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 |Ω \ 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