Two-Parameter Asymptotics in a Bisingular Cauchy Problem for a Parabolic Equation

The Cauchy problem for a quasilinear parabolic equation with a small parameter ε at the highest derivative is considered. The initial function, which has the form of a smoothed step, depends on a “stretched” variable x/ρ, where ρ is another small parameter. This problem statement is of interest for applications as a model of propagation of nonlinear waves in physical systems in the presence of small dissipation. In the case corresponding to a compression wave, asymptotic solutions of the problem are constructed in the parameters ε and ρ independently tending to zero. It is assumed that ε/ρ → 0. Far from the line of discontinuity of the limit solution, asymptotic solutions are constructed in the form of series in powers of ε and ρ. In a small domain of linear approximation, an asymptotic solution is constructed in the form of a series in powers of the ratio ρ/ε. The coefficients of the inner expansion are determined from a recursive chain of initial value problems. The asymptotics of these coefficients at infinity is studied. The time of reconstruction of the scale of the internal space variable is determined.     

Mappings connected with the gradient of conformal radius

In this paper we prove the following conformity criterion for the gradient of conformal radius ?R(D, z) of a convex domain D: the boundary ?D has to be a circumference. We calculate coefficients K(r) for K(r)-quasiconformal mappings ?R(D(r), z), D(r) ? D, 0 < r < 1, and complete the results obtained by F. G. Avkhadiev and K.-J. Wirths for the structure of boundary elements of quasiconformal mappings of the domain D.     

Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II

We consider a (generally, noncoercive) mixed boundary value problem in a bounded domain D of Rⁿ for a second order elliptic differential operator A(x, ?). The differential operator is assumed to be of divergent form in D and the boundary operator B(x, ?) is of Robin type on ?D. The boundary of D is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset Y ? ?D and control the growth of solutions near Y. We prove that the pair (A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set Y. Moreover, we prove the completeness of root functions related to L.     

On Bounded Positive (m,p)-Circle Domains

Let D be a bounded positive (m, p)-circle domain in ?². The authors prove that if dim(Iso(D)⁰) = 2, then D is holomorphically equivalent to a Reinhardt domain; if dim(Iso(D)⁰) = 4, then D is holomorphically equivalent to the unit ball in ?². Moreover, the authors prove the Thullen’s classification on bounded Reinhardt domains in ?² by the Lie group technique.     

The class of solenoidal planar-helical vector fields

The class of solenoidal vector fields whose lines lie in planes parallel to R ² is constructed by the method of mappings. This class exhausts the set of all smooth planarhelical solutions of Gromeka’s problem in some domain D ? R ³. In the case of domains D with cylindrical boundaries whose generators are orthogonal to R ², it is shown that the choice of a specific solution from the constructed class is reduced to the Dirichlet problem with respect to two functions that are harmonic conjugates in D ² = D∩R ²; i.e., Gromeka’s nonlinear problem is reduced to linear boundary value problems. As an example, a specific solution of the problem for an axisymmetric layer is presented. The solution is based on solving Dirichlet problems in the form of series uniformly convergent in \(\bar D^2\) in terms of wavelet systems that form bases of various spaces of functions harmonic in D ².     

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

Let D be an open connected subset of the complex plane C with sufficiently smooth boundary ?D. Perturbing the Cauchy problem for the Cauchy–Riemann system ??u = f in D with boundary data on a closed subset S ? ?D, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter ε ∈ (0, 1] in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on ?D\S, each of them has a unique solution in some appropriate Hilbert space H ⁺(D) densely embedded in the Lebesgue space L ₂(?D) and the Sobolev–Slobodetski? space H ^1/2?δ(D) for every δ > 0. The corresponding family of the solutions {u _ε} converges to a solution to the Cauchy problem in H ⁺(D) (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in H ⁺(D) is equivalent to boundedness of the family {u _ε} in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.     

Characterization of convex domains in ℂ<Superscript>2</Superscript> with non-compact automorphism group

In the field of several complex variables, the Greene-Krantz Conjecture, whose consequences would be far reaching, has yet to be proven. The conjecture is as follows: Let D be a smoothly bounded domain in ?ⁿ. Suppose there exists {g _j} ? Aut(D) such that {g _j(z)} accumulates at a boundary point p ∈ ?D for some z ∈ D. Then ?D is of finite type at p. In this paper, we prove the following result, yielding further evidence to the probable veracity of this important conjecture: Let D be a bounded convex domain in ?² with C ² boundary. Suppose that there is a sequence {g _j} ? Aut(D) such that {g _j(z)} accumulates at a boundary point for some point z ∈ D. Then if p ∈ ?D is such an orbit accumulation point, ?D contains no non-trivial analytic variety passing through p.     

On the mechanics of helical flows in an ideal incompressible nonviscous continuous medium

We find a general solution to the problem on the motion in an incompressible continuous medium occupying at any time a whole domain D ? R ³ under the conditions that D is an axially symmetric cylinder and the motion is described by the Euler equation together with the continuity equation for an incompressible medium and belongs to the class of helical flows (according to I.S. Gromeka’s terminology), in which sreamlines coincide with vortex lines. This class is constructed by the method of transformation of the geometric structure of a vector field. The solution is characterized in Theorem 2 in the end of the paper.     

Global Well-posedness of the Stochastic Generalized Kuramoto-Sivashinsky Equation with Multiplicative Noise

Global well-posedness of the initial-boundary value problem for the stochastic generalized Kuramoto- Sivashinsky equation in a bounded domain D with a multiplicative noise is studied. It is shown that under suitable sufficient conditions, for any initial data u₀ ∈ L₂(D × Ω), this problem has a unique global solution u in the space L₂(Ω, C([0, T], L₂(D))) for any T >0, and the solution map u₀ ? u is Lipschitz continuous.     

On the problem of determining the parameters of a layered elastic medium and an impulse source

Considering the linear system of elasticity equations describing the wave propagation in the half-space ? ₊ ³ = {x ∈ ?³ | x ₃ > 0} we address the problem of determining the density and elastic parameters which are piecewise constant functions of x ₃. The shape is unknown of a point-like impulse source that excites elastic oscillations in the half-space. We show that under certain assumptions on the source shape and the parameters of the elastic medium the displacements of the boundary points of the half-space for some finite time interval (0, T) uniquely determine the normalized density (with respect to the first layer) and the elastic Lamé parameters for x ₃ ∈ [0, H], where H = H(T). We give an algorithmic procedure for constructing the required parameters.     

The Cauchy problem for operators with injective symbol in the Lebesgue space <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> in a domain

Let D be a bounded domain in ? ⁿ (n ≥ 2) with infinitely smooth boundary ?D. We give some necessary and sufficient conditions for the Cauchy problem to be solvable in the Lebesgue space L ²(D) in D for an arbitrary differential operator A having an injective principal symbol. Furthermore, using bases with double orthogonality, we construct Carleman’s formula that restores a (vector-)function in L ²(D) from the Cauchy data given on a relatively open connected set Γ ? ?D and the values Au in D whenever the data belong to L ²(Γ) and L ²(D) respectively.     

On Riesz means of the coefficients of spinor zeta functions in genus two

Let F be a Siegel cusp form of integral weight k on the Siegel modular group Sp ₂(?) of genus 2. The Fourier coefficients of the spinor zeta function Z _F (s) are denoted by c _n . Let D _ρ (x;Z _F ) be the Riesz mean of c _n . In this paper, we obtain the truncated Voronoï-type formula of D _ρ (x;Z _F ) under the Ramanujan–Petersson conjecture.     

Asymptotic behavior of zeros of Mittag-Leffler functions of order 1/2

Let z _n denote the sequence of zeros of the Mittag-Leffler function E _ρ (z; μ), ρ > 0, μ ∈ ?, which is an entire function of order ρ. With the exception of the case ρ = 1/2, Re μ = 3 an asymptotic behavior of the sequence z _n ^ρ was known earlier up to infinitesimals o(1) having a sharply defined rate of decrease. In this paper the behavior of the sequence z _n ^1/2 is studied just in this exceptional case. Furthermore, for ρ = 1/2, μ > 3 we give the form of a curvilinear half-plane which is free of the points z _n .     

An upper bound for the number of uniformly packed binary codes

Under study are the binary codes uniformly packed (in the wide sense) of length n with minimum distance d and covering radius ρ. It is shown that every code of this kind is uniquely determined by the set of its codewords of weights ?n/2? ? ρ, …, ?n/2? + ρ. For odd d, the number of distinct codes is at most

$2^{2^{n - \tfrac{3}{2}\log n + o(log n)} } $

.

     

Totally real discs in non-pseudoconvex boundaries

LetD be a relatively compact domain inC² with smooth connected boundary ?D. A compact setK??D is called removable if any continuous CR function defined on ?D/K admits a holomorphic extension toD. IfD is strictly pseudoconvex, a theorem of B. Jöricke states that any compactK contained in a smooth totally real discS??D is removable. In the present article we show that this theorem is true without any assumption on pseudoconvexity.     

Valuation ideals and primary <Emphasis Type="Italic">w</Emphasis>-ideals

Let D be an integral domain, V (D) (resp., t-V (D)) be the set of all valuation (resp., t-valuation) ideals of D, and w-P(D) be the set of primary w-ideals of D. Let D[X] be the polynomial ring over D, c(f) be the ideal of D generated by the coefficients of f ∈ D[X], and N _v = {f ∈ D[X] | c(f) _v = D}. In this paper, we study integral domains D in which w-P(D) ? t-V (D), t-V (D) ? w-P(D), or t-V (D) = w-P(D). We also study the relationship between t-V (D) and \(V\left( {D{{\left[ X \right]}_{{N_v}}}} \right)\), and characterize when t-V (A + XB[X]) ? w-P(A + XB[X]) holds for a proper extension A ? B of integral domains.     

On the Distribution of Zero Sets of Holomorphic Functions

Let M be a subharmonic function with Riesz measure ν _M in a domain D in the n-dimensional complex Euclidean space ? _n , and let f be a nonzero function that is holomorphic in D, vanishes on a set Z ? D, and satisfies |f| ? expM on D. Then restrictions on the growth of ν _M near the boundary of D imply certain restrictions on the dimensions or the area/volume of Z. We give a quantitative study of this phenomenon in the subharmonic framework.     

The optimal solution set of the interval linear programming problems

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ _j , μ _ij , μ _i  ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ _ij  = μ _i  = λ _j  = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ _j , μ _ij and μ _i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.     

Estimates for Sobolev norms of triangular mappings

An increasing triangular mapping T on the n-dimensional cube Θ = [0, 1] ⁿ transforming a measure μ to a measure ν is considered, where μ and ν are absolutely continuous Borel probability measures having densities ρ _μ and ρ _ν . It is shown that if there exist positive constants ? and M such that ? < ρ _ν < M, ? < ρ _ν < M, there exist numbers α, β > 1 such that p = αβ(n ? 1)^?1 (α + β)^?1 > 1 and ρ _μ ∈ W ^1,α (Θ), ρ _ν ∈ W ^1,β ) (Θ), where W ^1,q denotes a Sobolev class, then the mapping T belongs to the class W ^1,p (Θ).     

Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. I

We consider (in general noncoercive) mixed problems in a bounded domain D in ? ⁿ for a second-order elliptic partial differential operator A(x, ?). It is assumed that the operator is written in divergent form in D, the boundary operator B(x, ?) is the restriction of a linear combination of the function and its derivatives to ?D and the boundary of D is a Lipschitz surface. We separate a closed set Y ? ?D and control the growth of solutions near Y. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, where the weight is a power of the distance to the singular set Y. Finally, we prove the completeness of the root functions associated with L.The article consists of two parts. The first part published in the present paper, is devoted to exposing the theory of the special weighted Sobolev–Slobodetskii? spaces in Lipschitz domains. We obtain theorems on the properties of these spaces; namely, theorems on the interpolation of these spaces, embedding theorems, and theorems about traces. We also study the properties of the weighted spaces defined by some (in general) noncoercive forms.