Alexei Filinkov

Abstract Cauchy Problems : Three Approaches

Preface Introduction ILLUSTRATION AND MOTIVATION Heat Equation The Reversed Cauchy Problem for the Heat Equation Wave Equation SEMIGROUP METHODS C0-Semigroups Integrated Semigroups k-Convoluted Semigroups C-Regularized Semigroups Degenerate Semigroups The Cauchy Problem for Inclusions Second Order Equations ABSTRACT DISTRIBUTION METHODS The Cauchy Problem The Degenerate Cauchy Problem Ultradistributions and New Distributions REGULARIZATION METHODS The Ill-Posed Cauchy Problem Regularization and C-Regularized Semigroups Bibliographic Remark Bibliography Glossary of Notation Index

Differential equations in spaces of abstract stochastic distributions

We develop the theory of stochastic distributions with values in a separable Hilbert space, and apply this theory to the investigation of abstract stochastic evolution equations with additive noise.

Generalized solutions to abstract stochastic problems

We prove existence and uniqueness of a solution to a generalized stochastic Cauchy problem in spaces of abstract ultra-distributions.

Laplace transform of k-semigroups and well-posedness of cauchy problems

We consider the abstract Cauchy problem has an exponential growth in some domain of the right halfplane. To study well-posedness of the problem we use the technique of ultradistributions and K-convoluted semigroups with K(t) defined by the resolvent estimate.

A rigorous description of optical phase

We represent the phase of an optical field by an operator valued distribution thus enabling a rigorous analysis of its statistics and new approaches to its approximation and measurement.

The Solution of a Free Boundary Problem Related to Environmental Management Systems

Abstract Explicit solutions of free boundary problems are notoriously difficult to find. In this article, we consider two log-normal diffusions. One represents the level of pollution, or degradation, in some environmental area. The second models the social, political, or financial cost of the pollution. A single control parameter is considered that reduces the rate of pollution. The optimal time to implement the change in the parameter is found by explicitly solving a free boundary problem. The novelty is that the smooth pasting conditions, which are difficult to justify, are not used in the derivation.