Page 140 - Textos de Matemática Vol. 34

130 JOSE´ MIGUEL URBANO
We can regard these cylinders as boxes moving inside Q(Rp,c0R) as the coor- dinates x of their centers range over the cube KR(ω). We may arrange L0 to be an integer and consider the cube Kc0R as the union, up to a set of measure zero, of LN0 disjoint cubes each of them congruent to KdR. Analogously, Q(Rp,c0R) is the disjoint union, up to a set of measure zero, of LN0 open boxes each congruent to Q(Rp, dR). Then we can view (x, 0) + Q(Rp, dR) as the blocks of a partition of Q(Rp,c0R). For a complete proof, see [7] or [8].
3.3. Stefan equations for the p-Laplacian. Consider now the equation
(3.5) vt − div |∇u|p−2∇u = 0 , v ∈ γ(u) ; p > 1 ,
where γ is a maximal monotone graph with a single jump at the origin; this equation generalizes to a nonlinear setting the modelling of the classical Stefan problem, that corresponds to the case p = 2 and describes a phase transition at constant temperature for a substance obeying Fourier’s law.
The case p > 2 was treated in [22] using the same geometric setting as in the case of the simple degenerate p-Laplacian, i.e., with γ(s) = s. Dealing with the maximal monotone graph involves a regularization procedure and the proof that the family of approximate solutions is equicontinuous. The consequence of estimating uniformly the regularization of the maximal monotone graph is the appearance of a third power (power 1) in the energy estimates. We thus have to deal with three powers (1, p and 2) and the price to be paid for the recovering of the homogeneity in the energy estimates is a dependence on the oscillation in the various constants that are determined along the proof. Owing to this fact we are no longer be able to exhibit a modulus of continuity but only to define it implicitly independently of the regularization. This is enough to obtain a continuous solution for the original problem, via Ascoli’s theorem, but the H¨older continuity, that holds in the case γ(s) = s, is lost.
For 1 < p < 2, the equation has a double singularity: its modulus of ellipticity |∇u|p−2 blows up at points where |∇u| = 0, and, concerning the time derivative, we have what can be heuristically described as γ′(0) = ∞. A bridge between the singularity in time and a degeneracy in space can be made through rewriting the equation in terms of v only, taking into account that u = γ−1(v) and γ−1 is now a well defined function, such that γ−1(s) = 0 for 0 ≤ s ≤ λ. It is clear that the time singularity in the u-equation becomes a space degeneracy for the v-equation. The appropriate geometry has to meet this mixed degenerate-singular character of the principal part of the equation. Given R > 0, sufficiently small such that Q(R,R1 ) ⊂ Ω , define
μ− := ess inf θ ; μ+ := ess sup θ ; ω := ess osc θ = μ+ − μ− 111
Q(R,R2 ) Q(R,R2 ) Q(R,R2 )
2T