Alexander G. Ramm

Kansas State University | KSU · Department of Mathematics

Let \(\rho =\rho (x,t)\) be the density of the fluid and v(x, t) be its velocity. If \(D\subset {\mathbb R}^3\) is a bounded domain with a smooth boundary S, then the fluid mass in D is the integral \(\int _D\rho \,dx\) and the amount of fluid flowing through the boundary is \(\int _S\rho v\cdot N\,dS\), where N is the outer unit normal to S, and \...

The NSP is formulated in Eq. (1.1).

The NSP consists of solving the following equations. $$\begin{aligned}&v'+(v,\nabla )v =-\nabla p+\nu \Delta v+f,{} & {} \quad \text {in }{\mathbb R}^3\times {\mathbb R}_+,\\&\nabla \cdot v =0,{} & {} \quad \text {in }{\mathbb R}^3\times {\mathbb R}_+,\\&v(x,0) =v_0(x),{} & {} \quad \text {in }{\mathbb R}^3,\\ \end{aligned}$$where \({\mathbb R}_+=[...

Let the assumption (1.15) p. 4 hold. In this chapter we prove that the NSP (3.1)–(3.3) implies the following.

There is at most one solution in \(W^1_2(\mathbb R^3)\times C(\mathbb R_+)\) of the NSP (3.1)–(3.3). To prove uniqueness of the solution to the NSP assume that \(\tilde{v}_1\) and \(\tilde{v}_2\) solve Eq. (3.56). Let \(w=\tilde{v}_1-\tilde{v}_2\). We have (with \(\tilde{G}=H\) and \(*\) the convolution in \(\mathbb {R}^3\)) \( w=-\int _{0}^{ t}ds\...

One of the a priori estimates was formulated and proved in Lemma 3.4, namely, estimate (3.59): \( \sup _{t\ge 0}\Vert v(x,t)\Vert <c,\) where v(x, t) is a solution to the NSP (3.1)–(3.3). It was proved under the assumption \( \Vert v_0(x,t)\Vert +\int _{0}^{\infty }\Vert f(x,s)\Vert \,ds<c. \) The other basic a priori estimate is formulated in Theo...

A theory of many-body wave scattering is developed under the assumption a << d << λ, where a is the characteristic size of the small body, d is the distance between neighboring bodies and λ is the wave-length in the medium in which the bodies are embedded. The multiple scattering is essential under these assumptions. The author’s theory is used for...

Formulas are derived for solutions of many-body wave scattering problem by small impedance particles embedded in a homogeneous medium. The limiting case is considered, when the size a of small particles tends to zero while their number tends to infinity at a suitable rate. The basic physical assumption is a << d << λ, where d is the minimal distanc...

Abstract:The problem discussed is the Navier-Stokes problem (NSP) inR3. Uniqueness of its solution isproved in a suitable spaceX. No smallness assumptions are used in the proof. Existence of the solution inXis proved fort∈[0,T], whereT>0 is sufficiently small. Existence of the solution inXis proved fort∈[0,∞)if some a priori estimate of the solutio...

The author's proof of the uniqueness of the solution to inverse scattering problem is given. This proof is based on the concept of Property C for PDE. This concept was introduced by the author and applied to many inverse problems, see the author's monograph "Inverse Problems", Springer, New York, 2005.

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.

The equation $v=v_0+\int_0^t(t-s)^{\lambda -1}v(s)ds$ is considered, $\lambda\neq 0,-1,-2...$ 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 $t^{-\frac 5 4}$.

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

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 f...

A new a priori estimate for solutions to Navier–Stokes equations is derived. Uniqueness and existence of these solutions in R³ for all t>0 is proved in a class of solutions locally differentiable in time with values in H¹(R³), where H¹(R³) is the Sobolev space. By the solution a solution to an integral equation is understood. No smallness restricti...

Consider the equation u’ (t) - ? u + | u |p u = 0, u(0) = u0(x), (1), where u’ := du/dt , p = const > 0, x E R3, t > 0. Assume that u0 is a smooth and decaying function, ||u0|| = sup |u(x, t)|. x E R3 ,t E R+ It is proved that problem (1) has a unique global solution and this solution satisfies the following estimate ||u(x, t)|| < c, where c > 0 do...

Consider the equation u’(t) = A (t, u (t)), u(0)= U0 ; u' := du/dt (1). Under some assumptions on the nonlinear operator A(t,u) it is proved that problem (1) has a unique global solution and this solution satisfies the following estimate ||u (t)|| < µ (t) -1 for every t belongs to R+ = [0,infinity). Here µ(t) > 0, µ belongs to C1 (R+), is a suitabl...

Formulas are derived for solutions of many‐body wave scattering problems by small impedance particles embedded in an inhomogeneous medium. The limiting case is considered when the size a of small particles tends to zero while their number tends to infinity at a suitable rate. Equations for the limiting effective (self‐consistent) field in the mediu...

t is proved that the scattering amplitude \(A(\beta, \alpha_0, k_0)\), known for all \(\beta\in S^2\), where \(S^2\) is the unit sphere in \(\mathbb{R}^3\), and fixed \(\alpha_0\in S^2\) 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 th...

P. Novikov in 1938 has proved that if u1(x)=u2(x) for |x|>R, where R>0 is a large number, uj(x)≔∫Djg0(x,y)dy,g0(x,y)≔14π|x−y|, and Dj⊂R3, j=1,2,Dj⊂BR, are bounded, connected, smooth domains, star-shaped with respect to a common point, then D1=D2. Here BR≔{x:|x|≤R}. Our basic results are: (a) the removal of the assumption about star-shapeness of Dj,...

The problem of practical preparing small impedance particles with a prescribed boundary impedance is formulated and its importance in physics and technology is discussed. It is shown that if this problem is solved then one can easily prepare materials with a desired refraction coefficient and materials with a desired radiation pattern.

The following conjecture has been known for many decades as Schiffer's symmetry problem (or Schiffer's conjecture): Assume that $\Delta u+k^2u=0$ 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...

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 $(v \cdot \nabla)v$....

It is proved that the set $\{e^{ik\beta \cdot s}\}|_{\forall \beta \in S^2}$, 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 $\mathbb{R}^3$.

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 $(v \cdot \nabla)v$....

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 e...

The book is important as it contains results many of which are not available in the literature, except in the author’s papers. Among other things, it gives uniqueness theorems for inverse scattering problems when the data are non-over-determined, numerical method for solving inverse scattering problems, a method (MRC) for solving direct scattering...

The goal of this chapter is to formulate some of the basic results on the theory of integral equations and mention some of its applications. The literature of this subject is very large. Proofs are not given due to space restriction. The results are taken from the works mentioned in the references.

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 $u(x)+\epsilon v(x)$ has also a closed smooth surface $S_\epsilon$ of zeros. Here $v$ is a smooth function and $\epsilon>0$ is a suff...

It is proved that the set of scattering amplitudes $\{A(\beta, \alpha, k)\}_{\forall \alpha \in S^2}$, known for all $\beta\in S^2$, 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 $A(\beta, \alpha, k)$ is the scattering amplitude correspondi...

A new numerical method is given for solving 3D inverse scattering problem with non-over-determined scattering data. The data are the values of the scattering amplitude $A(\beta, \alpha_0, k)$ for all $\beta\in S^2_{\beta}$, where $S^2_{\beta}$ is an open subset of the unit sphere $S^2$ in $\mathbb{R}^3$, $\alpha_0\in S^2$ is fixed, and all $k\in (a...

Let $u_t-a(t)u_{xx}=f(x, t)$ in $0\leq x \leq \pi,\,\,t\geq 0.$ Assume that $u(0,t)=u_1(t)$, $u(\pi,t)=u_2(t)$, $u(x,0)=h(x)$, 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...

It is proved that nonlinear integral equations of certain class have global solution and estimates of the solution are given as $t\to \infty$.

It is proved that nonlinear integral equations of certain class have global solution and estimates of the solution are given as $t\to \infty$.

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...

It is proved that the scattering amplitude , known for all , where is the unit sphere in , is fixed, is fixed, determines the surface of the obstacle and the boundary condition on uniquely. The boundary condition on is either the Dirichlet, or Neumann, or the impedance one. The uniqueness theorems for the solution of inverse scattering problems wit...

Scalar wave scattering by many small particles of arbitrary shapes with impedance boundary condition is studied. The problem is solved asymptotically and numerically under the assumptions a << d << lambda, where k = 2pi/lambda is the wave number, lambda is the wave length, a is the characteristic size of the particles, and d is the smallest distanc...

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...

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...

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.

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 = \Delta...

Silicon gravure patterns are engineered to have cells that are wettable and lands that are not wettable by aqueous inks. This strategy allows excess ink on the lands to be removed without using a doctor blade. Using an aqueous silica ink, continuous lines as narrow as 1.2 μm with 1.5 μm space are gravure printed.

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 ge...

It is proved that one can distribute many small particles in a given material so that in the resulting material heat propagates essentially along a given line.

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 → 0, where a is a characteristic size o...

A proof is given of the global existence and uniqueness of a weak solution to Navier-Stokes equations in unbounded exterior domains.

A proof is given of the global existence and uniqueness of a weak solution to Navier-Stokes boundary problem. The proof is short and essentially self-contained.

A simple proof is given for the explicit formula which allows one to recover a $C^2-$smooth vector field $A=A(x)$ in $\mathbb{R}^3$, decaying at infinity, from the knowledge of its $\nabla \times A$ and $\nabla \cdot A$. The representation of $A$ as a sum of the gradient field and a divergence-free vector fields is derived from this formula. Simila...

The goal of this Section is to formulate some of the basic results on the theory of integral equations and mention some of its applications. The literature of this subject is very large. Proofs are not given due to the space restriction. The results are taken from the works mentioned in the references.

There are several methods for proving the existence of the solution to the elliptic boundary problem $Lu=f \text{\,\, in\,\,} D,\quad u|_S=0,\quad (*)$. Here $L$ is an elliptic operator of second order, $f$ is a given function, and uniqueness of the solution to problem (*) is assumed. The known methods for proving the existence of the solution to (...

Assume that $D\subset \R^2$ is a strictly convex domain with $C^2-$smooth boundary. {\bf Theorem.} {\em If $\int_De^{ix}y^ndxdy=0$ for all sufficiently large $n$, then $D$ is a disc.}

A new proof is given of the existence of the solution to electromagnetic (EM) wave scattering problem for an impedance body of an arbitrary shape. The proof is based on the elliptic systems theory and elliptic estimates for the solutions of such systems.

Consider the Schrodinger operator -del(2) + q with a smooth compactly supported potential q, q = q(x), x epsilon R-3. Let A(beta, alpha, k) be the corresponding scattering amplitude, k(2) be the energy, alpha epsilon S-2 be the incident direction, beta epsilon S-2 be the direction of scattered wave, S-2 be the unit sphere in R-3. Assume that k = k(...

Scattering of electromagnetic (EM) waves by many small impedance particles (bodies), embedded in a homogeneous medium, is studied. Physical properties of the particles are described by their boundary impedances. The limiting equation is obtained for the effective EM field in the limiting medium, in the limit a→0, where a is the characteristic size...

The proposal deals with electromagnetic (EM) wave scattering by one and many small impedance particles of an arbitrary shape. Analytic formula is derived for EM wave scattering by one small impedance particle of an arbitrary shape and an integral equation for the effective field in the medium where many such particles are embedded. These results ar...

Assume that A is a bounded selfadjoint operator in a Hilbert space H. Then, the variational principle is obtained for some functional. As an application of this principle, a variational principle for the electrical capacitance of a conductor of an arbitrary shape is derived.

A formula for the electromagnetic (EM) field in the medium, in which many small perfectly conducting particles of an arbitrary shape are distributed, is derived.

Scattering of electromagnetic (EM) waves by small (ka 1) impedance particle D of an arbitrary shape, embedded in a homogeneous medium, is studied. Analytic, closed form, formula for the scattered field is derived. The scattered field is of the order O(a2-κ), where κ [0,1) is a number. This field is much larger than in the case of Rayleigh-type scat...

Consider an integral I(s): =∫0be−s(x2−icx)x²dx, where c > 0 and b > 0 are arbitrary positive constants. It is proved that as s →+∞. Possible applications of this result to the Pompeiu problem are outlined.

Wave scattering by many (M = M(a)) small bodies, at the boundary of which transmission boundary conditions are imposed, is studied. Smallness of the bodies means that ka ≪ 1, where a is the characteristic dimension of the body and k = 2π/λ is the wave number in the medium in which small bodies are embedded. Explicit asymptotic formula is derived fo...

Let $f \in L_{loc}^1 (\R^n)\cap \mathcal{S}$, where $\mathcal{S}$ is the Schwartz class of distributions, and $$\int_{\sigma (D)} f(x) dx = 0 \quad \forall \sigma \in G, \qquad (*)$$ where $D\subset \R^n$ is a bounded domain, the closure $\bar{D}$ of which is diffeomorphic to a closed ball, and $S$ is its boundary. Then the comp$ is connected and p...

Consider an abstract evolution problem in a Hilbert space H where A(t) is a linear, closed, densely defined operator in H with domain independent of t ≥ 0 and G(t,u) is a nonlinear operator such that ‖G(t,u)‖a(t) ‖u‖p, p = const > 1, ‖f(t)‖ ≤ b(t). We allow the spectrum of A(t) to be in the right half-plane Re(λ) < λ0(t), λ0(t) > 0, but assume tha...

Let $f \in L_{loc}^1 (\R^n)\cap \mathcal{S}'$,$\mathcal{S}'$ is the Schwartz class of distributions, and$$\int_{\sigma (D)} f(x) dx = 0 \quad \forall \sigma \in G, \qquad (*)$$where $D\subset \R^n$, $n\ge 2$, is a bounded domain, the closure $\bar{D}$ ofwhich is $C^1-$diffeomorphic to a closed ball. Then the complement of $\bar{D}$is connected and...

Consider an integral $I(s):=\int_0^T e^{-s(x^2-icx)}dx$, where $c>0$ and $T>0$ are arbitrary positive constants. It is proved that $I(s)\sim \frac{i}{sc}$ as $s\to +\infty$. The asymptotic behavior of the integral $J(s):=\int_0^Te^{s(x^2+icx)}dx$ is also derived. One has $J(s)\sim \frac{e^{sT^2+iscT}}{s(2T+ic)}$ as $s\to +\infty$.

Consider the Schr\"odinger operator $-\nabla^2+q$ $ $q$, $q=q(x), x \in \mathbf{R}^3$. Let $A(\beta,\alpha, k)$ be the corresponding scattering amplitude, $k^2$ be the energy, $\alpha \in S^2$ be the incident direction, $\beta \in S^2$ be the direction of scattered wave, $S^2$ be the unit sphere in $\mathbf{R}^3$. Assume that $k=k_0 >0$ is fixed, a...

The weak solution to the Navier–Stokes equations in a bounded domain D⊂R3D⊂R3 with a smooth boundary is proved to be unique provided that it satisfies an additional requirement. This solution exists for all t≥0t≥0. In a bounded domain DD the solution decays exponentially fast as t→∞t→∞ if the force term decays at a suitable rate.

Stability of stationary solutions of parabolic equations is conventionally studied by linear stability analysis, Lyapunov functions or lower and upper functions. We discuss here another approach based on differential inequalities written for the L 2 norm of the solution. This method is appropriate for the equations with time dependent coefficients....

An equation u̇=A(t)u+B(t)F(t,u(t−τ)), u(t)=v(t),−τ≤t≤0u(t)=v(t),−τ≤t≤0, is considered, where A(t)A(t) and B(t)B(t) are linear operators in a Hilbert space HH, u̇=dudt, F:H→HF:H→H is a non-linear operator, and τ>0τ>0 is a constant. Under some assumptions on A(t),B(t)A(t),B(t) and F(t,u)F(t,u) sufficient conditions are given for the solution u(t)u(t)...

If F:H→HF:H→H is a map in a Hilbert space HH, F∈Cloc2, and there exists a solution yy, possibly non-unique, such that F(y)=0F(y)=0, F′(y)≠0F′(y)≠0, then equation F(u)=0F(u)=0 can be solved by a DSM (Dynamical Systems Method) and the rate of convergence of the DSM is given provided that a source-type assumption holds. A discrete version of the DSM y...

Let $f \in L_{loc}^1 (\R^n)\cap \mathcal{S}'$, where $\mathcal{S}'$ is the Schwartz class of distributions, and $$\int_{\sigma (D)} f(x) dx = 0 \quad \forall \sigma \in G, \qquad (*)$$ where $D\subset \R^n$ is a bounded domain, the closure $\bar{D}$ of which is diffeomorphic to a closed ball. Then the complement of $\bar{D}$ is connected and path c...

A wide class of the operator equations F(u)=h in a Hilbert space is studied. Convergence of a Dynamical Systems Method (DSM), based on the continuous analog of the Newton method, is proved without any smoothness assumptions on the F′(u). It is assumed that F′(u) depends on u continuously. Existence and uniqueness of the solution to evolution equati...

Electromagnetic wave scattering by many parallel to $z-$axis, thin, impedance, circular infinite cylinders is studied asymptotically as $a\to 0$. Let $D_m$ be the crossection of the $m-$th cylinder, $a$ be its radius, and $\hat{x}_m=(x_{m1},x_{m2})$ be its center, $1\le m \le M$, $M=M(a)$. It is assumed that the points $\hat{x}_m$ are distributed s...

Formulas are derived for solutions of many-body wave scattering problems by small particles in the case of acoustically soft, hard, and impedance particles embedded in an inhomogeneous medium. The case of transmission (interface) boundary conditions is also studied in detail. The limiting case is considered, when the size $a$ of small particles ten...