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Methods Of Modern Mathematical Physics

The purpose of this chapter is to present the basic ideas of the theory of distributions. Distributions or generalized functions, as they are also known, have proved to be very useful in many branches of pure and applied mathematics. Many textbooks, monographs and articles have been written on their theory and their applications [12], [23], [53], [63], [64], [69], [71], [80], [97], [111]. Our present aim is to give a brief but solid introduction to the theory of distributions, particularly to those aspects that are important in the theory of asymptotic expansions.

Introduction to the Theory of Distributions

In this paper, we develop the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. There are no assumptions on the regularity of the boundary which is allowed to have arbitrary singularities. We give applications of the developed analysis to obtain a priori estimates for solutions of boundary value problems that are elliptic within the constructed calculus.

http://wwwf.imperial.ac.uk/~ruzh/papers/IMRN-Ruzhansky-Tokmagambetov-offprint-open-access.pdf

Nonharmonic Analysis of Boundary Value Problems

In this paper we develop the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. We give applications of the developed analysis to obtain a-priori estimates for solutions of operators that are elliptic within the constructed calculus.

Nonharmonic analysis of boundary value problems

In this paper, we study the Cauchy problem for the Landau Hamiltonian wave equation, with time-dependent irregular (distributional) electromagnetic field and similarly irregular velocity. For such equations, we describe the notion of a ‘very weak solution’ adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifier of the coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or distributional-type solutions under conditions when such solutions also exist.

https://biblio.ugent.be/publication/8585404/file/8585405.pdf

Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary

Given a Hilbert space $${\mathcal{H}}$$
 , we investigate the well-posedness of the Cauchy problem for the wave equation for operators with a discrete non-negative spectrum acting on $${\mathcal{H}}$$
 . We consider the cases when the time-dependent propagation speed is regular, Holder, and distributional. We also consider cases when it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of “very weak solutions” to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique “very weak solution” in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the harmonic and anharmonic oscillators, the Landau Hamiltonian on $${\mathbb{R}^n}$$
 , uniformly elliptic operators of different orders on domains, Hormander’s sums of squares on compact Lie groups and compact manifolds, operators on manifolds with boundary, and many others.

/pdf/wave-equation-for-operators-with-discrete-spectrum-and-3l7sb5kz8k.pdf

Niyaz Tokmagambetov

Papers

Nonharmonic Analysis of Boundary Value Problems

Nonharmonic analysis of boundary value problems

Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary

Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed