Page 142 - Textos de Matemática Vol. 34

132 JOSE´ MIGUEL URBANO
4. Conclusion
The examples presented are evidence of a pattern in the choice of the appropriate intrinsic geometric to deal with parabolic PDEs with a singular/degenerate char- acter: a singularity in the principal part of an equation requires a rescaling in the space variables while a degeneracy requires a rescaling in the time variable. In the case of equation (3.5) with p > 2, we are in the presence of two types of degeneracy and that explains why a rescaling in time is enough. Also, in the case 1 < p < 2 but with no jumps (i.e., for γ(s) = s), there is only a singularity, so a rescaling in space suffices. In the case of (3.5) with 1 < p < 2, we have the equivalent of both a singularity and a degeneracy in the principal part and so both rescalings are needed. The case of several jumps is still widely open and only for p = 2 there are some partial results in [9].
The table below sums up what is known, from the point of view of the local regularity properties of weak solutions of (3.5), in terms of the range of values p can assume and the number of jumps of the maximal monotone graph γ; the original reference for each result is also indicated:
range for p
# of jumps of γ
regularity
paper
p=2
one
continuity
[2] and [5]
p=2
several
continuity
[9]
p>2
none
H¨older continuity
[6]
p>2
one
continuity
[22]
1<p<2
none
H¨older continuity
[3]
1<p<2
one
continuity
[10]
[1] [2] [3] [4] [5] [6] [7]
References
D. G. Aronson and J. Serrin, Local behaviour of solutions of quasilinear parabolic equations, Arch. Ration. Mech. Anal. 25 (1967), 81-123.
L. A. Caffarelli and L. C. Evans, Continuity of the temperature in the two-phase Stefan problem, Arch. Ration. Mech. Anal. 81 (1983), 199-220.
Y. Z. Chen and E. DiBenedetto, On the local behaviour of solutions of singular parabolic equations, Arch. Ration. Mech. Anal. 103 (1988), 319-345.
E. DeGiorgi, Sulla differenziabilita´ e l’analiticita´ delle estremali degli integrali multipli rego- lari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25-43.
E. DiBenedetto, Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Pura Appl. (4) 130 (1982), 131-176.
, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), 487-535.
, Degenerate Parabolic Equations, Springer Verlag, Series Universitext, New York,
1993.