Continuity of parabolic $Q$-minima under the presence of irregular obstacles

Abstract

We study the pointwise continuity of functions satisfying a certain inequality, deducible from parabolic obstacle problems and valid also for parabolic Q-minima. Employing techniques going back to De Giorgi, we give conditions on the obstacles sufficient to imply continuity of the solution. These conditions are formulated as inequalities for the capacities of sub- and super-level sets of the obstacles, and thus also thin obstacles are considered.