CHAPTER 1
Introduction and Results
The aim of this paper is twofold: the first, and main one, is to establish a rather
satisfying regularity theory for general parabolic systems with degenerate diffusion;
the second one, of more technical and specialized character, is to introduce a suitable
analog of the classical harmonic approximation methods of DeGiorgi’s pioneering
work [14] from the elliptic setting, which in turn is the key to a regularity theory
for degenerate parabolic systems.
1.1. A short introduction to the regularity of parabolic systems
A brief description of the present status of the regularity theory for general
parabolic systems cannot begin but with the observation that already in the case
of elliptic systems the so called partial regularity also called almost everywhere
regularity is in general the best one can usually expect, and therefore the same
happens in the case of parabolic systems. Indeed, since the important counterex-
ample of DeGiorgi [15] see also [50, 51, 59] it is known that when dealing with
general elliptic systems of the type
div a(Du) = 0 or div(A(x)Du) = 0
considered in an open subset Ω Rn, solutions might possess singularities, and
therefore everywhere regularity fails to hold in general. Instead, one can show
partial regularity of solutions, i.e. they are regular outside a negligible closed subset,
thereby called the singular set of the solution:
(1.1) u Cloc
1,α
(Ωu,
RN
) and \ Ωu| = 0
and we refer to [33, 35, 48] for an account of the theory and a list of references.
Eventually estimates for the Hausdorff dimension and boundary regularity can be
inferred [46, 47, 24]. Let us mention that related results for integral functionals
in the calculus of variations are obtained in [40, 41]. The above partial regularity
results for elliptic systems have been extended to the case of parabolic systems of
the type
(1.2) ∂tu = div a(Du)
and we refer for instance to [6, 8, 27, 29, 53, 54] for the most recent and sharp
theorems on the issue. Let us meanwhile remark that the system in (1.2), as all
the other parabolic ones in this paper, will be considered in the cylindrical domain
ΩT := Ω × (0,T ) ,
where Ω
Rn
is an open bounded domain with n 2. Such partial regularity
results are however obtained under a non-degenerate ellipticity assumption, both
in the case of systems and in that of variational integrals, and this amounts to
1
Previous Page Next Page