcal{H}$ is a closed, densely defined, linear operator in a Hilbert space $\mathcal{H}$ and $G(t,u)$ is a nonlinear operator in $\mathcal{H}$ continuous with respect to $u$ and $t$. We assume that the spectrum of $A(t)$ lies in the half-plane $\Re λ\le γ(t)$, where $γ(t)$ is not necessarily negative and $\|G(t,u)\| \le α(t)\|u\|^p$, $p>1$, $t\ge 0$. Sufficient conditions for the solution to the equation to exist globally, to be bounded and to converge to zero as $t$ tends to $\infty$, under the non-classical assumption that $γ(t)$ can take positive values, are proposed and justified.

math.FA↗
arXiv2018

Proof of the Schiffer's conjecture

The following conjecture has been known for many decades as Schiffer's symmetry problem (or Schiffer's conjecture): Assume that $Δu+k^2u=0$ in $D$, $u|_S=0$, $u_N|_S=1$, where $D\subset \mathbb{R}^3$ is a bounded, connected, $C^2-$smooth domain, $S$ is its boundary, $N$ is a unit normal to $S$ pointing out of $D$, $k^2>0$ is a constant. Then $S$ is a sphere. In this paper the above conjecture is proved. It is also proved that the relation $\int_Se^{ikβ\cdot s}ds=0, \,\, \forall β\in S^2$ implies that $S$ is a sphere.

math.AP↗
arXiv2018

Inverse problem of potential theory

P. Novikov in 1938 has proved that if $u_1(x)=u_2(x)$ for $|x|>R$, where $R>0$ is a large number, $$u_j(x):=\int_{D_j}g_0(x,y)dy, \quad g_0(x,y):=\frac 1 {4π|x-y|},$$ and $D_j\subset \mathbb{R}^3$, $j=1,2,$ $D_j\subset B_R$, are bounded, connected, smooth domains, star-shaped with respect to a common point, then $D_1=D_2$. Here $B_R:= \{x: |x|\le R\}$. Our basic results are: a) the removal of the assumption about star-shapeness of $D_j$, b) a new approach to the problem, c) the construction of counter-examples for a similar problem in which $g_0$ is replaced by $g=\frac {e^{ik|x-y|}}{4π|x-y|}$, where $k>0$ is a fixed constant.

math.AP↗
arXiv2017

Inverse obstacle scattering with non-over-determined data

It is proved that the scattering amplitude $A(β, α_0, k_0)$, known for all $β\in S^2$, where $S^2$ is the unit sphere in $\mathbb{R}^3$, and fixed $α_0\in S^2$ and $k_0>0$, determines uniquely the surface $S$ of the obstacle $D$ and the boundary condition on $S$. The boundary condition on $S$ is assumed to be the Dirichlet, or Neumann, or the impedance one. The uniqueness theorem for the solution of multidimensional inverse scattering problems with non-over-determined data was not known for many decades. A detailed proof of such a theorem is given in this paper for inverse scattering by obstacles for the first time. It follows from our results that the scattering solution vanishing on the boundary $S$ of the obstacle cannot have closed surfaces of zeros in the exterior of the obstacle different from $S$. To have a uniqueness theorem for inverse scattering problems with non-over-determined data is of principal interest because these are the minimal scattering data that allow one to uniquely recover the scatterer.

math.NA↗
arXiv2017

Inverse obstacle scattering with non-over-determined data

It is proved that the scattering amplitude $A(β, α_0, k_0)$, known for all $β\in S^2$, where $S^2$ is the unit sphere in $\mathbb{R}^3$, and fixed $α_0\in S^2$ and $k_0>0$, determines uniquely the surface $S$ of the obstacle $D$ and the boundary condition on $S$. The boundary condition on $S$ is assumed to be the Dirichlet, or Neumann, or the impedance one. The uniqueness theorem for the solution of multidimensional inverse scattering problems with non-over-determined data was not known for many decades. Such a theorem is proved in this paper for inverse scattering by obstacles for the first time.

math-ph↗
arXiv2017

Global existence, uniqueness and estimates of the solution to the Navier-Stokes equations

The Navier-Stokes (NS) problem consists of finding a vector-function $v$ from the Navier-Stokes equations. The solution $v$ to NS problem is defined in this paper as the solution to an integral equation. The kernel $G$ of this equation solves a linear problem which is obtained from the NS problem by dropping the nonlinear term $(v \cdot \nabla)v$. The kernel $G$ is found in closed form. Uniqueness of the solution to the integral equation is proved in a class of solutions $v$ with finite norm $N_1(v)=\sup_{ξ\in \mathbb{R}^3, t\in [0, T]}(1+|ξ|)(|v|+|\nabla v|)\le c (*)$, where $T>0$ and $C>0$ are arbitrary large fixed constants. In the same class of solutions existence of the solution is proved under some assumption. Estimate of the energy of the solution is given.

math.AP↗
arXiv2017

Completeness of the set $\{e^{ikβ\cdot s}\}|_{\forall β\in S^2}$

It is proved that the set $\{e^{ikβ\cdot s}\}|_{\forall β\in S^2}$, where $S^2$ is the unit sphere in $\mathbb{R}^3$, $k>0$ is a fixed constant, $k^2$ is not a Dirichlet eigenvalue of the Laplacian in $D$, $s\in S$, is total in $L^2(S)$. Here $S$ is a smooth, closed, connected surface in $\mathbb{R}^3$.

math.AP↗
arXiv2017

Representation of big data by dimension reduction

Suppose the data consist of a set $S$ of points $x_j, 1 \leq j \leq J$, distributed in a bounded domain $D \subset R^N$, where $N$ and $J$ are large numbers. In this paper an algorithm is proposed for checking whether there exists a manifold $\mathbb{M}$ of low dimension near which many of the points of $S$ lie and finding such $\mathbb{M}$ if it exists. There are many dimension reduction algorithms, both linear and non-linear. Our algorithm is simple to implement and has some advantages compared with the known algorithms. If there is a manifold of low dimension near which most of the data points lie, the proposed algorithm will find it. Some numerical results are presented illustrating the algorithm and analyzing its performance compared to the classical PCA (principal component analysis) and Isomap.

cs.IT↗
arXiv2016

Inverse problems for parabolic equations 3

Let $u_t-a(t)u_{xx}=f(x, t)$ in $0\leq x \leq π,\,\,t\geq 0.$ Assume that $u(0,t)=u_1(t)$, $u(π,t)=u_2(t)$, $u(x,0)=h(x)$, and the extra data $u_x(0,t)=g(t)$ are known. The inverse problem is: {\it How does one determine the unknown $a(t)$?} The function $a(t)>a_0>0$ is assumed continuous and bounded. This question is answered and a method for recovery of $a(t)$ is proposed. There are several papers in which sufficient conditions are given for the uniqueness and existence of $a(t)$, but apparently there was no method proposed for calculating of $a$. The method given in this paper for proving the uniqueness and existence of the solution to inverse problem is new and it allows one to calculate the unknown coefficient $a(t)$.

math.AP↗
arXiv2016

On the denseness of the set of scattering amplitudes

It is proved that the set of scattering amplitudes $\{A(β, α, k)\}_{\forall α\in S^2}$, known for all $β\in S^2$, where $S^2$ is the unit sphere in $\mathbb{R}^3$, $k>0$ is fixed, $k^2$ is not a Dirichlet eigenvalue of the Laplacian in $D$, is dense in $L^2(S^2)$. Here $A(β, α, k)$ is the scattering amplitude corresponding to an obstacle $D$, where $D\subset \mathbb{R}^3$ is a bounded domain with a boundary $S$. The boundary condition on $S$ is the Dirichlet condition.

math-ph↗
arXiv2016

Perturbation of zero surfaces

It is proved that if a smooth function $u(x)$, $x\in \mathbb{R}^3$, such that $\inf_{s\in S}|u_N(s)|>0$, where $u_N$ is the normal derivative of $u$ on $S$, has a closed smooth surface $S$ of zeros, then the function $u(x)+εv(x)$ has also a closed smooth surface $S_ε$ of zeros. Here $v$ is a smooth function and $ε>0$ is a sufficiently small number.

math.CA↗
arXiv2016

Solution to the Pompeiu problem and the related symmetry problem

Assume that $D\subset \mathbb{R}^3$ is a bounded domain with $C^1-$smooth boundary. Our result is: {\bf Theorem 1.} {\em If $D$ has $P-$property, then $D$ is a ball.} Four equivalent formulations of the Pompeiu problem are discussed. A domain $D$ has $P-$property if there exists an $f\neq 0$, $f\in L^1_{loc}(\mathbb{R}^3)$ such that $\int_{D}f(gx+y)dx=0$ for all $y\in \mathbb{R}^3$ and all $g\in SO(2)$, where $ SO(2)$ is the rotation group. The result obtained concerning the related symmetry problem is: {\bf Theorem 2.} {\em If $(\nabla^2 +k^2)u=0$ in $D$, $u|_S=1$, $u_N|_S=0$, and $k>0$ is a constant, then $D$ is a ball.}

math.AP↗
arXiv2016

Scattering of EM waves by many small perfectly conducting or impedance bodies

A theory of electromagnetic (EM) wave scattering by many small particles of an arbitrary shape is developed. The particles are perfectly conducting or impedance. For a small impedance particle of an arbitrary shape an explicit analytical formula is derived for the scattering amplitude. The formula holds as $a\to 0$, where $a$ is a characteristic size of the small particle and the wavelength is arbitrary but fixed. The scattering amplitude for a small impedance particle is shown to be proportional to $a^{2-κ}$, where $κ\in [0,1)$ is a parameter which can be chosen by an experimenter as he/she wants. The boundary impedance of a small particle is assumed to be of the form $ζ=ha^{-κ}$, where $h=$const, Re$h\ge 0$. The scattering amplitude for a small perfectly conducting particle is proportional to $a^3$, it is much smaller than that for the small impedance particle. The many-body scattering problem is solved under the physical assumptions $a\ll d\ll λ$, where $d$ is the minimal distance between neighboring particles and $λ$ is the wavelength. The distribution law for the small impedance particles is $\mathcal{N}(δ)\sim\int_δN(x)dx$ as $a\to 0$. Here $N(x)\ge 0$ is an arbitrary continuous function that can be chosen by the experimenter and $\mathcal{N}(δ)$ is the number of particles in an arbitrary sub-domain $Δ$. It is proved that the EM field in the medium where many small particles, impedance or perfectly conducting, are distributed, has a limit, as $a\to 0$ and a differential equation is derived for the limiting field. On this basis the recipe is given for creating materials with a desired refraction coefficient by embedding many small impedance particles into a given material.

math-ph↗
arXiv2016

Heat transfer in a complex medium

The heat equation is considered in the complex medium consisting of many small bodies (particles) embedded in a given material. On the surfaces of the small bodies an impedance boundary condition is imposed. An equation for the limiting field is derived when the characteristic size $a$ of the small bodies tends to zero, their total number $\mathcal{N}(a)$ tends to infinity at a suitable rate, and the distance $d = d(a)$ between neighboring small bodies tends to zero: $a << d$, $\lim_{a\to 0}\frac{a}{d(a)}=0$. No periodicity is assumed about the distribution of the small bodies. These results are basic for a method of creating a medium in which heat signals are transmitted along a given line. The technical part for this method is based on an inverse problem of finding potential with prescribed eigenvalues.

math-ph↗
arXiv2016

A simple proof of the closed graph theorem

Assume that $A$ is a closed linear operator defined on all of a Hilbert space $H$. Then $A$ is bounded. A new short proof of this classical theorem is given on the basis of the uniform boundedness principle. The proof can be easily extended to Banach spaces.

math.FA↗
arXiv2016

Inverse scattering on the half-line revisited

The inverse scattering problem on the half-line has been studied in the literature in detail. V. Marchenko presented the solution to this problem. In this paper, the invertibility of the steps of the inversion procedure is discussed and a new set of necessary and sufficient conditions on the scattering data is given for the scattering data to be generated by a potential suitable potential. Our proof is new and in contrast with known proof it does not use equations on the negative half-line.

math-ph↗