Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia.

Convexity and weak closeness of the set of Φ-superharmonic functions in a bounded Lipschitz domain in Rn is considered. By using the fact of that Φ-superharmonic functions are just the solutions to an obstacle problem and establishing some special properties of the obstacle problem, it is shown that if Φ satisfies Δ2-condition, then the set is not convex unless Φ(r) = Cr2 or n = 1. Nevertheless, it is found that the set is still weakly closed in the corresponding Orlicz-Sobolev space.

We discuss the potential theory of optimal stopping for a standard process and an unbounded reward function. This is applied to Brownian motion constrained to a N(m, σ2) distribution at time 1. Boyce [2] has discovered, via computer, various interesting features of this example. We provide direct proofs for some of them, in particular for the qualitative jump of the optimal strategy as the variance σ2 passes the critical value 1.