Let B(0)(s, t) be a Brownian pillow with continuous sample paths, and let h, u : [0, 1](2) -> R be two measurable functions. In this paper we derive upper and lower bounds for the boundary non-crossing probability
psi (u; h) := P{B(0)(s, t) + h(s, t) <= u(s, t), for all s, t is an element of [0, 1]}.
Further we investigate the asymptotic behaviour of psi(u; gamma h) with gamma tending to infinity and solve a related minimisation problem.