Pliska Studia Mathematica Bulgarica

Volume 21, 2012

Proceedings of the International Conference on Partial Differential Equations and Applications Sofia, September 14-16, 2011 (In honor of 65-th Anniversary of Prof. Petar Popivanov)

GUEST EDITOR: T. Gramchev

Sofia, 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 α₁-modulation spaces in α₂-modulation spaces for 0 ≤ α₁ ≤ α₂ ≤ 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 L²-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 C²(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.