Analytic Coefficients Research Articles

AN EXPRESSION FOR THE SOLUTION OF A DIFFERENTIAL EQUATION IN TERMS OF ITERATES OF DIFFERENTIAL OPERATORS

We obtain theorems on the expression of A−1 in terms of iterates of the operator A, which is the reproducing operator of a 1-parameter group of linear transformations of a Banach space, and whose spectrum does not surround 0. These results are applied to first order differential equations with analytic coefficients and right-hand sides (symmetric first order systems on a compact manifold without boundary), and to second order elliptic equations (equations with a real principal part on a manifold without boundary, selfadjoint equations degenerate on the boundary of the domain, and the Dirichlet problem for a selfadjoint equation in a domain with an analytic boundary). We obtain formulas expressing the value of the solution at a point in terms of the derivatives of the coefficients and the right-hand side at this point. Bibliography: 9 titles.

Integral Operators for Fourth Order Linear Parabolic Equations

An integral operator is constructed that maps analytic functions of two complex variables onto analytic solutions of fourth order linear parabolic equations of two space variables with analytic coefficients. The operator reduces to Bergman’s operator for the fourth order elliptic equation when the solution is independent of the time variable. In the case of radial coefficients, the kernel functions of the operator are independent of the dimension and by a method of ascent, analytic solutions of three or more dimensions may be represented by a simple modification of the operator.

On Continuation of Regular Solutions of Linear Partial Differential Equations

Here we briefly introduce the speaker's recent works on continuation of regular solutions of linear partial differential equations with real analytic coefficients. The method of argument is deeply concerned with the non-characteristic boundary value problem for hyperfunction solutions. First we intuitively compare this new method with the old one which owes much to Grusin [2] and was employed in the case of constant coefficients. Then we give results on hyperfunction boundary value problem as our main tool. Finally we give the main results and prospects on continuation of real analytic solutions.

Extension of Solutions of Systems of Linear Differential Equations

The purpose of this paper is to prove some theorems on the extensibility of hyperfunction solutions and real analytic solutions of systems of linear differential equations with real analytic coefficients. It is Ehrenpreis [2] that revealed the intimate relations between the algebraic character of overdetermin ed systems of linear differential equations and Hartogs' theorem on the removable singularity of holomorphic functions of several complex variables. In fact, the essential part of his results can be summarized as follows: if a (generalized) function satisfies an overdetermin ed system of linear differential equations with constant coefficients outside a compact set in Rn, then it must be extended uniquely all over Rn as a solution of the same system. Note that the notion of overdetermin ed system is purely algebraic and that the Cauchy-Riemann equation, which holomorphic functions of n complex

Analytic perturbation theory for screened Coulomb potentials: Nonrelativistic case

We have developed an analytic perturbation theory for screened Coulomb radial wave functions, based on an expansion of the potential of the form $V(r)=\ensuremath{-}(\frac{a}{r})[1+{V}_{1}\ensuremath{\lambda}r+{V}_{2}{(\ensuremath{\lambda}r)}^{2}+{V}_{3}{(\ensuremath{\lambda}r)}^{3}+\ensuremath{\cdots}]$, where $\ensuremath{\lambda}\ensuremath{\simeq}\ensuremath{\alpha}{Z}^{\frac{1}{3}}$ is a small parameter characterizing the screening. The coefficients ${V}_{k}$ may be chosen such that the above form converges rapidly and gives a good approximation to realistic numerical potentials, such as those of Herman and Skillman, in the interior of the atom. The screened radial wave functions are obtained as a series in $\ensuremath{\lambda}$ with simple analytic coefficients, owing to the special symmetries of the unperturbed Coulomb problem. Both bound and continuum shapes are correctly treated in the region $\ensuremath{\lambda}rl1$. For inner bound states, this includes all of the region where the wave function is large. Similarly, high-energy continuum wave functions will have completed several oscillations in this interval so that by $r\ensuremath{\simeq}{\ensuremath{\lambda}}^{\ensuremath{-}1}$ one has reached the asymptotic region. Consequently, expressions for bound-state normalizations can be given as series in $\ensuremath{\lambda}$, which are accurate, in general, for the $K$ shell and for other low-lying levels of high-$Z$ elements. The continuum normalizations which we obtain are valid for energies on the order of the $K$-shell binding energy above threshold. Bound-state energies and continuum phase shifts are also obtained in these circumstances.

A necessary condition for a power series to be a formal solution of a singular linear differential equation of order k

We obtain a necessary condition on the coefficients of a formal power series, which is a formal solution of a nontrivial singular linear differential equation of order k, with analytic coefficients and prove a “uniqueness” theorem for the differential equation.

Complete Families of Solutions for Parabolic Equations with Analytic Coefficients

A complete family of solutions is constructed for the general linear second order parabolic equation in one space variable with entire coefficients defined in a domain with moving boundary and for a class of second order parabolic equations in two space variables with entire coefficients defined in a cylindrical domain. The construction is based on the use of integral operators and results on the analytic continuation of solutions to partial differential equations with analytic coefficients. A numerical example is given which uses a complete family of solutions to approximate the solution to the first initial-boundary value problem for a parabolic equation in one space variable defined in a cylindrical domain.

CORRECTNESS OF THE CAUCHY PROBLEM IN GEVREY CLASSES FOR NONSTRICTLY HYPERBOLIC OPERATORS

The Cauchy problem for any nonstrictly hyperbolic operator with analytic coefficients is correct in some Gevrey class if the domain is a lens of spatial type.Bibliography: 21 items.

22.—Analytic, Generalized, Hyperanalytic Function Theory and an Application to Elasticity

SynopsisThough it is still an open problem for which class of first-order elliptic systems Carleman's theorem holds, this is proven here for a certain class of systems (with analytic coefficients) for which Douglis introduced the hypercomplex algebra and hyperanalytic functions. The proof is based on a representation formula generalising Vekua's approach with Volterra integral equations in C2 to more than two unknowns. The representation formula is of its own interest because it provides the generation of complete families of solutions. The equations of plane inhomogeneous elasticity problems lead to a system of the desired class.

On continuation of regular solutions of linear partial differential equations with real analytic coefficients

On continuation of regular solutions of linear partial differential equations with real analytic coefficients

On Continuation of Regular Solutions of Linear Partial Differential Equations with Real Analytic Coefficients

On Continuation of Regular Solutions of Linear Partial Differential Equations with Real Analytic Coefficients

The Noncharacteristic Cauchy Problem for Parabolic Equations in One Space Variable

An integral operator is constructed which maps ordered pairs of analytic functions onto analytic solutions of linear second order parabolic equations in one space variable with analytic coefficients. This operator is then used to construct a solution to the noncharacteristic Cauchy problem for parabolic equations in one space variable. Applications are made to the inverse Stefan problem and the analytic continuation of solutions to parabolic equations.

Duality and isomorphism of the spaces of solutions of certain linear elliptic equations and systems with analytic coefficients

Duality and isomorphism of the spaces of solutions of certain linear elliptic equations and systems with analytic coefficients

On micro-analyticity of the elementary solutions of hyperbolic differential equations with real analytic coefficients

On micro-analyticity of the elementary solutions of hyperbolic differential equations with real analytic coefficients

Cauchy Problem and Analytic Continuation for Systems of First Order Elliptic Equations with Analytic Coefficients

Let a, b, c, d, f, g be analytic functions of two real variables x, y in the $z = x + iy$ plane. Consider the elliptic equation (M) $\partial u/\partial x - \partial v/\partial y = au + bv + f,\partial u/\partial y + \partial v/\partial x = cu + dv + g$. The following areas will be investigated: (1) the integral respresentations for solutions of (M) to the boundary $\partial G$ of a simply connected domain G; (2) reflection principles for (M) under nonlinear analytic boundary conditions; (3) the sufficient conditions for the nonexistence and analytic continuation for the solutions of the Cauchy problem for (M).

Non-uniqueness and uniqueness in the Cauchy problem at simple characteristic points

The subject of this paper is uniqueness and non-uniqueness in the Cauchy problem at simple characteristic points of the initial hypersurface. We only consider linear partial differential equations with analytic coefficients. We use the notation from the book [1] by H6rmander with the exception that D~ = t3/~xj, 1 0 in a neighbourhood of x °, and if H is of a type covered by [6, Theorem 3] then there is a neighbourhood V of x ° and a function u e C ~ ( V ) such that P ( x , D ) u = O , x ° ~ suppu C {x; ~0(x) =>0, x ~ V}." When we want to construct characteristic hypersurfaces that are not hyperplanes we have to rely on integration of Hamilton's equations. So we restrict ourselves to operators with real principal part and to simple characteristic points on the initial hypersurface. The three points of departure of this paper is the method to prove non-uniqueness by Malgrange [3], the uniqueness theorem [2, Theorem 8.1] by H6rmander, see Theorem 3 below, and the technique of proving uniqueness taken from Persson [5]. The part played by Theorem 3 is a passive one leaving to us to study the case when a priori the bicharacteristic curve does not leave the support of the solution. In [9] Zachmanoglou proves the following theorem.

The Cauchy Problem for Douglis-Nirenberg Elliptic Systems of Partial Differential Equations

Several partial answers are given to the question: Suppose U is a solution of the Douglis-Nirenberg elliptic system $LU = F$ where F is analytic and L has analytic coefficients. If $U = 0$ in some appropriate sense on a hyperplane (or any analytic hypersurface) must U vanish identically? One answer follows from introducing a so-called formal Cauchy problem for Douglis-Nirenberg elliptic systems and establishing existence and uniqueness theorems. A second Cauchy problem, in some sense a more natural one, is discussed for an important subclass of the Douglis-Nirenberg elliptic systems. The results in this case give a second partial answer to the original question. The methods of proof employed are largely algebraic. The systems are reduced to systems to which the Cauchy-Kowalewski theorem applies.

Solutions of Partial Differential Equations with Support on Leaves of Associated Foliations

Suppose that the linear partial differential operator $P(x,D)$ has analytic coefficients and that it can be written in the form $P(x,D) = R(x,D)S(x,D)$ where $S(x,D)$ is a polynomial in the homogeneous first order operators ${A_1}(x,D), \cdots ,{A_r}(x,D)$. Then in a neighborhood of any point ${x^0}$ at which the principal part of $S(x,D)$ does not vanish identically, there is a solution of $P(x,D)u = 0$ with support the leaf through ${x^0}$ of the foliation induced by the Lie algebra generated by ${A_1}(x,D), \cdots ,{A_r}(x,D)$. This result yields necessary conditions for hypoellipticity and uniqueness in the Cauchy problem. An application to second order degenerate elliptic operators is also given.

Integral operators and the first initial boundary value problem for pseudoparabolic equations with analytic coefficients

Integral operators and the first initial boundary value problem for pseudoparabolic equations with analytic coefficients

ON THE ANALYTICITY OF SOLUTIONS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS

We study partial differential equations and systems all of whose solutions are analytic, and obtain a priori estimates on the analytic continuations of these solutions to the complex plane. Using these estimates, we obtain conditions which are necessary for the analyticity of all solutions of linear differential equations and systems with analytic coefficients. Bibliography: 20 items.