The paper considers the existence of constructive solutions for Dirichlet's boundary value problem for open, bounded integrable sets \(\Omega\subseteq\mathbb{R}^n\) and uniformly continuous boundary conditions \(f : \partial\Omega \to \mathbb{R}\). It is shown that: in presence of Markov's Principle, the existence of a weak solution for the Dirichlet problem is equivalent to the Limited Principle of Omniscience (Theorem 4); in the absence of Markov's Principle, the existence of a strong solution for the Dirichlet problem implies the Weak Limited Principle of Omniscience (Proposition 9). It is in this sense that a proof of existence of solutions to the Dirichlet problem, and also to Navier-Stokes equations, of which the Dirichlet problem is a special case, cannot be constructive. It is left open whether the converse of Proposition 9, as well as the variant of Proposition 9 for weak existence, hold. Finally, in Section 5, conditions that ensure the constructive existence of a weak solution for the Dirichlet problem are isolated.