An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary

Abstract

Concerning the obstacle-problem-like equation [Delta]u = [lambda]+/2 [chi]{u>0} [minus] [lambda][minus]/2 [chi]{u<0}, where [lambda]+ [ge] 0 and [lambda]+ + [lambda][minus] > 0, we prove regularity of the solution as well as an estimate on the Hausdorff dimension of the free boundary [Omega][cap] ([part]{u > 0} [cup] [part] {u < 0}). We apply this result to the Stefan-like equation [alpha][part]t max(v, 0) + [beta][part]t min(v, 0) [minus] [Delta]v = 0.