Pliska Studia Mathematica Bulgarica
Volume 21, 2012
C O N T E N T S
- Gramchev, T. Preface. (pp. 3-4)
- Gramchev, T. Curriculum vitae of P. Popivanov. (pp. 5-6)
- Calvo, D., De Donno, G., Rodino, L. Operators with Polynomial Coefficients and Generalized Gelfand-Shilov Classes. (pp. 7-24)
- Toft, J., Wahlberg, P. Embeddings of ?-Modulation Spaces. (pp. 25-46)
- Anedda, C., Cadeddu, L., Porru, G. Problems for P-Monge-Ampere Equations. (pp. 47-70)
- Garello, G., Morando, A. L^{p} Microlocal Properties for Multi-Quasi-Elliptic Pseudodifferential Operators. (pp. 71-96)
- Boggiatto, P., Carypis, E., Oliaro, A. Windowed-Wigner Representations, Interferences and Operators. (pp. 97-112)
- Cappiello, M., Nicola, F. On the Holomorphic Extension of Solutions of Elliptic Pseudodifferential Equations. (pp. 113-126)
- van der Mee, C. Closed Form Solutions of Integrable Nonlinear Evolution Equations. (pp. 127-146)
- Kounchev, O., Render, H. Polyharmonic Hardy Spaces on the Klein-Dirac Quadric with Application to Polyharmonic Interpolation and Cubature Formulas. (pp. 147-176)
- Ganchev, G., Milousheva, V. An Invariant Theory of Surfaces in the Four-Dimensional Euclidean or Minkowski Space. (pp. 177-200)
- Gerdjikov, V. Riemann-Hilbert Problems with Canonical Normalization and Families of Commuting Operators. (pp. 201-216)
- Gerdjikov, V., Grahovski, G. On the 3-Wave Equations with Constant Boundary Conditions. (pp. 217-236)
- Fabricant, A., Kutev, N., Rangelov, T. New Hardy-Type Inequalities with Singular Weights. (pp. 237-246)
- Chobanov, G., Kutev, N. Interior Boundaries for Degenerate Elliptic Equations of Second Order Some Theory and Numerical Observations. (pp. 247-256)
- Vitanov, N. On Modified Method of Simplest Equation for Obtaining Exact Solutions of Nonlinear PDEs: Case of Elliptic Simplest Equation. (pp. 257-266)
- Apostolova, L. Hyperbolic Double-Complex Laplace Operator. (pp. 267-278)
- Dimiev, S. Hyperbolic Fibrations and PDE. (pp. 279-286)
- Christov, O. Canonically Conjugate Variables for the ?CH Equation. (pp. 287-298)
- Zapryanova, T. Best Approximation and Moduli of Smoothness. (pp. 299-306)
- Petrova, Z. Oscillation Properties of Some Functional Fourth Order Ordinary Differential Equations. (pp. 307-314)
- Boyadzhiev, G. Existence Theorems for Non-Cooperative Elliptic Systems. (pp. 315-320)
A B S T R A C T S
OPERATORS WITH POLYNOMIAL COEFFICIENTS AND GENERALIZED GELFAND-SHILOV CLASSES
Daniela Calvo
: dnlclv7@gmail.com
Giuseppe De Donno
giuseppe.dedonno@unito.it
Luigi Rodino
luigi.rodino@unito.it
2010 Mathematics Subject Classification: 35S05, 35J60, 35A20, 35B08,
35B40.
Key words: global hypoellipticity, Gelfand-Shilov spaces, multi-quasi-elliptic operators.
We study the problem of the global regularity for linear partial
differential operators with polynomial coefficients. In particular for multiquasi-elliptic operators we prove global regularity in generalized Gelfand-Shilov classes. We also provide counterexamples of globally regular operators
which are not multi-quasi-elliptic.
EMBEDDINGS OF α-MODULATION SPACES
Joachim Toft
joachim.toft@lnu.se
Patrik Wahlberg
patrik.wahlberg@unito.it
2010 Mathematics Subject Classification: 42B35, 46E35.
Key words: α-modulation spaces, embeddings, sharpness.
We show upper and lower embeddings of α_{1}-modulation spaces
in α_{2}-modulation spaces for 0 ≤ α_{1} ≤ α_{2} ≤ 1, and prove partial results on
the sharpness of the embeddings.
PROBLEMS FOR P-MONGE-AMPERE EQUATIONS
Claudia Anedda
canedda@unica.it
Lucio Cadeddu
cadeddu@unica.it
Giovanni Porru
porru@unica.it
2010 Mathematics Subject Classification: 35A23, 35B51, 35J96, 35P30, 47J20, 52A40.
Key words: Generalized Monge-Ampere equations, Rearrangements, Eigenvalues, Isoperimetric inequalities.
We study the problem of the global regularity for linear partial
differential operators with polynomial coefficients. In particular for multiquasielliptic operators we prove global regularity in generalized Gelfand-
Shilov classes. We also provide counterexamples of globally regular operators
which are not multi-quasi-elliptic.
L^{p} MICROLOCAL PROPERTIES FOR MULTI-QUASI-ELLIPTIC PSEUDODIFFERENTIAL
Operators
Gianluca Garello
gianluca.garello@unito.it
Alessandro Morando
alessandro.morando@ing.unibs.it
2010 Mathematics Subject Classification: 35S05, 35A17.
Key words: pseudodifferential operators, weighted Sobolev spaces, microlocal properties.
In the present paper microlocal properties of a class of suitable
L^{p} bounded pseudodifferential operators are stated in the framework of
weighted Sobolev spaces of L^{p} type. Applications to microlocal regularity of
solutions to multi-quasi-elliptic partial differential equations are also given.
WINDOWED-WIGNER REPRESENTATIONS, INTERFERENCES AND OPERATORS
Paolo Boggiatto
paolo.boggiatto@unito.it
Evanthia Carypis
evanthia.carypis@unito.it
Alessandro Oliaro
alessandro.oliaro@unito.it
2010 Mathematics Subject Classification: 42B10, 47A07, 35S05.
Key words: Wigner type representations, pseudodifferential operators, interference.
"Windowed-Wigner" representations, denoted by Wig_{ψ} and
Wig^{∗}_{ψ}, were introduced in [2] in connection with uncertainty principles and
interferences problems. In this paper we present a more precise analysis of
their behavior obtaining an estimate of the L^{2}-norm of interferences of couples
of "model" signals. We further define a suitable functional framework
for the associated operators and show that they form a class of pseudodifferential
operators which define a natural "path" between the multiplication,
Weyl and Fourier multipliers operators.
ON THE HOLOMORPHIC EXTENSION OF SOLUTIONS OF ELLIPTIC PSEUDODIFFERENTIAL EQUATIONS
Marco Cappiello
marco.cappiello@unito.it
Fabio Nicola
fabio.nicola@polito.it
2010 Mathematics Subject Classification: 35B65, 35S05, 35A20.
Key words: Elliptic equations, holomorphic extension, pseudodifferential operators.
We derive analytic estimates and holomorphic extensions for
the solutions of a class of elliptic pseudodifferential equations on R^{d}.
CLOSED FORM SOLUTIONS OF INTEGRABLE NONLINEAR EVOLUTION EQUATIONS
Cornelis van der Mee
cornelis@krein.unica.it
2010 Mathematics Subject Classification: 35Q55.
Key words: nonlinear evolution equations, closed form solutions.
In this article we obtain closed form solutions of integrable nonlinear
evolution equations associated with the nonsymmetricmatrix Zakharov-
Shabat system by means of the inverse scattering transform. These solutions
are parametrized by triplets of matrices. Alternatively, the time evolution
of the Marchenko integral kernels and direct substitution are employed in
deriving these solutions.
POLYHARMONIC HARDY SPACES ON THE KLEIN-DIRAC QUADRIC WITH APPLICATION TO POLYHARMONIC INTERPOLATION AND CUBATURE FORMULAS
Ognyan Kounchev
kounchev@math.bas.bg
Hermann Render
hermann.render@ucd.ie
2010 Mathematics Subject Classification: 65D30, 32A35, 41A55.
Key words: Hardy spaces, numerical integration, cubature formulas, error estimate.
In the present paper we introduce a new concept of Hardy
type space naturally defined on the Klein-Dirac quadric. We study different
properties of the functions belonging to these spaces, in particular boundary
value problems. We apply these new spaces to polyharmonic interpolation
and to interpolatory cubature formulas.
AN INVARIANT THEORY OF SURFACES IN THE FOUR-DIMENSIONAL EUCLIDEAN OR MINKOWSKI
Space
Georgi Ganchev
ganchev@math.bas.bg
Velichka Milousheva
vmil@math.bas.bg
2010 Mathematics Subject Classification: 53A07, 53A35, 53A10.
Key words: Key words: Surfaces in Euclidean or Minkowski 4-space, Weingarten-type linear map, tangent indicatrix, normal curvature ellipse, Bonnet-type fundamental theorems, general rotational
surfaces, meridian surfaces.
The present article is a survey of some of our recent results
on the theory of two-dimensional surfaces in the four-dimensional Euclidean
or Minkowski space. We present our approach to the theory of surfaces in
Euclidean or Minkowski 4-space, which is based on the introduction of an
invariant linear map of Weingarten-type in the tangent plane at any point of
the surface under consideration. This invariant map allows us to introduce
principal lines and an invariant moving frame field at each point of the
surface. Writing derivative formulas of Frenet-type for this frame field, we
obtain a system of invariant functions, which determine the surface up to a
motion.
We formulate the fundamental theorems for the general classes of surfaces
in Euclidean or Minkowski 4-space in terms of the invariant functions.
We show that the basic geometric classes of surfaces, determined by
conditions on their invariants, can be interpreted in terms of the properties
of two geometric figures: the tangent indicatrix and the normal curvature
ellipse.
We apply our theory to some special classes of surfaces in Euclidean or
Minkowski 4-space.
RIEMANN-HILBERT PROBLEMS WITH CANONICAL NORMALIZATION AND FAMILIES OF COMMUTING OPERATORS
V. S. Gerdjikov
gerjikov@inrne.bas.bg
2010 Mathematics Subject Classification: 35Q15, 31A25, 37K10, 35Q58.
Key words: Riemann-Hilbert Problems, commuting operators, integrable 3-wave interactions.
We start with a Riemann-Hilbert Problems (RHP) with canon-
ical normalization whose sewing functions depends on several additional vari-
ables. Using Zakharov-Shabat theorem we are able to construct a family of
ordinary differential operators for which the solution of the RHP is a common
fundamental analytic solution. This family of operators obviously commute.
Thus we are able to construct new classes of integrable nonlinear evolution
equations.
ON THE 3-WAVE EQUATIONS WITH CONSTANT BOUNDARY CONDITIONS
V. S. Gerdjikov
gerjikov@inrne.bas.bg
G. G. Grahovski
grah@inrne.bas.bg
2010 Mathematics Subject Classification: 37K40, 35Q15, 35Q51, 37K15.
Key words: Inverse Scattering Transform, 3-wave equations, Lax representation, Spectral
Theory, Fundamental Analytic Solutions.
The inverse scattering transform for a special case of the 3-
wave resonant interaction equations with non-vanishing boundary conditions
is studied. The Jost solutions and the fundamental analytic solutions (FAS)
for the associated spectral problem are constructed. The inverse scattering
problem for the Lax operator is formulated as a Riemann-Hilbert problem
on a Riemannian surface. The spectral properties of the Lax operator are
formulated.
NEW HARDY-TYPE INEQUALITIES WITH SINGULAR WEIGHTS
Alexander Fabricant
fabrican@math.bas.bg
Nikolai Kutev
kutev@math.bas.bg
Tsviatko Rangelov
rangelov@math.bas.bg
2010 Mathematics Subject Classification: 26D10.
Key words: Hardy inequality, Weights, Sharp estimates.
We prove a new Hardy-type inequality with weights that are
possibly singular at internal point and on the boundary of the domain. As
an illustration some applications and examples are given.
INTERIOR BOUNDARIES FOR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER SOME THEORY AND NUMERICAL OBSERVATIONS
G. Chobanov
chobanov@math.bas.bg
N. Kutev
kutev@math.bas.bg
2010 Mathematics Subject Classification: 35J70, 35J15, 35D05.
Key words: Linear degenerate elliptic equations, viscosity solutions, visualization.
For boundary value problems for degenerate-elliptic equations
of second order in Ω ⊂ R_{n} there are cases when a closed surface Γ exists,
dividing Ω into two subdomains in such a manner that two new correct
boundary value problems can be formulated without introducing new boundary
conditions. Such surfaces are called interior boundaries. Some theoretical
results regarding the connections between the solutions of the original problem
and the two new problems are given. Some numerical experiments using the
finite elements method are carried out trying to visualize the effects of the
presence of such interior boundary when n = 2. Also some more precise
study of the solutions in the case n = 2 is presented.
ON MODIFIED METHOD OF SIMPLEST EQUATION FOR OBTAINING EXACT SOLUTIONS OF NONLINEAR PDEs:
CASE OF ELLIPTIC SIMPLEST EQUATION
Nikolay K. Vitanov
vitanov@imbm.bas.bg
2010 Mathematics Subject Classification: 74J30, 34L30.
Key words: nonlinear partial differential equations, method of simplest equation, exact
travelling-wave solutions, elliptic equation.
The modified method of simplest equation is useful tool for
obtaining exact and approximate solutions of nonlinear PDEs. These so-
lutions are constructed on the basis of solutions of more simple equations
called simplest equations. In this paper we study the role of the simplest
equation for the application of the modified method of simplest equation.
As simplest equation we discuss the elliptic equation.
HYPERBOLIC DOUBLE-COMPLEX LAPLACE OPERATOR
Lilia N. Apostolova
liliana@math.bas.bg
2010 Mathematics Subject Classification: 35G35, 32A30, 30G35.
Key words: hyperbolic double-complex Laplace operator, hyperbolic decomplexification.
In this paper is introduced the hyperbolic double-complex
Laplace operator. The hyperbolic decomplexification of the hyperbolic doublecomplex
Laplace operator and its characteristic set is found. The exponential
eigenfunctions of the zero eigenvalue of the hyperbolic double-complex
Laplace operator are found as well.
HYPERBOLIC FIBRATIONS AND PDE
Stancho Dimiev
sdimiev@math.bas.bg
2010 Mathematics Subject Classification: 35L10, 35L90.
Key words: PDO, hyperbolic fibration, Euclidean fibration, Minkowski metrics.
In this note we try to distinguish the hyperbolic fibrations from
the Euclidean one with the help of the invariant action of partial differential
operators on the fibration. Two examples are given.
CANONICALLY CONJUGATE VARIABLES FOR THE μCH EQUATION
Ognyan Christov
christov@fmi.uni-sofia.bg
2010 Mathematics Subject Classification: 35Q35, 37K10.
Key words: μCH equation, Poisson bracket, canonically conjugate variables.
We consider the μCH equation which arises as an asymptotic
rotator equation in a liquid crystal with a preferred direction if one takes
into account the reciprocal action of dipoles on themselves. This equation
is closely related to the periodic Camassa-Holm and the Hunter-Saxton
equations. The μCH equation is also integrable and bi-Hamiltonian, that is,
it is Hamiltonian with respect to two compatible Poisson brackets. We give
a set of conjugated variables for both brackets.
BEST APPROXIMATION AND MODULI OF SMOOTHNESS
Teodora Dimova Zapryanova
teodorazap@abv.bg
2010 Mathematics Subject Classification: 41A25, 41A10.
Key words: Moduli of smoothness, Best algebraic approximation, K-functional, Linear
operator.
The aim of this note is to present moduli of smoothness which
are introduced by different schools of approximation for characterization of
the best algebraic approximation. We observe that Potapov's generalized
moduli are equivalent to the error in approximation by the algebraic version
of the trigonometric Jackson integrals in uniform norm and in weighted
integral metric.
OSCILLATION PROPERTIES OF SOME FUNCTIONAL FOURTH ORDER ORDINARY DIFFERENTIAL EQUATIONS
Zornitza A. Petrova
zap@tu-sofia.bg
2010 Mathematics Subject Classification: 34A30, 34A40, 34C10.
Key words: Oscillation, functional ordinary differential equation, eventually positive solution,
eventually negative solution.
In this paper are considered oscillation properties of some
classes of functional ordinary differential equations, namely equations of the type
EXISTENCE THEOREMS FOR NON-COOPERATIVE ELLIPTIC SYSTEMS
G. Boyadzhiev
gpb@math.bas.bg
2010 Mathematics Subject Classification: 35J65, 35K60, 35B05, 35R05.
Key words: Elliptic systems, non-cooperative, existence, sub- and super-solution.
Existence of classical C^{2}(Ω) ∩
C(Ω^{‾}) solutions of non-cooperative
weakly coupled systems of elliptic second-order PDE is proved via the method
of sub- and super-solutions.