A criterion for unitary equivalence of unbounded operators to integral operators

Necessary and sufficient conditions are established for unitary equivalence of unbounded non-self-adjoint linear operators with Carleman and B-Carleman integral operators.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 271–280, February, 1978.     

A comparison principle for quasilinear operators in unbounded domains

We establish a comparison principle for quasilinear elliptic operators in unbounded cylindrical domains by combining the traditional test function method with nonlinear capacity techniques. Some applications to monotonicity and symmetry of solutions to nonlinear elliptic problems are given.     

Nonlinear eigenvalue problems for quasilinear operators on unbounded domains

We prove several existence results for eigenvalue problems involving the p-Laplacian and a nonlinear boundary condition on unbounded domains. We treat the non-degenerate subcritical case and the solutions are found in an appropriate weighted Sobolev space. Received May 2000     

On Stokes operators with variable viscosity in bounded and unbounded domains

We consider a generalization of the Stokes resolvent equation, where the constant viscosity is replaced by a general given positive function. Such a system arises in many situations as linearized system, when the viscosity of an incompressible, viscous fluid depends on some other quantities. We prove that an associated Stokes-like operator generates an analytic semi-group and admits a bounded H _∞ -calculus, which implies the maximal L ^q -regularity of the corresponding parabolic evolution equation. The analysis is done for a large class of unbounded domains with -boundary for some r > d with r ≥ q, q′. In particular, the existence of an L ^q -Helmholtz projection is assumed.     

Random fixed points of asymptotically nonexpansive random operators on unbounded domains

The aim of this paper is to prove some random fixed point theorems for asymptotically nonexpansive random operator defined on an unbounded closed and starshaped subset of a Banach space.      

Singular integral operators on product domains along twisted surfaces

We introduce a class of singular integral operators on product domains along twisted surfaces.We prove that the operators are bounded on L^p provided that the kernels satisfy weak conditions.     

Cauchy-type integral formula and Plemelj formula of hypermonogenic functions on unbounded domains

In this paper, we discuss the Cauchy-type integral formula of hypermonogenic functions on unbounded domains in real Clifford analysis, then we extend the Plemelj formula and Cauchy–Pompeiu formula of hypermonogenic functions on bounded domains to unbounded domains. We also deal with the Green-type formula on unbounded domains and get several important corollaries.     

Chaotic unbounded differentiation operators

We construct dense sets of hypercyclic vectors for unbounded differention operators, including differentiation operators on the Hardy spaceH ₂, and the Laplacian operator onL ₂((), for any bounded open subset of ². Furthermore, we show that these operators are chaotic, in the sense of Devaney.     

Semilinear problems in unbounded domains

As formulated by Silva [E.A. de B.e. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Anal. 16 (1991) 455-477] and Schechter [M. Schechter, A generalization of the saddle point method with applications, Ann. Polon. Math. 57 (3) (1992) 269-281; M. Schechter, New saddle point theorems, in: Generalized Functions and Their Applications, Varanasi, 1991, Plenum, New York, 1993, pp. 213-219], the sandwich theorem has become a very useful tool in finding critical points of functionals leading to solutions of partial differential equations. In the present paper, this theorem is strengthened to apply to more general situations. We present some applications.     

Some remarks on index of matrix singular integral operators in non-smooth domains

On manifolds with non-smooth boundaries the Mikhlin-Calderon-Zygmund operators with a matrix symbol are considered. The Noetherian property conditions (wave factorization) and reduction to Atiyah-Singer theorem are described. These results were obtained through joint work with I.V. Scherbenko. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On unbounded Bergman operators

This paper explores properties of the Bergman operator on unbounded open subsets of the plane. In addition to the characterization of the bounded commutant of such operators it proves the Berger-Shaw theorem and gives some general criteria under which the operator and its self-commutator are densely defined.