Page 132 - Textos de Matemática Vol. 34

122 JOSE´ MIGUEL URBANO
respect to |∇u|. The latter is of particular interest since the equation in (1.2) might be either degenerate or singular.
A striking result of Moser [17] is that the Harnack estimate continues to hold for non-negative, local weak solutions of the parabolic analog of (1.1). As before, it can be used to prove that weak solutions are locally H¨older continuous. Since the linearity is immaterial to the proof, one might expect, as in the elliptic case, an extension of these results to quasilinear equations of the type
(1.4) ut−divA(x,t,u,∇u)+B(x,t,u,∇u)=0 in ΩT ≡Ω×(0,T),
where the structure condition is as in (1.3). Surprisingly however, Moser’s proof could be extended only for the case p = 2, i.e., for equations whose principal part has a linear growth with respect to |∇u|. This appears in the work of Aronson and Serrin [1] and Trudinger [21]. The methods of DeGiorgi also could not be extended: Ladyzenskaja et al. [14] proved that solutions of (1.4) are H¨older continuous, pro- vided the principal part has exactly a linear growth with respect to |∇u|. Analogous results were established by Kruˇzkov [11], [12] and by Nash [18] by entirely different methods. Thus it appears that unlike the elliptic case, the degeneracy or singular- ity of the principal part plays a peculiar role, and for example, for the p-Laplace equation
ut − div |∇u|p−2∇u = 0 , p > 1
one could not establish whether non-negative weak solutions satisfy the Harnack estimate or whether a solution is locally H¨older continuous.
In the mid-1980s, some progress was made in the theory of degenerate PDEs
of the type of (1.4), for p > 2: it was shown by DiBenedetto [6] that solutions
are locally H¨older continuous. Surprisingly, the same techniques can be suitably
modified to establish the local H¨older continuity of any local solution of quasilin-
ear porous medium-type equations. These modified methods in turn, are crucial in
proving that weak solutions of the p-Laplace equation are of class C1,α(ΩT ). There- loc
fore understanding the local structure of the solutions of (1.4) has implications to the theory of equations with degeneracies quite different from (1.4). In the early 1990s the theory was completed and it was established that solutions of (1.4) are H¨older continuous for all p > 1 (see [3], and [7] for a complete account). These results follow, one way or another, from a single unifying idea which can be called intrinsic scaling. The diffusion processes in the equations evolve in a time scale determined instant by instant by the solution itself, so that, loosely speaking, they can be regarded as the heat equation in their own intrinsic time-configuration. A precise description of this fact as well as its effectiveness is linked to its technical implementations. The purpose of this note is to illustrate the relation between the structure of some nonlinear PDEs, that exhibit some form of degeneracy or singularity, and the geometric configuration in which they have to be analyzed.