A~formula expressing the solution of a~differential equation with analytic coefficients on a~manifold without boundary in terms of the data of the problem

A. V. Babin

Sbornik. Mathematics, Tome 30 (1976) no. 4, pp. 539-563

Résumé

On a compact manifold $\Omega$ we examine the equation \begin{equation} A_2u=h. \end{equation} We assume that $A_2$ is a second-order elliptic selfadjoint positive definite differential operator and that the coefficients of the operator and of the function $h$ are analytic on $\Omega$. It is well known that equation (1) has a unique global solution $u(\omega)$ defined on the whole $\Omega$ (as a consequence of the Cauchy–Kowalewski theorem there are many local solutions). In this paper we obtain an explicit expression for the value of $u(\omega)$ at a point $\omega_0$ in terms of the Taylor coefficients of the right-hand side at $\omega_0$, and of the coefficients of the operator. By the same token we obtain an expression for the solution of the global problem in terms of the local data of this problem. Bibliography: 7 titles.