Existence of a Classical Solution for a Multidimensional Filtration Problem

The authors consider the filtration problem 0(u)ut = u in T = ×(0, T), u(x, 0) = u0(x) in , u(x, t) = '(x) on @ ×(0, T), under the following assumptions: () 2W21 (R), kk (2) 1,R M, () 0, 0() > 0 if 2 (−1, 0), () = 1 if 2 [0,1), u0(x) 2 W21( ), '(x, t) 2 W2,1 1 (@ ×(0, T)). The main theorem of this paper is an existence theorem in a suitably small time interval [0, T1] for the solution u 2 H1+ ,(1+ )/2( T ) \W2,1 p ( T ), p 2, such that u has a regular free boundary (i.e. the level set {u = 0}). The key point to obtain the regularity of a free boundary is the proof of an a priori estimate of kutk1.