Boundary values of solutions of some classes of differential equations

V. I. Gorbachuk ; M. L. Gorbachuk

Sbornik. Mathematics, Tome 31 (1977) no. 1, pp. 109-133

Résumé

Differential equations of the following forms are considered: $$ y'+Ay=0\quad\text{and}\quad-y''+A^2y=0, $$ where $A$ is a positive selfadjoint operator in a Hilbert space $H$. The question of whether the solutions of such equations have boundary values at the end points of the interval $(a,b)$ on which they are considered is investigated, as well as the problem of recovering a solution from its boundary values. A characterization of the boundary values is given in terms of the behavior of the solution near the end points $a$ and $b$. A number of examples are cited in which $A$ is realized as a differential operator in various function spaces. When applied to these concrete situations, the abstract theorems yield the existence and characteristics of the boundary values for certain classes of elliptic and parabolic equations; in particular, the well-known results of F. Riesz, Köthe and Komatsu are obtained and sharpened in this way. The approach is based on the spectral theory of selfadjoint operators. Bibliography: 18 titles.