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.
About the Author :
Alexander L. Skubachevskii is a renowned expert in partial differential equations and functional differential equations, who has made a decisive contribution to the development of new directions in these areas. He was the first to study a number of open problems: the existence of unbounded oscillating solutions of second-order functional differential equations, the solvability and regularity of solutions of elliptic problems with nonlocal boundary conditions, the existence of Feller semigroups in the non-transversal case, the solvability and regularity of solutions of elliptic and parabolic functional differential equations, the existence of classical solutions of mixed problems for the Vlasov-Poisson system of equations with an external magnetic field, the Kato square root problem for regular accretive functional differential operators. In 2016, he was awarded the I.G. Petrovsky Prize of the Russian Academy of Sciences for his achievements in the study of nonclassical boundary value problems. A.L. Skubachevskii is the Head of the scientific school at the RUDN University, originated from the first results on elliptic differential-difference equations in the late 1970s. The research by A.L. Skubachevskii and his disciples on boundary-value problems for elliptic and parabolic functional-differential equations made a significant contribution to the construction of a general theory of functional-differential equations and found various applications in elasticity theory, control theory, and nonlinear optics. The activities of the school are also associated with recent achievements in the field of elliptic differential equations with nonlocal boundary conditions that arise in plasma theory and biophysics in the study of multidimensional diffusion processes, as well as the Vlasov kinetic equations playing an important role in the issues of controlled thermonuclear fusion. A.L. Skubachevskii is an author of 269 papers, 3 monographs, and 1 textbook. Leonid E. Rossovskii is an expert in the field of elliptic functional differential equations. He obtained his PhD degree (1996) and Dr. Sci. degree (2013) under the supervision of A.L. Skubachevskii. L.E. Rossovskii constructed the theory of boundary-value problems for elliptic functional differential equations with compression and expansion of independent variables in the principal part (necessary and sufficient conditions for the Garding-type inequality, unique solvability and smoothness of solutions of the Dirichlet and the Neumann problems, the Fredholm property of the general boundary-value problems in the Sobolev spaces, conditions of the unique solvability in weighted spaces, spectral stability of the Neumann problem with respect to small perturbations of the domain, the influence of multiplicatively incommensurable compressions on the solvability and properties of solutions etc.). L.E. Rossovskii is an author of 33 papers, 1 monograph, and 1 textbook.
