Classic solutions of boundary value problems for partial differential equations with operator of finite index in the main part of equation

N. A. Sidorov

The Bulletin of Irkutsk State University. Series Mathematics, Tome 27 (2019), pp. 55-70

Résumé

This paper is an attempt to give the review of a part of our results in the area of singular partial differential equations. Using the results of the theory of complete generalized Jordan sets we consider the reduction of the PDE with the irreversible linear operator $B$ of finite index in the main differential expression to the regular problems. Earlier we and other authors applied similar methods to the development of Lyapunov alternative method in singular analysis and numerous applications in mechanics and mathematical physics. In this paper, we show how the problem of the choice of boundary conditions is connected with the $B$-Jordan structure of coefficients of PDE. The estimation of various approaches shows that the most efficient approach for solving this problem is the functional approach combined with the alternative Lyapunov method, Jordan structure coefficients and skeleton decomposition of irreversible linear operator in the main part of the equation. On this base, the problem of the correct choice of boundary conditions for a wide class of singular PDE can be solved. The aggregated theorems of existence and uniqueness of classical solutions can be proved with continuously depending of experimental definite function. The theory is illustrated by considering the solution of some integro-differential equations with partial derivatives.

Détail

Keywords: degenerate PDE, Banach space, Noether operator, boundary value problems.
Mots-clés : Jordan set