Everywhere regularity of solutions to a class of strongly coupled degenerate parabolic systems

DISCRETE AND CONTINUOUS
DYNAMICAL SYSTEMS
Supplement Volume 2005

Website:
pp. 576–586

PARTIAL REGULARITY OF SOLUTIONS TO A CLASS OF
STRONGLY COUPLED DEGENERATE PARABOLIC SYSTEMS

Dung Le
Department of Applied Mathematics
University of Texas at San Antonio
6900 North Loop 1604 West
San Antonio, TX 78249, USA
Abstract. Using the method of heat approximation, we will establish partial regularity results for bounded weak solutions to certain strongly coupled degenerate
parabolic systems.

1. Introduction. The aim of this paper is to study the partial regularity for weak
solutions of nonlinear parabolic systems of the form
ut = div(a(x, t, u)Du) + f (x, t, u, Du),
(1.1)
n+1
n
in a domain Q = Ω × (0, T ) ⊂ R
, with Ω being an open subset of R , n ≥ 1.
The vector valued functions u, f take values in Rm , m ≥ 1. Du denotes the spatial
nm
derivative of u. Here, a(x, t, u) = (Aαβ
, Rnm ).
ij ) is a matrix in Hom(R

1,0
A weak solution u to (1.1) is a function u ∈ W2 (Q, Rm ) such that
f (x, t, u, Du)φ dz

[−uφt + a(x, t, u)DuDφ] dz =
Q

Q

for all φ ∈ Cc1 (Q, Rm ). Here, we write dz = dxdt.
It has been known that, in the case of systems of equations (i.e. m > 1), one
cannot expect that bounded weak solutions of (1.1) will be Hăolder continuous everywhere (see [7]). Partial regularity for (1.1), when a is regularly elliptic, was
considered by Giaquinta and Struwe in [5].
In this paper, we study the partial regularity for (1.1) when certain degeneracy
is present. In particular, we consider the case when a ceases to be regular elliptic at
certain values of u. Strongly coupled systems of porous medium type are included
here.
For the sake of simplicity, we will only consider the homogeneous case f ≡ 0,
and assume that a(x, t, u) depends only on u. The nonhomogeneous case can be
treated similarly modulo minor modifications. In fact, we will assume the following
structural conditions on (1.1).
(A.1): There exists a C 1 map g : Rm → Rm , with Φ(u) = Du g(u), such that
for some positive constants λ, Λ > 0 there hold
a(u)Du · Du ≥ λ|Dg(u)|2 ,

|a(u)Du| ≤ Λ|Φ(u)||Dg(u)|.

2000 Mathematics Subject Classification. Primary: 35K65; Secondary: 35B65.
Key words and phrases. Parabolic systems, Degenerate systems, Partial Hă
older regularity.

The author is partially supported by NSF Grant #DMS0305219, Applied Mathematics
Program.

576


PARTIAL REGULARITY OF SOLUTIONS

577

(A.2): (Degeneracy condition) Φ(0) = 0. There exist positive constants C1 , C2
such that
C1 (|Φ(u)| + |Φ(v)|)|u − v| ≤ |g(u) − g(v)| ≤ C2 (|Φ(u)| + |Φ(v)|)|u − v|.

(A.3): (Comparability condition) For any β ∈ (0, 1), there exist constants
C1 (β), C2 (β) such that if u, v ∈ Rm and β|u| ≤ |v| ≤ |u|, then C1 (β)|Φ(u)| ≤
|Φ(v)| ≤ C2 (β)|Φ(u)|.
(A.4): (Continuity condition) Φ(u) is invertible for u = 0. The map a(u)Φ(u)−1
is continuous on Rm \{0}. Moreover, there exists a monotone nondecreasing
concave function ω : [0, ∞) → [0, ∞) such that ω(0) = 0, ω is continuous at
0, and
|a(v)Φ(v)−1 − a(u)Φ(u)−1 | ≤ (|Φ(u)| + |Φ(v)|)ω(|u − v|2 ),

(1.2)

|Φ(u) − Φ(v)| ≤ (|Φ(u)| + |Φ(v)|)ω(|u − v|2 )
(1.3)
m
for all u, v ∈ R .
In (A.4), we use ω to quantify our continuity hypothesis on a(u)Φ−1 (u). We

would like to remark that the existence of the function ω also comes from the continuity of a(u)Φ−1 (u) and Φ(u) ( see [4, page 169]). To avoid certain technicalities
in the presentation of our proof, we assume here (1.3). We will see at the end of
this paper that it is not necessary.
Systems that satisfy the above structural conditions include the porous medium
type systems: a behaves like certain powers of the norm |u|. In this case, one may
consider g(u) = |u|α/2 u for some α > 0.
As we mentioned before, partial regularity results for the regular case, when g
is the identity map, were established in [5]. There are three essential ingredients in
their proof. The first element is an inequality of Caccioppoli, or reverse-Poincar´e,
type. The second element of the proof is Campanato’s decay estimate for the
averaged mean square deviation of solutions to systems of equations with constant
coefficients. Finally, the technique of freezing the coefficients was used. One then
had to control the deviation of the original solution of (1.1) from that of system of
constant coefficients. In this step, a variant version of the famous Gehring reverse
Hăolder inequality played a crucial role.
Given the above assumptions, Cacciopoli’s inequality is not available. More
seriously, the crucial Gehring reverse inequality, and thus higher integrability of
Du, does not seem to hold anymore. However, we are able to prove the following
partial regularity result.
Theorem 1. Let u be a bounded weak solution to (1.1). Set
Reg(u) = {(x, t) ∈ Ω × (0, T ) : u is Hă
older continuous in a neighborhood of (x, t)}

and Sing(u) = Ω × (0, T )\Reg(u). Then Sing(u) ⊆ Σ1

Σ2 , where

Σ1 = {(x, t) ∈ Ω × (0, T ) : lim inf |(u)QR (x,t) | = 0},
R→0

Σ2 = {(x, t) ∈ Ω × (0, T ) : lim inf
R→0

QR

|u − (u)QR (x,t) |2 dz > 0}.


578

DUNG LE

Here, for each R > 0, QR (x, t) = BR (x) × (t − R2 , t) and (u)QR (x,t) =

u dz.
QR (x,t)

Moreover, H n (Σ2 ) = 0, where H n is the n-dimensional Hausdorff measure.

We would like to describe briefly our approach here. In order to deal with this
degenerate situation, one needs to find another way to avoid the unavailable Gehring
lemma. Recently, the method of A-harmonic approximation has been successfully
used by Duzaar et al. [2, 3] to treat regular elliptic systems and p-Laplacian systems.
One of the advantages of this method is that it is more elementary and avoids the
technical difficulties associated with the use of the Gehring lemma, the missing
stone in our case. Inspired by this method, we introduce in Section 2 its parabolic
variance: the heat approximation. Basically, the point is to show that a function
which is approximately a solution to a heat equation in a parabolic cylinder will be
L2 -close to some heat solution in a smaller cylinder. We also present a parabolic
version of Giaquinta’s lemma [2, Lemma A.1].

Finally, we prove Theorem 1 in Section 3. We apply a scaling argument to reflect
the degeneracy of (1.1) and make use of the above heat approximation lemma in
each scaled cylinders. We start with the key assumption that the solution u is
not averagely (in a scaled cylinder) too close to the singular point u = 0. We
then derive a decay estimate for the averaged mean square deviation of g(u). This
estimate allows us to show that u is still averagely far away from the singular
point in a smaller cylinder, and the argument then can be repeated. Moreover, we
then obtain the decay estimates for the averaged mean square deviation of u in a
sequence of scaled and nested cylinders. It is then standard to conclude that u is
locally Hăolder continuous from these estimates.
2. A-heat approximation. In this section, we present the parabolic version of the
A-harmonic approximation lemma formulated in [2]. The proof is a straightforward
modification of that for the elliptic case modulo some careful choice of the pertinent
norms. We present some details here for the sake of completeness.
Fix (x0 , t0 ) ∈ Rn+1 . For each R > 0, we consider the cylinder QR = BR (x0 ) ×
(t0 − R2 , t0 ). Let V (QR ) be the space of functions g ∈ W21,0 (QR , Rm ) with norm
g

V (QR )

g 2 (x, t) dx +

= sup R−n−2
t

BR

QR

|Dg|2 dz.

(2.4)

For A ∈ Bil(Hom(Rnm , Rnm )) and φ ∈ C 1 (QR ) = C 1 (QR , Rm ), we define
LA (g, φ, QR ) =
QR

[A(Dg, Dφ) − gφt ] dz,

and
∆(LA , g, QR ) = sup{|LA (g, φ, QR )| : φ ∈ Cc1 (QR ), sup |Dφ| ≤
QR

1
1
, |φt | ≤ 2 }.
R
R

We shall consider the set of A-heat functions
H(A, QR ) = {H ∈ V (QR ) : LA (H, φ, QR ) = 0, ∀φ ∈ Cc1 (QR )}.

The following A-heat approximation lemma is the parabolic version of [2, Lemma
2.1].


PARTIAL REGULARITY OF SOLUTIONS

579

Lemma 2.1. Consider fixed λ, Λ > 0. For any given ε > 0 there exists δ ∈ (0, 1]
that depends on λ, Λ, ε and has the following property:
for any A ∈ Bil(Hom(Rnm , Rnm )) satisfying
A(u, u) ≥ λ|u|2 ,

|A(u, v)| ≤ Λ|u||v| for all u, v ∈ Rnm ,

(2.5)

≤ 1, and |∆(LA , g, Qρ )| ≤ δ,

(2.6)

for any g ∈ V (Qρ ) satisfying
g

V (Qρ )

then there exists v ∈ H(A, Qρ ) such that
ρ−2
Qρ

|v − g|2 dz ≤ ε and

Qρ

|Dv|2 dz ≤ 1.

(2.7)

Proof. We assume first that ρ = 1. If the conclusion is false, we can find ε > 0,
{Ak } each satisfying (2.5) and {gk } ⊂ V (Q1 ) such that

Q1

|vk − gk |2 dz ≥ ε for all vk ∈ H(Ak , Q1 ) with

Q1

|Dvk |2 dz ≤ 1,

(2.8)

and
gk

|LAk (gk , φ, Q1 )| ≤

V (Q1 )

≤ 1,

(2.9)

1
sup(|Dφ| + |φt |), any φ ∈ Cc1 (Q1 ).
k Q1

(2.10)

By (2.9), we can extract a weakly convergent sequence still denoted by {gk },
g ∈ V (Q1 ) and A such that:
gk → g weakly in V (Q1 ), gk → g in L2 (Q1 ), Ak → A, and g

V (Q2 )

≤ 1.

For φ ∈ Cc1 (Q1 ), we have
LA (g, φ, Q1 ) = LA (g − gk , φ, Q1 ) +

Q1

(A − Ak )(Dgk , Dφ) dz + LAk (gk , φ, Q1 ).

Letting k → ∞ and using (2.10), we see that g ∈ H(A, Q1 ). We then consider
the solution vk in Q1 of the problem
LAk (vk , φ, Q1 ) = 0 for all φ ∈ Cc1 (Q1 ),
with vk = g on the parabolic boundary of Q1 . Set φk = vk − g. We have
λ
Q1

On the other hand,

|Dvk − Dg|2 dz ≤

Ak (Dφk , Dφk ) dz.
Q1

580

DUNG LE

Ak (Dφk , Dφk ) dz =
Q1

Q1
B1

= −

B1

vk (φk )t dz −

Q1

Q1

vk (φk )t dz −

Q1

vk φk dx +

+
Q1

(A − Ak )(Dg, Dφk ) dz −

Q1

(A − Ak )(Dg, Dφk ) dz −

=

A(Dg, Dφk ) dz

(A − Ak )(Dg, Dφk ) dz

Q1

=

Ak (Dg, Dφk ) dz
Q1

Ak (Dg, Dφk ) dz

Q1

vk φk dx +

= −

Ak (Dvk , Dφk ) dz −

≤ |(A − Ak )|

Q1

B1

1
2

φ2k dx +

B1

φk (φk )t dz
Q1

φ2k dx

|Dg||Dφk | dz.

We conclude that Dvk −Dg L2 (Q1 ) → 0. Since vk = g on the parabolic boundary
of Q1 , a simple use of the Poincar´e inequality shows that vk − g L2 (Q1 ) → 0.
Let Vk = vk /mk , where mk = max{ Dvk L2 (Q1 ) , 1}. Since lim Dvk L2 (Q1 ) =
Dg

L2 (Q1 )

≤ 1, we have mk → 1. Thus, Vk − g

L2 (Q1 )

→ 0 and

Q1

|DVk |2 dz ≤

1. This and the fact that gk − g L(Q1 ) → 0 contradict (2.8). The proof for the
case ρ = 1 is complete.
Finally, for ρ = 1, we will use the following scalings:
x = x0 + RX, t = t0 + R2 T ⇒ |dx| = Rn |DX|, dt = R2 dT.
1
1
g(x0 + RX, t0 + R2 T ) = g(x, t), φ(X, T ) = φ(x, t).
R
R
Therefore DX g¯ = Dx g, g¯T = Rgt , DX φ = RDx φ, φT = R2 φt ,
g¯(X, T ) =

g¯

V (Q1 )

= g

V (QR ) ,

Q1

¯ − g¯|2 dz = R−2
|h

QR

|h − g|2 dz,

and
[−¯
g φt + A(DX g¯, DX φ] dz = RLA (g, φ, QR ).

LA (¯
g , φ, Q1 ) =
Q1

From these relations, it is easy to see that the case ρ = 1 follows from the above
proof.
We then have the following parabolic version of Giaquinta’s lemma [2, Lemma
A.1]
Lemma 2.2. Suppose that A satisfies the conditions of Lemma 2.1. For any ǫ > 0,
there exists a constant C depending on λ, Λ, ε such that

inf





1
2

Qρ
2

|H − g|2 dz

≤ Cρ ∆(LA , g, Qρ ) + ǫρ g

: H ∈ H(A, Qρ ) and
V (Qρ ) .

Qρ

|DH|2 dz ≤ g

2
V (Qρ )






PARTIAL REGULARITY OF SOLUTIONS

581

Proof. Assume first that ρ = 1. Let δ be the constant found in Lemma 2.1. We
consider first the case when
∆(LA , g, Q1 ) ≤ δ g

We then have
|H − g|2 dz

inf
Q1

= g

V (Q1 )
Q1

≤ǫ g

|

1
2

: H ∈ H(A, Q1 ) and
H

g

V (Q1 ) .

V (Q1 )

−

g
g

V (Q1 )

|2 dz

Q1

|DH|2 dz ≤ g

2
V (Q1 )

1
2

: H ∈ H(A, Q1 )

V (Q1 ) .

Otherwise, by the Poincar´e inequality, we have

inf
Q1

|H − g|2 dz

1
2

1/2

: H ∈ H(A, Q1 )

≤

Q1

|(g)Q1 − g|2 dz

1/2

≤ supτ
≤C g

Q1
V (Q1 )

|(g)Q1 − (g(x, τ ))B1 |2 dz
≤

1/2

+

C
δ ∆(LA , g, Q1 ).

Q1

|(g(x, t))B1 − g|2 dz

Combining these estimates, we prove the lemma when ρ = 1. The case ρ = 1
follows from the scaling argument used in Lemma 2.1.
3. Partial Regularity Theorem. We now consider the system (1.1). For R > 0,
let µ = supQ |u|. We can assume that µ > 0. Let V0 be a constant vector in Rm
with 34 µ ≤ |V0 | ≤ µ. We will make a change of variables
x
¯ = x − x0 ,

s = Φ2µ t − t0 , with Φµ = sup|u|≤µ |Φ(u)|.

(3.11)

From now on we will work with the new variables (¯
x, s) in the cylinders QR =
BR × JR , BR = {¯
x : |¯
x| ≤ R}, JR = (−R2 , 0). The system (1.1) becomes
us =

1
divx (a(u)Du).
Φ2µ

(3.12)

Let φ be a cut-off function in Q2R . That is, φ ∈ Cc1 (Q2R ) with φ ≡ 1 in QR and
|Dφ| ≤ 1/R, |φs | ≤ 1/R2 . Testing (3.12) with (u − V0 )φ and using the fact that
|a(u)Du(u − V0 )Dφ| ≤ ε|Dg(u)|2 + C(ε)|Dφ|2 Φ2µ |u − V0 |2 ,

which holds due to (A.1) and (A.3), we get the following degenerate Cacciopoli’s

type estimate.
Φ2µ sup

s∈JR

BR

|u − V0 |2 dx +

λ
2

QR

|Dg(u)|2 d¯
z≤

CΦ2µ
R2

Q2R

|u − V0 |2 d¯
z,
(3.13)

where d¯
z = d¯
xds. By (A.2), the above yields
R2 g(u) − g(V0 )

2
V (QR )

≤ CΦ2µ

Q2R

|u − V0 |2 d¯
z.

(3.14)


582

DUNG LE

In order to apply Lemma 2.2, we define
1
Φ(V0 )a(V0 )Φ−1 (V0 ).
(3.15)
Φ2µ
Thanks to (A.3), there is a positive constant λ such that λΦµ ≤ |Φ(V0 )| ≤ Φµ .
This and (A.1) show that A satisfies the assumptions of Lemma 2.2.
A=

We also set IR =
QR

that

|u − V0 |2 d¯
z for each R > 0. Our first step is to show

Lemma 3.3. For φ ∈ Cc1 (QR ), with |Dφ| ≤ 1/R and |φt | ≤ 1/R2 , we have
1

R2 |LA (g(u), φ, QR )| ≤ CΦµ ω(I2R ) (I2R ) 2 .

(3.16)

Proof. We note that
[−(g(u) − g(V0 ))φs + A(Dg(u), Dφ)] d¯
z.

LA (g(u), φ, QR ) =
QR

Multiplying (3.12) by Φ(V0 ) and testing by φ, we get

QR

[−(Φ(V0 )(u − V0 )φs +

1
Φ(V0 )a(x, u)DuDφ] d¯
z = 0.
Φ2µ

Hence,
ADg(u) −

LA (g(u), φ, QR ) =
QR

−

QR

1
Φ(V0 )a(x, u)Du, Dφ d¯
z
Φ2µ

[g(u) − g(V0 ) − Φ(V0 )(u − V0 )]φs d¯
z.

Using (A.4), for u = 0, we estimate the first integrand on the right by
1
Φ2µ

Φ(V0 ){a(V0 )Φ(V0 )−1 − a(x, u)Φ(u)−1 }Dg(u), Dφ
≤ Φ1µ |Dφ||Dg(u)||a(V0 )Φ(V0 )−1 − a(u)Φ(u)−1 | ≤ 2|Dφ||Dg(u)|ω(|u − V0 |2 ).
Similarly, the second integrand can be estimated by
1
0

(Φ(tu + (1 − t)V0 ) − Φ(V0 ))(u − V0 )dt sup |φs | ≤ C

Φµ
ω(|u − V0 |2 )|u − V0 |.
R2
(3.17)

Thus,
|LA (g(u), φ, QR )|

≤

2
R

+
≤

QR
CΦµ
R2

1
R
CΦ
+ R2µ

|Dg(u)ω(|u − V0 |2 )| d¯
z
QR

QR

ω(|u − V0 |2 )|u − V0 | d¯
z

z
|Dg(u)|2 d¯
2

QR

1
2

QR
2

ω (|u − V0 | ) d¯
z

This, (3.13) and the concavity of ω give the lemma.

z
ω 2 (|u − V0 |2 )| d¯
1
2

2

QR

1
2

|u − V0 | d¯
z

1
2


PARTIAL REGULARITY OF SOLUTIONS

Next, we have a decay estimate for g(u).

583

Hereafter, we will denote fR =

f d¯
z.
QR

Lemma 3.4. For ǫ > 0 and σ ∈ (0, 1/4), we have

QσR

|g(u) − g(u)σR |2 d¯
z ≤ C[σ 2 + σ −n−2 (ω 2 (IR ) + ǫ2 )]Φ2µ IR .

(3.18)

Proof. From (A.1), we see that A satisfies the assumption of Lemma 2.2. Therefore,
we can find H ∈ H(A, QR ) such that
Q2R

Q2R

|H − g(u)|2 d¯
z ≤
≤

|DH|2 d¯
z ≤ g(u) − g(V0 )

2
V (Q2R )

and

CR2 |LA (g(u), φ, Q2R )|2
+Cε2 R2 g(u) − g(V0 ) 2V (Q2R )
CΦ2µ [ω 2 (I4R )I4R + ǫ2 I4R ],

using (3.14), (3.16). For σ ∈ (0, 1), using a decay result in [1] for the function H,
we have

QσR

|g(u) − g(u)σR |2 d¯
z ≤

QσR

|g(u) − HσR |2 d¯
z

≤

QσR

|g(u) − H|2 d¯
z+

≤

QσR

|g(u) − H|2 d¯
z + σ2

QσR

|H − HσR |2 d¯
z

QR

|H − HR |2 d¯
z.

By the general Poincar´e inequality [8, Lemma 3], we also have

QR

|H − HR |2 d¯
z ≤ CR2

Q2R

|DH|2 d¯
z ≤ CR2 g(u) − g(V0 )

2
V2R

2
≤ CΦ2µ I4R
.

Combining these estimates and the fact that

QσR

|g(u) − H|2 d¯
z ≤ Cσ −n−2

Q2R

|g(u) − H|2 d¯
z,

we obtain the lemma.
Finally, we will show that u is still averagely far away from the singular point in
smaller cylinders so that the above argument can be repeated. More importantly,
this will allow us to derive the decay estimate for u from that of g(u).
Lemma 3.5. Assume that
3
µ ≤ |V0 | ≤ µ, IR ≤ ε2 µ2 .
(3.19)
4
For any given θ ∈ (0, 1), there exist ε, σ > 0 sufficiently small such that there is
a sequence of vectors {Vi } satisfying
i): Vi = (u)Qσi R and |Vi | ≥ 21 µ for all i > 1.
ii):
Qσi+1 R

|u − Vi+1 |2 d¯
z ≤ θ2

Qσ i R

|u − Vi |2 d¯
z,

(3.20)


584

DUNG LE

|Vi+1 − Vi |2 ≤ (σ −n−2 + θ)2

Qσi R

|u − Vi |2 d¯
z.

(3.21)

Proof. Given any θ ∈ (0, 1). By (3.18), (3.19) and the continuity of ω, we can find
σ, ǫ, ε, in that order, sufficiently small such that

QσR

|g(u) − g(u)σR |2 d¯
z ≤ [θεΦµ µ]2 .

(3.22)

We will fix such σ and make ε even smaller later on. Let V¯ be a vector such that
¯
g(V ) = g(u)σR . Thanks to (A.2) and by choosing ε sufficiently small, we have
|g(V¯ ) − g(V0 )|2 ≤

QσR

(|g(u) − g(V¯ )|2 + |g(u) − g(V0 )|2 ) d¯
z ≤ (C + θ2 )[εΦµ µ]2 .

We also have |g(V¯ ) − g(V0 )| ≥ CΦµ |V¯ − V0 |. Therefore, |V¯ − V0 | ≤ (C + θ)εµ, so
that |V¯ | ≥ |V0 | − (C + θ)εµ ≥ 12 µ, if ε is sufficiently small.
By (A.2) again, we have |g(u) − g(V¯ )| ≥ CΦµ |u − V¯ | for some universal constant
C. If ε ≤ C, we derive from (3.22) that
Φ2µ

QσR

|u − uσR |2 d¯
z ≤ Φ2µ

QσR

|u − V¯ |2 d¯
z ≤ θ2 Φ2µ IR ,

which implies
|u − uσR |2 d¯
z ≤ θ2

QσR

QR

|u − V0 |2 d¯
z.

(3.23)

Hence,

|V0 − uσR |2 ≤

QσR

(|u − uσR |2 + |u − V0 |2 ) d¯
z ≤ (σ −n−2 + θ2 )

QR

|u − V0 |2 d¯
z.

(3.24)
We obtain ii) for i = 0. If we can establish that |Vi | ≥ 21 µ, then it is clear
that the above argument could be repeated with V0 being replaced by Vi = uσi R .
In particular, we take A = Φ12 a(Vi ) in (3.15) and replace V0 with Vi in the proof
µ
of Lemma 3.3. Using (A.3) we easily see that (3.18) continues to hold. Thus, let
assume that i) and ii) are true up to i − 1. The above argument applies to give
(3.20). We then have
1/2
2

Qσi R

|u − Vi | d¯
z

≤ θi εµ,

|Vi − Vi−1 | ≤

σ −(n+2) + θ2 θi−1 εµ.

Hence,
∞

|Vi | ≥ |V0 | −

σ −(n+2) +
i=0

θ2 θi εµ

3
≥ µ−
4

√

σ −(n+2) + θ2
µ
εµ ≥ ,
1−θ
2

if ε is sufficiently small. Our proof is complete by induction.
Combining Lemma 3.5, Lemma 3.4 and (A.2), we obtain the following decay
estimate for u in a sequence of nested cylinders.

PARTIAL REGULARITY OF SOLUTIONS

585

Proposition 1. Suppose that there exist R > 0 and V0 such that
and

QR

3
4µ

≤ |V0 | ≤ µ

|u − V0 |2 d¯
z ≤ ε2 µ2 .

Let ǫ > 0 be given. If ε is sufficiently small, then there exist σ0 ∈ (0, 1) and a
constant C such that for ρ = σ i R and Vi = uQρ , with 0 < σ < σ0 and i ≥ 1, there
holds

Qσρ

|u − Vi |2 d¯
z ≤ CK(σ, ǫ)

where K(σ, ǫ) = σ 2 + σ −n−2 [ω 2
Qρ

|u − Vi−1 |2 d¯
z,

Qρ

(3.25)

|u − Vi−1 |2 d¯
z + ǫ2 ].

This decay estimate allows us to give the proof of our main partial regularity
result.
Σ2 . For some β > 0 we

Proof of Theorem 1. We consider a point (x0 , t0 ) ∈ Σ1
have
βµ ≤ |(u)QR (x0 ,t0 ) | ≤ µ for all R > 0, and lim inf
R→0

QR

|u − uR |2 dz = 0.

Let V0 = (u)QR (x0 ,t0 ) . It is clear that we can replace the condition 34 µ ≤ |V0 | ≤ µ
in Proposition 1 by βµ ≤ |V0 | ≤ µ, where β can be any real in (0, 1). Hence, in
terms of the new variables (¯
x, s) defined in (3.11), we will verify that the conditions
of Proposition 1 are fulfilled and (3.25) holds.
Denote QR,à = BR (x0 ) ì [t0 12 R2 , t0 ]. We see that the conditions of Proposiµ

tion 1 are satisfied if we can show that
QR

|u − V0 |2 d¯
z is small. To see this, let

¯ = kR. By the choice of k, we see that QR,µ
⊆ QR . Hence,
k = min{1, Φµ } and R
¯
QR
¯

|u − V0 |2 d¯
z=

Therefore, lim inf R→0
QR

QR,µ
¯

|u − V0 |2 dz ≤

Φ2µ
k n+2

QR

|u − V0 |2 dz.

|u − uR |2 d¯
z = 0 and the conditions of Proposition 1

are verified. It is then standard to follow the argument of [5] and iterate (3.25)
to assert that u is Hăolder continuous in a neighborhood of (x0 , t0 ). The Hăolder
exponent may depend on (x0 , t0 ) and β, µ. Going back to the original variables
(x, t), we see that u is Hăolder continuous in a neighborhood of (x0 , t0 ).
To see that the singular set Σ2 is small. We simply use the following general
Poincar´e inequality by Struwe [8, Lemma 3].

QR

|u − uR |2 dz ≤ CR2

Q2R

|Du|2 dz.

We remark that this inequality was proven in [8] using only the fact that u satisfies
(1.1) with |a(u)Du| ≤ C|Du| for some constant C. This is true in our situation
because of (A.1) and the fact that u is bounded. Thus, Σ2 is a subset of
Σ∗2 = {(x, t) ∈ Ω × (0, T ) : lim inf
R→0

1
Rn

QR

|Du|2 dz > 0},


586

DUNG LE

which satisfies H n (Σ∗2 ) = 0 by a result of [6, page 70]. Our proof is then complete.
Finally, as we remarked in the introduction, the continuity condition (1.3) is not
necessary. A careful reading reveals that this condition is only used in the estimate
(3.17) of the proof of Lemma 3.3. Without (1.3), the right hand side of (3.17) could
ˆ µ = inf{|Φ(V )| : 1 µ ≤ |V | ≤ µ}.
ˆ µ |Φ02 | ω(|u−V0 |2 )|u−V0 |, where Φ
be replaced by C Φ
R
2
ˆ µ ω, and continue as before.
We then need only to replace the function ω with Φ
REFERENCES
[1] S. Campanato, Equazioni paraboliche del second ordine e spazi L2,θ (Ω, δ), Ann. mat. Pura
Appl., 73 (1966), 55–102.
[2] F. Duzaar and J. Grotowski, Optimal interior partial regularity for nonlinear elliptic systems:
The method of A-harmonic approximation, Manuscripta Math., 103 (2000), 267-298.
[3] F. Duzaar and G. Mingione, The p-harmonic approximation and the regularity of p-harmonic
maps, Calc. Var. to appear.
[4] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,
Princeton University Press, Princeton, NJ, 1983.
[5] M. Giaquinta and M. Struwe, On the partial regularity of weak solutions of nonlinear parabolic
systems, Math. Z., 179 (1982), 437–451.
[6] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003.

[7] O. John and J. Stara, Some (new) counterexamples of parabolic systems, Commentat math.
Univ. Carol., 36 (1995), 503–510.
[8] M. Struwe., On the Hă
older continuity of bounded weak solutions of quasilinear parabolic
systems, Manuscripta Math., 35 (1981), 125–145.

Received September, 2004; in revised February, 2005.
E-mail address: