Partial Differential Equations Theory (Sobolev Space, Weak Solution, Semigroup Theory, Fourier and Galerkin Methods)

Most modern textbooks on partial differential equations either contain proofs of fundamental results only in special cases, or proof outlines, or do not contain proofs at all. This textbook, despite its relatively small volume, provides all the main results with complete proofs.

The foundation for the material in the textbook is the functional approach associated with the concept of a weak solution and Sobolev spaces. In this textbook, such an approach is demonstrated on second-order model equations: the Poisson equation, the heat equation, and the wave equation. However, it allows one to generalize the main results presented in the textbook to the case of more general equations of the corresponding type with variable coefficients.

The theory of partial differential equations constructed on the basis of Sobolev spaces was earlier presented in a number of remarkable books. Although many of these sources have been used in preparing this textbook, its main difference however is the presentation of the semigroup theory with the application to evolutionary equations (thanks to this, mixed problems and the Cauchy problem are studied using a unified method), as well as the use of modern techniques, which made it possible to simplify the majority of proofs.