Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients

We give the asymptotics at infinity of a Green function for an elliptic equation with periodic coefficients on R^d. Basic ingredients in establishing the asymptotics are an integral representation of the Green function and the saddle point method. We also completely determine the Martin compactification of R^d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the Green function.     

On the Representation of Band-Dominant Functions on the Sphere Using Finitely Many Bits

A band-dominant function on the Euclidean sphere embedded in R ^q+1 is the restriction to this sphere of an entire function of q+1 complex variables having a finite exponential type in each of its variables. We develop a method to represent such a function using finitely many bits, using the values of the function at scattered sites on the sphere. The number of bits required in our representation is asymptotically the same as the metric entropy of the class of such functions with respect to any of the L ^p norms on the sphere.     

Locating peaks of a Schrödinger equation with sign-changing nonlinearity

In this paper, we study the concentration phenomenon of a positive ground state solution of a nonlinear Schrödinger equation on R^N. The coefficient of the nonlinearity of the equation changes sign. We prove that the solution has a maximum point at x₀∈Ω⁺={x∈R^N:Q(x)>0} where the energy attains its minimum.     

Study of the cosmic rays transport problems using second order parabolic type partial differential equation

It has been exactly 100 years since Hess’s historical discovery: an extraterrestrial origin of cosmic rays [1]. Galactic cosmic rays (GCR) being charged particles, penetrate the heliosphere and are modulated by the solar magnetic field. The propagation of cosmic rays is described by Parker’s transport equation [2], which is a second order parabolic type partial differential equation. It is time dependent 4-variables (with r, θ, φ, R, meaning: distance from the Sun, heliolatitudes, heliolongitudes and particles’ rigidity, respectively) equation which is a well known tool for studying problems connected with solar modulation of cosmic rays. Transport equation contains all fundamental processes taking place in the heliosphere: convection, diffusion, energy changes of the GCR particles owing to the interaction with solar wind’s inhomogeneities, drift due to the gradient and curvature of the regular interplanetary magnetic field and on the heliospheric current sheet.     

A class of solutions for the inverse diffusion problem

The close relation between Hermitian wavelets transforms and the diffusion equation is used to derive a one-parameter family of distributed sources as solutions to the inverse diffusion problem in R^N × R_. The class of solutions is interpreted in terms of energetically dominant events in the wavelet representation, where the scale of the event is proportional to its age. The construction procedure is a straightforward extension of the inverse wavelet transform formula. Simple examples illustrate the method.     

Associative solutions of a composite generalization of addition formulae

Let X be a real vector space and J be a nontrivial real interval. We determine all solutions (g, M, H) of the equation $$ g(x + M(g(x))y) = H(g(x),g(y)) for x,y \in X, $$ under the assumptions that g: X → J is continuous on rays, M: J → R is continuous and H: J ² → J is associative.     

A sufficient condition for the complete reducibility of the regular representation

The “Mackey machine” is heavily employed to prove the following theorem. Let G be a separable locally compact group. Suppose that every positive definite function p on G which vanishes at infinity is associated with the regular representation R, i.e., p(g) = (R_g?, ?) for some L² function ?. Then R decomposes into a direct sum of irreducible representations. This generalizes the theorem of Figà-Talamanca for unimodular groups. Although we use his result several times, our techniques are basically very different, the most difficult part occurring in a connected Lie group context.     

About the p-adic Yosida equation inside a disk

Let K be a complete ultrametric algebraically closed field and let ?(d(0, R^?)) be the field of meromorphic functions inside the disk d(0,R⁻) = {x ∈ K ∣ ∣x∣ < R}. Let ?_b(d(0, R^?)) be the subfield of bounded meromorphic functions inside d(0,R⁻) and let ?_u(d(0, R^?)) = ?(d(0, R^?)) ? ?_b(d(0, R^?)) be the subset of unbounded meromorphic functions inside d(0,R⁻). Initially, we consider the Yosida Equation: , where m ∈ ?^* and F(X) is a rational function of degree d with coefficients in ?_b(d(0, R^?)). We show that, if d ≥ 2m + 1, this equation has no solution in ?_u(d(0, R^?)).Next, we examine solutions of the above equation when F(X) is apolynomial with constant coefficients and show that it has no unbounded analytic functions in d(0,R⁻). Further, we list the only cases when the equation may eventually admit solutions in ?_u(d(0, R^?)). Particularly, the elliptic equation may not.     

Does Huygens' principle hold for small perturbations of the wave equation?

The perturbed wave equation □u + q(x)u = 0 in $\text{R}$³ × $\text{R}$ with C∞ ($\text{R}$³) compactly supported initial data at t = 0 is considered. It is proven that the Huygens' principle does not hold for this equation if the potential is (essentially) non-negative, well-behaved at infinity and small in a suitable sense. The treatment is elementary and based on energy estimates and the positivity of the Riemann function for the wave equation in three space dimensions. The result still holds if the solution u is “small” over some space-time propagation cone. In the ease in which q has compact support, stronger results of this type for the above equation are obtained.     

On the stochastic Benjamin-Ono equation

We discuss the Cauchy problem for the stochastic Benjamin-Ono equation in the function class Hs(R), s>3/2. When there is a zero-order dissipation, we also establish the existence of an invariant measure with support in H²(R). Many authors have discussed the Cauchy problem for the deterministic Benjamin-Ono equation. But our results are new for the stochastic Benjamin-Ono equation. Our goal is to extend known results for the deterministic equation to the stochastic equation.     

Asymptotics of solutions of a differential equation of second order with two turning points and a complex parameter. II

Asymptotic formulas are constructed and rigorously justified for linearly independent solutions of a second-order differential equation with a coefficient possessing the property of finite smoothness and containing a complex parameter ζ (forTmζ=0 the equation has two real turning points). A perturbation method is applied which consists in extending the coefficient of the equation to the complex Z plane and approximating it in an ε-neighborhood of the real axis of this plane by a quadratic polynomial. It is proved that the leading terms of the constructed formulas expressed in terms of parabolic cylinder functions are uniform with respect to arg ζ and that the error admitted under the approximation indicated above can be estimated by the quantityO(K^?1/2, (K→∞ is the second parameter, in addition to S, on which the coefficient of the differential equation depends).     

SOME PROPERTIES OF HOLOMORPHIC CLIFFORDIAN FUNCTIONS IN COMPLEX CLIFFORD ANALYSIS

In this article, we mainly develop the foundation of a new function theory of several complex variables with values in a complex Clifford algebra defined on some subdomains of Cn+1, so-called complex holomorphic Cliffordian functions. We define the complex holomorphic Cliffordian functions, study polynomial and singular solutions of the equation D△mf=0, obtain the integral representation formula for the complex holomorphic Cliffordian functions with values in a complex Clifford algebra defined on some submanifolds of Cn+1, deduce the Taylor expansion and the Laurent expansion for them and prove an invariance under an action of Lie group for them.     

Some well-posed functional equations which generate semigroups

Consider the abstract linear functional equation (FE) (Dx)(t) = f(t) (t ? 0), x(t) = ?(t) (t ? 0) in a Banach space B. A theorem is proven which contains the following result as a special case. Let Y(R; B; η) be a L^p-space or C₀-space on R = (?t8, ∞), with a suitable weight function η, and with values in B. Let D be a closed (unbounded) causal linear operator in Y(R; B; η), which commutes with translations. Suppose that D + λI has a continuous causal inverse for some complex λ, and that D restricted to those functions in Y(R;B;η) which vanish on R^? = (?∞, 0] has a continuous causal inverse. Then (FE) generates a strongly continuous semigroup of translation type on a Banach space, which is essentially the cross product of the restriction of the domain of D to R^? and Y(R⁺; B; η). Examples with B = Cⁿ on how the theory applies to a neutral functional differential equation, a difference equation, a Volterra integrodifferential equation (with nonintegrable kernel but integrable resolvent), and a fractional order functional differential equation are given. Also, an abstract neutral functional differential equation in a Hilbert space is studied and applications to an abstract Volterra integrodifferential equation in a Banach space are indicated.     

Convergence towards attractors for a degenerate Ginzburg-Landau equation

We study a real Ginzburg-Landau equation, in a bounded domain of \mathbbR^N ,\mathbb{R}^N , with a variable, generally non-smooth diffusion coefficient having a finite number of zeroes. By using the compactness of the embeddings of the weighted Sobolev spaces involved in the functional formulation of the problem, and the associated energy equation, we show the existence of a global attractor. The extension of the main result in the case of an unbounded domain is also discussed, where in addition, the diffusion coefficient has to be unbounded. Some remarks for the case of a complex Ginzburg-Landau equation are given.     

The analytic continuation of solutions to elliptic boundary value problems in two independent variables

An explicit representation is derived for the continuation across an analytic boundary of the solution to a boundary value problem for an analytic elliptic equation of second order in two independent variables. The representation is in terms of Cauchy data on the boundary and the complex Riemann function. This is equivalent to a representation for the solution to Cauchy's problem given by Henrici in 1957. It is confirmed that the method of complex characteristics is satisfactory for locating real singularities in the solution provided that the Riemann function is entire in its four arguments. Applications to Laplace's and Helmholtz's equations are discussed. By inserting known, simple solutions to the latter equation into the representation formula, several nontrivial integral relations involving the Bessel function J₀, and a possibly new series expansion for J_μ(x), are found.     

Construction of an analytic solution for one class of Langevin-type equations with orthogonal random actions

We find an analytic representation of a solution of the Itô-Langevin equations in R ³ with orthogonal random actions with respect to the vector of the solution. We construct a stochastic process to which the integral of the solution weakly converges as a small positive parameter with the derivative in the equation tends to zero.     

The trace class conjecture without the K-finiteness assumption

Let G be the group of real points of a reductive algebraic ℚ-group satisfying the same assumptions as in [5], Chapter I, and let Γ be a discrete subgroup of G. Let R_Γ be the right regular representation of G in L²(Γ\G). We prove in this Note that, for any integrable rapidly decreasing function ƒ on G, the restriction of R_Γ(ƒ) to the discrete spectrum of R_Γ is a trace class operator.     

Conditional limiting distribution of Type III elliptical random vectors

In this paper we consider elliptical random vectors in R^d,d≥2 with stochastic representation RAU where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R^d and A∈R^d×d is a non-singular matrix. When R has distribution function in the Weibull max-domain of attraction we say that the corresponding elliptical random vector is of Type III. For the bivariate set-up, Berman [Sojurns and Extremes of Stochastic Processes, Wadsworth & Brooks/ Cole, 1992] obtained for Type III elliptical random vectors an interesting asymptotic approximation by conditioning on one component. In this paper we extend Berman's result to Type III elliptical random vectors in R^d. Further, we derive an asymptotic approximation for the conditional distribution of such random vectors.     

Boundedness of solutions of a system of integro-differential equations

Let b: [?1, 0] →$\text{R}$ be a nondecreasing, strictly convex C²-function with b(? 1) = 0, and let g: $\text{R}$ⁿ → $\text{R}$ⁿ be a locally Lipschitzian mapping, which is the gradient of a function G: $\text{R}$ⁿ → $\text{R}$. Consider the following vector-valued integro-differential equation of the Levin-Nohel type

$\text{x}\text{?}\text{(t)=?∝}{}_{\mathrm{?1}}{}^0\text{b(θ)g(x(t + θ))dθ}$

. (E) This equation is used in applications to model various viscoelastic phenomena. By LaSalle's invariance principle, every bounded solution x(t) goes to a connected set of zeros of g, as time t goes to infinity. It is the purpose of this paper to give several geometric criteria assuring the boundedness of solutions of (E) or some of its components.