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Alexander G. Ramm

Publications and source records attributed to Alexander G. Ramm.

At least 19 recordsLinked to original sources
arXiv2024

On the Laplace transform

Sufficient conditions are given for a function $F(p)$ to be the Laplace transform of a function $f(t)$ or a distribution $f$. No assumption on $f$ is given a priori. It is not even assumed that $f=0$ for $t<0$.

math.CV↗
arXiv2023

Boundary values of analytic functions

Let $D$ be a connected bounded domain in $\R^2$, $S$ be its boundary which is closed, connected and smooth. Let $Φ(z)=\frac 1 {2πi}\int_S\frac{f(s)ds}{s-z}$, $f\in L^1(S)$, $z=x+iy$. Boundary values of $Φ(z)$ on $S$ are studied. The function $Φ(t)$, $t\in S$, is defined in a new way. Necessary and sufficient conditions are given for $f\in L^1(S)$ to be boundary value of an analytic in $D$ function. The Sokhotsky-Plemelj formulas are derived for $f\in L^1(S)$.

math.CV↗
arXiv2022

On the analytic functions

Let $Ω$ be a connected bounded domain on the complex plane, $S$ be its boundary, which is closed, star-shaped, $C^1$-smooth, and $H(Ω)$ is the set of analytic (holomorphic) in $Ω$ functions. The aim of this paper is to prove that an arbitrary $f\in L^1(S)$, satisfying the condition $\int_Sf(s)ds=0$, can be boundary value of an $f\in H(Ω)$.

math.CV↗
arXiv2021

When does a double-layer potential equal to a single-layer one?

Let $D$ be a bounded domain in $\mathbb{R}^3$ with a closed, smooth, connected boundary $S$, $N$ be the outer unit normal to $S$, $k>0$ be a constant, $u_{N^{\pm}}$ are the limiting values of the normal derivative of $u$ on $S$ from $D$, respectively $D':=\mathbb{R}^3\setminus D$; $g(x,y)=\frac{e^{ik|x-y|}}{4π|x-y|}$, $w:=w(x,μ):=\int_S g_{N}(x,s)μ(s)ds$ be the double-layer potential, $u:=u(x,σ):=\int_S g(x,s)σ(s)ds$ be the single-layer potential. In this paper it is proved that for every $w$ there is a unique $u$, such that $w=u$ in $D$ and vice versa. Necessary and sufficient conditions are given for the existence of $u$ and the relation $w=u$ in $D'$, given $w$ in $D'$, and for the existence of $w$ and the relation $w=u$ in $D'$, given $u$ in $D'$.

math.AP↗
arXiv2021

Theory of hyper-singular integrals and its application to the Navier-Stokes problem

In this paper the convolution integrals $\int_0^t(t-s)^{λ-1}b(s)ds$ with hyper-singular kernels are considered, where $λ\le 0$ and $b$ is a smooth or $b$ is in $L^1(\mathbb{R}_+)$. For such $λ$ these integrals diverge classically even for smooth $b$. These convolution integrals are defined in this paper for $λ\le 0$, $λ\neq 0,-1,-2,...$. Integral equations and inequalities are considered with the hyper-singular kernels $(t-s)^{λ-1}_+$ for $λ\le 0$, where $t^λ_+:=0$ for $t<0$. In particular, one is interested in the value $λ=-\frac 14$ because it is important for the Navier-Stokes problem (NSP). Integral equations of the type $b(t)=b_0(t)+ \int_0^t(t-s)^{λ-1}b(s)ds$, $λ\le 0$, are studied. The solution of these equations is investigated, existence and uniqueness of the solution is proved for $λ=-\frac 1 4$. This special value of $λ$ is of basic importance for a study of the Navier-Stokes problem (NSP). The above results are applied to the analysis of the NSP in the space $\mathbb{R}^3$ without boundaries. It is proved that the NSP is contradictory in the following sense: even if one assumes that the initial data $v_0(x):=v(x,0)\not\equiv 0$, $\nabla \cdot v_0(x)=0$ one proves that the solution $v(x,t)$ to the NSP has the property $v(x,0)=0$. This paradox shows that the NSP is not a correct description of the fluid mechanics problem and it proves that the NSP does not have a solution.

math.GM↗
arXiv2020

On a hyper-singular equation

The equation $v=v_0+\int_0^t(t-s)^{λ-1}v(s)ds$ is considered, $λ\neq 0,-1,-2...$ and $v_0$ is a smooth function rapidly decaying with all its derivatives. It is proved that the solution to this equation does exist, is unique and is smoother than the singular function $t^{-\frac 5 4}$.

math.AP↗
arXiv2019

Symmetry problems in harmonic analysis

Symmetry problems in harmonic analysis are formulated and solved. One of these problems is equivalent to the refined Schiffer's conjecture which was recently proved by the author. Let $k=const>0$ be fixed, $S^2$ be the unit sphere in $\mathbb{R}^3$, $D$ be a connected bounded domain with $C^2-$smooth boundary $S$, $j_0(r)$ be the spherical Bessel function. The harmonic analysis symmetry problems are stated in the following theorems: {\bf Theorem A.} {\em Assume that $\int_S e^{ikβ\cdot s}ds=0$ for all $β\in S^2$. Then $S$ is a sphere of radius $a$, where $j_0(ka)=0$. } {\bf Theorem B.} {\em Assume that $\int_D e^{ikβ\cdot x}dx=0$ for all $β\in S^2$. Then $D$ is a ball.

math.AP↗
arXiv2019

Estimating the size of the scatterer

Formula for the size of the scatterer is derived explicitly in terms of the scattering amplitude corresponding to this scatterer. By the scatterer either a bounded obstacle $D$ or the support of the compactly supported potential is meant

math.AP↗
arXiv2019

Global existence and uniqueness of the solution to a nonlinear parabolic equation

Consider the equation $$ u'(t)-Δu+|u|^ρu=0, \quad u(0)=u_0(x), (1), $$ where $ u':=\frac {du}{dt}$, $ ρ=const >0, $ $x\in \mathbb{R}^3$, $t>0$. Assume that $u_0$ is a smooth and decaying function, $$\|u_0\|\:=\sup_{x\in \mathbb{R}^3, t\in \mathbb{R}_+} |u(x,t)|.$$ It is proved that problem (1) has a unique global solution and this solution satisfies the following estimate $$\|u(x,t)\| 0$ does not depend on $x,t$.

math.AP↗
arXiv2017

Creating materials in which heat propagates along a line: theory and numerical results

In this paper the theory is developed for creating a material in which the heat is transmitted along a given line. This gives a possibility to transfer information using heat signals. This seems to be a novel idea. The technical part of the theory is the construction of the potential $q(x)$. This potential describes the heat equation $u_t = Δu - q(x)u$ in the limiting medium which is obtained after the small impedance particles are distributed in a given domain. A numerical method is also established to construct numerically such a potential.

math.NA↗
arXiv2015

Scattering of electromagnetic waves by small impedance particles of an arbitrary shape

An explicit formula is derived for the electromagnetic (EM) field scattered by one small impedance particle $D$ of an arbitrary shape. If $a$ is the characteristic size of the particle, $λ$ is the wavelength, $a<<λ$ and $ζ$ is the boundary impedance of $D$, $[N,[E,N]]=ζ[N,H]$ on $S$, where $S$ is the surface of the particle, $N$ is the unit outer normal to $S$, and $E$, $H$ is the EM field, then the scattered field is $E_{sc}=[\nabla g(x,x_1), Q]$. Here $g(x,y)=\frac{e^{ik|x-y|}}{4π|x-y|}$, $k$ is the wave number, $x_1\in D$ is an arbitrary point, and $Q=-\frac{ζ|S|}{iωμ}τ\nabla \times E_0$, where $E_0$ is the incident field, $|S|$ is the area of $S$, $ω$ is the frequency, $μ$ is the magnetic permeability of the space exterior to $D$, and $τ$ is a tensor which is calculated explicitly. The scattered field is $O(|ζ| a^2)>> O(a^3)$ as $a\to 0$ when $λ$ is fixed and $ζ$ does not depend on $a$. Thus, $|E_{sc}|$ is much larger than the classical value $O(a^3)$ for the field scattered by a small particle. It is proved that the effective field in the medium, in which many small particles are embedded, has a limit as $a\to 0$ and the number $M=M(a)$ of the particles tends to $\infty$ at a suitable rate. Thislimit solves a linear integral equation. The refraction coefficient of the limiting medium is calculated analytically. This yields a recipe for creating materials with a desired refraction coefficient.

physics.optics↗
arXiv2014

Symmetry problem

A novel approach to an old symmetry problem is developed. A new proof is given for the following symmetry problem, studied earlier.

math-ph↗