Uniqueness criteria for solutions of singular boundary value problems

Abstract:

In this paper we consider the equation \begin{equation}\tag {$1$} u''(t) + (k/t)u’(t) + Au(t) = 0,\quad 0 < t < T,\quad u(T) = 0,\end{equation} where $u:(0,T) \to D(A) \subset B$ is a Banach space valued function taking values in a dense subdomain $D(A)$ of the Banach space B. Here A is a closed (possibly unbounded) linear operator on $D(A)$ while k is a real constant. The differential equation is an abstract Euler-Poisson-Darboux equation. We give necessary and sufficient conditions on the point spectrum of A to insure uniqueness of the strong solution $u \equiv 0$ as well as sufficient conditions on the point spectrum to insure uniqueness of weak solutions. u is only required to satisfy (a) ${t^k}\left \| {u’(t)} \right \| \to {0^ + }$ as $t \to {0^ + }$ if $k > 1$, (b) ${t^k}\left \| {u’(t)} \right \| + {t^{k + 1}}\left \| {u(t)} \right \| \to 0$ as $t \to {0^ + },0 < k \leqslant 1$, (c) $t\left \| {u’(t)} \right \| + \left \| {u(t)} \right \| \to 0$ as $t \to {0^ + },k < 0$. The operator A need not possess a complete set of eigenvectors nor need one have a backward uniqueness theorem available for (1) for the Cauchy final value problem. Our techniques extend to the n-axially symmetric abstract equation \begin{equation}\tag {$2$} \sum \limits _{i = 1}^n {[{\partial ^2}u/\partial t_i^2 + ({k_i}/{t_i})\partial u/\partial {t_i}] + Au = 0.} \end{equation} The proofs rest upon an application of the Hahn-Banach Theorem and the consequent separation properties of ${B^\ast }$, the dual of B, as well as the completeness properties of the eigenfunctions of certain Bessel equations associated with (1).