Boundary regularity for a parabolic obstacle type problem

Abstract

We study the regularity of the free boundary, near contact points with the fixed boundary, for a parabolic free boundary problem Δu
–∂u/∂t = χ{u≠0} in Qr+ = {(x, t) ∈ Br x (–_r_2, 0); _x_1 > 0}, u = f(x, t) on {_x_1 = 0} ∩ Qr. We will show that under certain regularity assumptions on the boundary data f the free boundary is a _C_1 manifold up to the fixed boundary. We also show that the _C_1 modulus of continuity is uniform for a certain, and specified, subclass of solutions.