The equivalence of the uniqueness of the Dirichlet problem and the maximum principle for elliptic systems (Q1329324)

The main result of this paper is stated in the following theorem: Let \(D\) be a bounded domain in \(\mathbb{R}^ n\). Let \(U\) be a complex-valued vector function satisfying \((*)\) \(\Delta U + C(x)U = 0\) in \(D\), where \(C(x)\) is a complex-valued bounded Hölder continuous matrix. Then a necessary and sufficient condition for existence of a positive constant \(K\) such that \(\| U \|_ D \leq K \| U \|_{\partial D}\) for all \(C^ 2(D) \cap C(\overline D)\) solutions \(U\) of \((*)\) is that the Dirichlet problem \(\Delta U + C(x)U = 0\) in \(D\), \(U = 0\) on \(\partial D\), has only the trivial solution \(U \equiv 0\).