Numerical pseudodifferential operator and Fourier regularization

The concept of numerical pseudodifferential operator, which is an extension of numerical differentiation, is suggested. Numerical pseudodifferential operator just is calculating the value of the pseudodifferential operator with unbounded symbol. Many ill-posed problems can lead to numerical pseudodifferential operators. Fourier regularization is a very simple and effective method for recovering the stability of numerical pseudodifferential operators. A systematically theoretical analysis and some concrete examples are provided.     

The Edge Algebra Structure of Boundary Value Problems

Boundary value problems for pseudodifferential operators (with orwithout the transmission property) are characterised as a substructureof the edge pseudodifferential calculus with constant discreteasymptotics. The boundary in this case is the edge and the inner normalthe model cone of local wedges. Elliptic boundary value problems fornoninteger powers of the Laplace symbol belong to the examples as wellas problems for the identity operator in the interior with a prescribednumber of trace and potential conditions. Transmission operators arecharacterised as smoothing Mellin and Green operators with meromorphicsymbols.     

Pseudodifferential Operators and Equations of Variable Order

A new class of elliptic pseudodifferential equations of variable order is considered. Local Sobolev–Slobodetskii spaces are introduced and used to obtain theorems on the continuity of these pseudodifferential operators and describe the Fredholm properties of the corresponding pseudodifferential equations and boundary value problems.     

Interaction between thermoelastic and scalar oscillation fields

Three-dimensional mathematical problems of the interaction between thermoelastic and scalar oscillation fields in a general physically anisotropic case are studied by the boundary integral equation methods. Uniqueness and existence theorems are proved by the reduction of the original interface problems to equivalent systems of boundary pseudodifferential equations. In the non-resonance case the invertibility of the corresponding matrix pseudodifferential operators in appropriate functional spaces is shown on the basis of the generalized Sommerfeld-Kupradze type thermoradiation conditions for anisotropic bodies. In the resonance case the co-kernels of the pseudodifferential operators are analysed and the efficient conditions of solvability of the original interface problems are established.     

Boundary value problems on manifolds with fibered boundary

We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah–Patodi–Singer problems in subspaces (it contains both as special cases). The boundary conditions in this theory are taken as elements of the C *‐algebra generated by pseudodifferential operators and families of pseudodifferential operators in the fibers. We prove the Fredholm property for elliptic boundary value problems and compute a topological obstruction (similar to Atiyah–Bott obstruction) to the existence of elliptic boundary conditions for a given elliptic operator. Geometric operators with trivial and nontrivial obstruction are given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some boundary value problems for pseudodifferential equations with degeneration

Boundary value problems for a new class of degenerate pseudodifferential equations containing a variable-symbol degenerate pseudodifferential operator based on a special integral transform and the first derivative with respect to one of the variables are studied. Existence theorems for these problems are proved. A priori estimates for their solutions are obtained in special weighted spaces similar to Sobolev ones.     

The equation of the p-adic closed strings for the scalar tachyon field

We investigate the structure of solutions of boundary value problems for a non-linear pseudodifferential equation describing the dynamics(rolling)of p-adic closed strings for a scalar tachyon field.     

Commutator Criteria for Magnetic Pseudodifferential Operators

The gauge covariant magnetic Weyl calculus has been introduced and studied in previous works. We prove criteria in terms of commutators for operators to be magnetic pseudodifferential operators of suitable symbol classes; neither the statements nor the proofs depend on a choice of a vector potential. We apply this criteria to inversion problems, functional calculus, affiliation results and to the study of the evolution group generated by a magnetic pseudodifferential operator.     

The Nonclassical Boundary-contact Problems of Elasticity for Homogeneous Anisotropic Media

The existence and uniqueness of solutions of the nonclassical boundary-contact problems (i.e., problems with a contact on some part of the boundaries) of elasticity for homogeneous anisotropic media are investigated in Besov and Bessel potential spaces using methods of potential theory and the theory of pseudodifferential equations on manifolds with boundary. The smoothness of the solutions obtained is studied.     

A residual–based error estimator for BEM–discretizations of contact problems

Summary. We develop an a posteriori error estimate for boundary element solutions of static contact problems without friction. The presented result is based on an error estimate for linear pseudodifferential equations and on a certain commutator property for pseudodifferential operators. A heuristic extension of the obtained error estimate to frictional contact problems is presented, too. Numerical examples indicate a good performance of the error estimator for both the frictionless and the frictional problem. Mathematics Subject Classification (1991):35J85, 65N38, 73T05Dedicated to Hans Grabmüller on the occasion of his 60th birthday     

The equation of the <Emphasis Type="Italic">p</Emphasis>-adic closed strings for the scalar tachyon field

We investigate the structure of solutions of boundary value problems for a non-linear pseudodifferential equation describing the dynamics (rolling) of p-adic closed strings for a scalar tachyon field. The work was partially supported by grant of the President RF (No. NSh-1542.2003.1)     

Traces of <Emphasis Type="Italic">G</Emphasis>-Operators Concentrated on Submanifolds

We consider the traces on submanifolds of G-operators generated by pseudodifferential operators and operators of shift along the orbits of a discrete group G. Such operators arise in various problems in differential equations and mathematical physics, for example, in Sobolev problems. We show that the trace of a G-operator on a submanifold is the sum of a pseudodifferential operator on the submanifold and a G-operator concentrated on a sub-submanifold.     

Uniform Asymptotic Solution in the Form of an Airy Function for Semiclassical Bound States in One-Dimensional and Radially Symmetric Problems

Theoretical and Mathematical Physics - We consider stationary scalar and vector problems for differential and pseudodifferential operators leading to the appearance of asymptotic solutions of...     

Explicit algebras with the Leibniz–Mellin translation product

A sub‐calculus of the calculus (“algebra”) of all complete conormal symbols arising in the edge pseudodifferential calculus is constructed. This calculus of complete conormal symbols is suitable for constructing sub‐calculi of the general edge pseudodifferential calculus, for which the edge‐degenerate pseudodifferential operators involved map conormal asymptotics of distributions near the edges in a prescribed manner. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Well-posedness of general nonlocal nonhomogeneous boundary value problems for pseudodifferential equations with partial derivatives

In the article we study the questions of well-posedness of general nonlocal boundary value problems for pseudodifferential equations in the Besov-type limit spaces.     

Spectral Invariance of Pseudodifferential Boundary Value Problems on Manifolds with Conical Singularities

Journal of Fourier Analysis and Applications - We prove the spectral invariance of the algebra of classical pseudodifferential boundary value problems on manifolds with conical singularities in the...     

Generation of Semigroups for Vector-Valued Pseudodifferential Operators on the Torus

We consider toroidal pseudodifferential operators with operator-valued symbols, their mapping properties and the generation of analytic semigroups on vector-valued Besov and Sobolev spaces. Here, we restrict ourselves to pseudodifferential operators with x-independent symbols (Fourier multipliers). We show that a parabolic toroidal pseudodifferential operator generates an analytic semigroup on the Besov space \(B_{pq}^s({\mathbb T}^n,E)\) and on the Sobolev space \(W_p^k({\mathbb T}^n,E)\), where E is an arbitrary Banach space, \(1\le p,q\le \infty \), \(s\in {\mathbb R}\) and \(k\in {\mathbb N}_0\). For the proof of the Sobolev space result, we establish a uniform estimate on the kernel which is given as an infinite parameter-dependent sum. An application to abstract non-autonomous periodic pseudodifferential Cauchy problems gives the existence and uniqueness of classical solutions for such problems.     

Boundary value problems for complete partial differential equations of variable order

We prove the well-posedness of new boundary value problems for partially pseudodifferential complete nonclassical equations of variable order in space variables with higher derivatives of odd order in time.     

Geometrical properties of nonlinear maps and their application, Part I: The topological degree of quasiruled Fredholm maps on quasicylindrical domains

In this paper we introduce the notion of quasicylindrical domains in Banach spaces and develop a concept of a degree for quasiruled Fredholm mappings on quasicylindrical domains. Note that this quasicylindrical structure appears in a rather natural way, whenever “analytically” given nonlinear pseudodifferential operators are investigated in spaces of sufficiently smooth functions. Moreover, the class of quasiruled Fredholm mappings on quasicylindrical domains is sufficiently large, so that within this framework one can study a quite large class of nonlinear boundary value problems which are related to pseudodifferential operators.