Regularity Results for a Nonlinear Elliptic-Parabolic System with Oscillating Coefficients

In this paper we study the initial boundary value problem for the system div(sigma(u)del phi) = 0, u(t) Delta u = sigma(u)vertical bar del(phi)vertical bar(2). This problem is known as the thermistor problem which models the electrical heating of conductors. Our assumptions on sigma(u) leave open the possibility that lim inf(u ->infinity)(u) = 0, while lim sup(u ->infinity) sigma(u) is large. This means that s(u) can oscillate wildly between 0 and a large positive number as (u ->infinity). Thus our degeneracy is fundamentally different from the one that is present in porous medium type of equations. We obtain a weak solution (u, phi) with vertical bar del phi vertical bar, vertical bar del u vertical bar is an element of(2) L-infinity by first establishing a uniform upper bound for e epsilon(u) for some small epsilon. This leads to an inequality in del(,)(phi) from which the regularity result follows. This approach enables us to avoid first proving the Holder continuity of phi in the space variables, which would have required that the elliptic coefficient sigma(u) be an A(2) weight. As it is known, the latter implies that ln sigma(u) is "nearly bounded".