arXiv subjects
Publications and source records attributed to A. G. Ramm.
The aim is to formulate a discrete analog of L'Hospital's rule and describe some of its applications. While the usual L'Hospital's rule is very well known, its discrete analog apparently was not in the literature. Since the L'Hospital's rule proved to be very useful in many applications, one may think that its discrete analog will also be useful. ======================== This paper is withdrawn: I have learned that the result is known.
Equation $(-Δ+k^2)u+f(u)=0$ in $D$, $u\mid_{\partial D}=0$, where $k=\const>0$ and $D\subset\R^3$ is a bounded domain, has a solution if $f:\R\to\R$ is a continuous function in the region $|u|\geq a$, piecewise-continuous in the region $|u|\leq a$, with finitely many discontinuity points $u_j$ such that $f(u_j\pm 0)$ exist, and $uf(y)\geq 0$ for $|u|\geq a$, where $a\geq 0$ is an arbitrary fixed number.
It is proved that the measurement of the acoustic pressure on the ear membrane allows one to determine the shape of the ear channel uniquely.
The following result is proved: {\bf Theorem.} Let $D\subset \R^3$ be a bounded domain homeomorphic to a ball, $|D|$ be its volume, $|S|$ be the surface area of its smooth boundary $S$, $D\subset B_R:=\{x:|x|\leq R\}$, and $H_R$ is the set of all harmonic in $B_R$ functions. If $$\frac 1 {|D|}\int_Dhdx=\frac 1 {|S|}\int_Shds\quad \forall h\in H_R,$$ then $D$ is a ball.
In the first part of the paper boundary-value problems are considered under weak assumptions on the smoothness of the domains. We assume nothing about smoothness of the boundary $\partial D$ of a bounded domain $D$ when the homogeneous Dirichlet boundary condition is imposed; we assume boundedness of the embedding $i_{1}:H^{1}(D)\to L^{2}(D)$ when the Neumann boundary condition is imposed; we assume boundedness of the embeddings $i_{1}$ and of $i_{2}:H^{1}(D)\to L^{2}(\partial D)$ when the Robin boundary condition is imposed, and, if, in addition, $i_{1}$ and $i_{2}$ are compact, then the boundary-value problems with the spectral parameter are of Fredholm type. Several examples of the classes of rough domains for which the embedding $i_2$ is compact are given. Applications to scattering by rough obstacles are mentioned.
Modified Rayleigh conjecture (MRC) in scattering theory was proposed and justified by the author (J.Phys A, 35 (2002), L357-L361). MRC allows one to develop efficient numerical algorithms for solving boundary-value problems. It gives an error estimate for solutions. In this paper the MRC is formulated and proved for static problems.
Let $F:X\to X$ be a $C^2_\loc$ map in a Banach space $X$, and $A$ be its Frèchet derivative at the element $w:=w_\ve$, which solves the problem $(\ast) \dotw=-A^{-1}_\ve(F(w)+\ve w)$, $w(0)=w_0$, where $A_\ve:=A+\ve I$. Assume that $\|A^{-1}_\ve\|\leq c \ve^{-k}$, $0 \ve_0$. Then $(\ast)$ has a unique global solution, $w(t)$, there exists $w(\infty)$, and $(\ast\ast) F(w(\infty))+\ve w(\infty)=0$. Thus the DSM (Dynamical Systems Method) is justified for equation $(\ast\ast)$. The limit of $w_\ve$ as $\ve\to 0$ is studied.
Let $A(k)u(k)=f(k) (1)$ be an operator equation, $X$ and $Y$ are Banach spaces, $k\inΔ\subset\C$ is a parameter, $A(k):X\to Y$ is a map, possibly nonlinear. Sufficient conditions are given for continuity of $u(k)$ with respect to $k$. Necessary and sufficient conditions are given for the continuity of $u(k)$ with respect to $k$ in the case of linear operators $A(k)$.
Consider the equation $-s^2Δu_s+q(x)u_s=f(u_s)$ in $\R^3$, $|u(\infty)|<\infty$, $s=const>0$. Under what assumptions on $q(x)$ and $f(u)$ can one prove that the solution $u_s$ exists and $\lim_{s\to 0} u_s=u(x)$, where $u(x)$ solves the limiting problem $q(x)u=f(u)$? These are the questions discussed in the paper.
Consider an operator equation (*) $B(u)-f=0$ in a real Hilbert space. Let us call this equation ill-posed if the operator $B'(u)$ is not boundedly invertible, and well-posed otherwise. The DSM (dynamical systems method) for solving equation (*) consists of a construction of a Cauchy problem, which has the following properties: 1) it has a global solution for an arbitrary initial data, 2) this solution tends to a limit as time tends to infinity, 3) the limit is the minimal-norm solution to the equation $B(u)=f$. A global convergence theorem is proved for DSM for equation (*) with monotone $C_{loc}^2$ operators $B$.
The Rayleigh conjecture on the representation of the scattered field in the exterior of an obstacle $D$ is widely used in applications. However this conjecture is false for some obstacles. AGR introduced the Modified Rayleigh Conjecture (MRC), and in this paper we present successful numerical algorithms based on the MRC for various 2D and 3D obstacle scattering problems. The 3D obstacles include a cube and an ellipsoid. The MRC method is easy to implement for both simple and complex geometries. It is shown to be a viable alternative for other obstacle scattering methods.
Cubature formulas, asymptotically optimal with respect to accuracy, are derived for calculating multidimensional weakly singular integrals. They are used for developing a universal code for calculating capacitances of conductors of arbitrary shapes.
Consider an operator equation $F(u)=0$ in a real Hilbert space. Let us call this equation ill-posed if the operator $F'(u)$ is not boundedly invertible, and well-posed otherwise. If $F$ is monotone $C^2_{loc}(H)$ operator, then we construct a Cauchy problem, which has the following properties: 1) it has a global solution for an arbitrary initial data, 2) this solution tends to a limit as time tends to infinity, 3) the limit is the minimum norm solution to the equation $F(u)=0$. Example of applications to linear ill-posed operator equation is given.
Let $F$ be a nonlinear map in a real Hilbert space $H$. Suppose that $\sup_{u\in B(u_0,R)}$ $\|[F'(u)]^{-1}\|\leq m(R)$, where $B(u_0,R)=\{u:\|u-u_0\|\leq R\}$, $R>0$ is arbitrary, $u_0\in H$ is an element. If $\sup_{R>0}\frac{R}{m(R)}=\infty$, then $F$ is surjective. If $\|[F'(u)]^{-1}\|\leq a\|u\|+b$, $a\geq 0$ and $b>0$ are constants independent of $u$, then $F$ is a homeomorphism of $H$ onto $H$. The last result is known as an Hadamard-type theorem, but we give a new simple proof of it based on the DSM (dynamical systems method).
A numerical algorithm is presented for solving the direct scattering problems by the Modified Rayleigh Conjecture Method (MRC) introduced by A.G.Ramm. Some numerical examples are given. They show that the method is numerically efficient.
An optimal algorithm is described for solving the deconvolution problem of the form ${\bf k}u:=\int_0^tk(t-s)u(s)ds=f(t)$ given the noisy data $f_δ$, $||f-f_δ||\leq δ.$ The idea of the method consists of the representation ${\bf k}=A(I+S)$, where $S$ is a compact operator, $I+S$ is injective, $I$ is the identity operator, $A$ is not boundedly invertible, and an optimal regularizer is constructed for $A$. The optimal regularizer is constructed using the results of the paper MR 40#5130.
The aim of this note is to prove a new discrepancy principle. The advantage of the new discrepancy principle compared with the known one consists of solving a minimization problem approximately, rather than exactly, and in the proof of a stability result.
Let F(u_\ve)+\ve(u_\ve-w)=0 \eqno{(1)} where $F$ is a nonlinear operator in a Hilbert space $H$, $w\in H$ is an element, and $\ve>0$ is a parameter. Assume that $F(y)=0$, and $F'(y)$ is not a boundedly invertible operator. Sufficient conditions are given for the existence of the solution to \eqref{e1.1} and for the convergence $\lim_{\ve\to 0}\|u_\ve-y\|=0$. An example of applications is considered. In this example $F$ is a nonlinear integral operator.