Parabolic comparison principle and quasiminimizers in metric measure spaces

0.16 MB·English

by  Juha Kinnunen

PARABOLIC COMPARISON PRINCIPLE AND
QUASIMINIMIZERS IN METRIC MEASURE SPACES
JUHA KINNUNENANDMATHIASMASSON
Abstract. Wegiveseveralcharacterizationsofparabolic(quasisuper)-
3 minimizersinametricmeasurespaceequippedwithadoublingmeasure
1 and supporting a Poincar´e inequality. We also prove a version of com-
0 parison principle for super- and subminimizers on parabolic space-time
2 cylinders and a uniqueness result for minimizers of a boundary value
problem. We also give an example showing that the corresponding re-
n
a sultsdonot hold, in general, for quasiminimizers evenin theEuclidean
J case.

6
1

]
P 1. Introduction
A
Thisnotestudies propertiesof quasiminimimizers of parabolicvariational
.
h
inequalities of the p-Laplacian type in metric measure spaces. In the Eu-
t
a clidean case the prototype equation is
m
∂u
[ −div(|Du|p−2Du) = 0, 1 < p < ∞,
∂t
1
v and the corresponding variational inequality
4
7 ∂φ
7 p u dxdt+ |Du|pdxdt ≤ |Du+Dφ|pdxdt
∂t
3 ZZ ZZ ZZ
.
1 for every compactly supported test function φ. Roughly speaking, a quasi-
0 minimizer is a function which satisfies this condition up to multiplicative
3
constants. The precise definitions will begiven later. Our results and meth-
1
: ods also hold for more general variational kernels of the p-Laplacian type,
v
but as we shall see, in the class of quasiminimizers it seems to be enough to
i
X consider the prototype case.
r Themainpurposeofthestudyofquasiminimizersistoprovidearguments
a
in the calculus of variations that are only based on energy estimates and,
consequently, areapplicableinmoregeneralcontexts thanEuclideanspaces.
In particular, we do not always have the correspoding non-linear partial
differential equation available. This is a challenging problem already in
the Euclidean case with the Lebesgue measure and our results are relevant
already in that case. In the elliptic case quasiminimizers were introduced
by Giaquinta and Giusti in [GG1] and [GG2] and in the parabolic case by
Wieser in [W]. See also [Z1] and [Z2]. First they were used a tool in the
regularity theory, but later it has turned out that they have theory which is
of independent interest, see [B], [KM] and [KS].
2010 Mathematics Subject Classification. 30L99, 35K92.
1
2 JUHAKINNUNENANDMATHIASMASSON
Until recently, most of the results have been obtained for elliptic quasi-
minimizers. Recent papers [KMMP], [MMPP] and [MS] extend the regu-
larity theory of quasiminimizers to metric measure spaces with a doubling
measure and a Poincar´e inequality. These are rather standard assumptions
in analysis on metric measure spaces and they are also assumed to hold
throughout this note. We point out that there is a large literature on time
independent variational problems in metric measure spaces, but parabolic
quasiminimizers open an opportunuty to consider more general time depen-
dent variational problems. So far there are only few references and many
fundamental open problems remain.
In this note, we shall focus on two aspects. First, we give several char-
acterizations of parabolic quasisupermininimizers extending the results in
[KM] and [B]. We show, by an example, that quasiminimizers do not have
comparison principle and a boundary value problem does not have a unique
solution, in general. However, if we restrict our attention to the minimizers,
that are quasiminimizers with the constant one, then we have a parabolic
comparison principle for super- and subminimizers and a uniqueness result
for a boundary value problem for the minimizers. We prove these results in
the second part of the paper. It has been shown in [W] that the minimizers
have a certain amount of regularity in time. More precisely, the time deriv-
ative of a quasiminimizer belongs to the dualof the correspondingparabolic
Sobolev space. The corresponging result does not hold for super- and sub-
minimizers and a delicate smoothing argument is needed. In the Euclidean
case with the Lebesgue measure, the comparision principle is not only suffi-
cient butalso necessary condition for afunctionto bea superminimizer,but
in the metric space the theory for the parabolic obstacle problems is cur-
rently missing, see [KL], [KKP] and [KKS]. Our uniqueness result extends
results of [W] to the metric setting.
2. preliminaries
2.1. Doubling measure. Let (X,d) be a complete metric space. A mea-
sure µ is said to be doubling if there exists a universal constant C ≥ 1 such
µ
that
µ(B(x,2r)) ≤ C µ(B(x,r)),
µ
for every r > 0 and x ∈ X. Here B(x,r) denotes the open ball with the
center x andradiusr. We assumethatthemeasureis nontrivial in thesense
that 0< µ(B(x,r)) < ∞ for every x ∈ X and r > 0. Recall that a complete
metric space with a doubling measure is proper, that is, every closed and
bounded set is compact.
2.2. Upper gradients. We assume that Ω is an open and bounded subset
of X, although the boundedness is not always needed. Following [HK], a
non-negativeBorelmeasurablefunctiong :Ω → [0,∞]issaidtobeanupper
gradient of a function u : Ω → [−∞,∞] in Ω, if for all compact rectifiable
paths γ joining x and y in Ω we have
(2.1) |u(x)−u(y)| ≤ gds.
Zγ
3
In case u(x) = u(y) = ∞ or u(x) = u(y) = −∞, the left side is defined
to be ∞. Throughout the paper we assume that 1 < p < ∞. A property
is said to hold for p-almost all paths, if the set of non-constant paths for
which the property fails is of zero p-modulus. Following [S1], if (2.1) holds
for p-almost all paths γ in X, then g is said to be a p-weak upper gradient
of u. This refinement is related to the the existence of a minimimal weak
upper gradient and otherwise it does not play any other role in our study.
When u is a measurable function and it has an uppergradient g ∈ Lp(Ω),
it can beshown [S2] that there exists a minimal p-weak upper gradient of u,
we denote it by g , in the sense that g is a p-weak upper gradient of u and
u u
for every p-weak upper gradient g of u it holds g ≤ g µ-almost everywhere
u
in Ω. Moreover, if v = u µ-almost everywhere in a Borel set E ⊂ Ω, then
g = g µ-almost everywhere in E. For more on upper gradients in metric
v u
spaces we refer to [BB] and [H].
2.3. Newtonian spaces. Following [S1], for u∈ Lp(Ω), we define
kuk = kuk +kg k ,
1,p,Ω Lp(Ω,µ) u Lp(Ω,µ)
and N1.p(Ω) = {u : kuk < ∞}. An equivalence relation in N1,p(Ω) is
1,p,Ω
defined by saying that u ∼ v if ku−vkNe1,p(Ω) = 0. The Newtonian space
N1,p(eΩ)isdefinedtobethespaceN1,p(Ω)/ ∼,withthenormdefineedabove.
1,p
AfunctionubelongstothelocalNewtonian spaceN (Ω)ifitbelongsto
loc
N1,p(Ω′) for every Ω′ ⋐ Ω. Thenoteation Ω′ ⋐ Ω means that Ω′ is a compact
subset of Ω. The Newtonian space with zero boundary values is defined as
N1,p(Ω) = {u| : u∈ N1,p(X),u = 0 in X \Ω}.
0 Ω
1,p
In practice, this means that a function belongs to N (Ω) if and only if
0
its zero extension to X \ Ω belongs to N1,p(X). For more properties of
Newtonian spaces, see [S1, KKM, BB, H].
2.4. Poincar´e’s inequality. The space X is said to support a weak (1,p)-
Poincar´e inequality with 1 ≤ p < ∞, if there exist constants P > 0 and
0
τ ≥ 1 such that
1/p
1 1
|u−u |dµ ≤ P r gpdµ ,
µ(B(x,r)) B(x,r) 0 µ(B(x,τr)) u
ZB(x,r) ZB(x,τr) !
for every u ∈ N1,p(X), x ∈ X and r > 0. If the measure is doubling and
the space supports a (1,p)-Poincar´e inequality, then Lipschitz continuous
functions are dense in the Newtonian space. This will be useful for us, since
the test functions in the definition of quasiminimizers are assumed to be
Lipschitz continuous.
2.5. General assumptions. Throughoutthispaperweassume(X,d,µ)to
beacompletemetricspace,equippedwithapositivedoublingBorelmeasure
µ which supports a weak (1,p)-Poincar´e inequality for some 1≤ p < ∞.
4 JUHAKINNUNENANDMATHIASMASSON
2.6. Parabolic upper gradients and Newtonian spaces. From now on
we will denote the product measure by dν = dµdt. Whenever t is such that
u(·,t) ∈ N1,p(Ω), we define the parabolic minimal p-weak upper gradient of
u in a natural way by setting
g (x,t) = g (x),
u u(·,t)
at ν-almost every (x,t) ∈ Ω×(0,T)= Ω and wecall theparabolicminimal
T
p-weak upper gradient as the upper gradient.
WedefinetheparabolicNewtonianspaceLp(0,T;N1,p(Ω))tobethespace
of functions u(x,t) such that for almost every 0 < t < T the function u(·,t)
belongs to N1,p(Ω), and
T
p
ku(·,t)k dt < ∞.
1,p,Ω
Z
0
We say that u∈ Lp (0,T;N1,p(Ω)) if for every 0 < t < t < T and Ω′ ⋐ Ω
loc loc 1 2
we have u ∈ Lp(t ,t ;N1,p(Ω′)). Finally, we say that u ∈ Lp(0,T;N1,p(Ω))
1 2 c
if for some 0 < t < t < T, we have u(·,t) = 0 outside [t ,t ].
1 2 1 2
The next Lemma on taking limits of upper gradients will also be needed
later in this paper.
Lemma 2.1. Let u be such that g ∈ Lp (Ω ). Then the following state-
u loc T
ments hold:
(a) As s → 0, we have g → 0 in Lp (Ω ).
u(x,t−s)−u(x,t) loc T
(b) As ε → 0, we have g → 0 pointwise ν-almost everywhere in Ω
uε−u T
and in Lp (Ω ).
loc T
Proof. See Lemma 6.8 in [MS]. (cid:3)
3. Parabolic quasiminimizers
In this section we define parabolic quasiminimizers in metric measure
spaces and give several characterizations for them. We begin with a brief
discussion about the Euclidean case.
3.1. Euclidean case. There is a variational approach to the p-parabolic
equation
∂u
(3.1) −div(|Du|p−2Du) = 0, 1 < p < ∞.
∂t
To see this, first assume that u ∈ Lp (0,T;W1,p(Ω)) ∩ L2(Ω ) is a weak
loc loc T
solution of (3.1) and denote dz = dxdt. Let U ⋐ Ω and φ ∈ C∞(U). Then
T 0
|Du|pdz = |Du|p−2Du·Dudz
ZU ZU
∂φ
= |Du|p−2Du·(Du+Dφ)dz − u dz.
∂t
ZU ZU
from which it follows that
∂φ
(3.2) p u dz+ |Du|pdz ≤ |Du+Dφ|pdz.
∂t
ZU ZU ZU
In the last step we used Young’s inequality and rearranged terms.
5
Assume then that (3.2) holds for every U ⋐ Ω and let φ ∈ C∞(Ω ).
T 0 T
Then εϕ ∈ C∞(U) with ε> 0 for some U ⋐ Ω and we have
0 T
∂φ
εp u dz+ |Du|pdz ≤ |Du+εDφ|pdz
∂t
ZU ZU ZU
from which we conclude that
∂φ 1
p u dz+ (|Du|p−|Du+εDφ|p)dz ≤ 0.
∂t ε
ZU ZU
As ε → 0, by the dominated convergence theorem we arrive at
∂φ
u dz− |Du|p−2Du·Dφdz ≤ 0.
∂t
ZU ZU
Thereverse inequality follows by choosing −εφ as the test function and con-
sequently u is a weak solution of (3.1). If u∈ Lp (0,T;W1,p(Ω))∩L2 (Ω )
loc loc loc T
satisfies (3.2) for every U ⋐ Ω , we say that u is a parabolic minimizer.
T
Hence every weak solution of the p-parabolic equation is a parabolic mini-
mizer and, conversely, every parabolic minimizer is a weak solution of the
p-parabolic equation.
Let us then consider more general equations
∂u
(3.3) −divA(x,t,Du) = 0, 1 < p < ∞,
∂t
of the the p-Laplacian type, where A : Rn × R × Rn → R satisfies the
following assumptions:
(1) (x,t) 7→ A(x,t,ξ) is measurable for every ξ,
(2) ξ 7→ A(x,t,ξ) is continuous for almost every (x,t), and
(3) there exist 0 < c ≤ c < ∞ such that for every ξ and almost every
1 2
(x,t), we have
A(x,t,ξ)·ξ ≥ c |ξ|p and |A(x,t,ξ)| ≤ c |ξ|p−1.
1 2
As above, we can show that a weak solution of (3.3) satisfies
∂φ
u dz+c |Du|pdz ≤ c |Du|p−1|Du+Dφ|dz
1 2
∂t
ZU ZU ZU
for every U ⋐ Ω and φ ∈ C∞(U). Young’s inequality implies that there
T 0
are constants α > 0 and K ≥ 1, depending only on p, c and c , such that
1 2
∂φ
α u dz+ |Du|pdz ≤ K |Du|pdz
∂t
ZU ZU ZU
for every U ⋐ Ω and φ ∈ C∞(U). A function u ∈ Lp (0,T;W1,p(Ω))∩
T 0 loc loc
L2 (Ω ), which satisfies this property is called a parabolic quasiminimizer.
loc T
In contrast with the elliptic case, two parameters α and K are required in
the parabolic case to be able to obtain a notion of a quasiminimizer which
includes the whole class of equations of the type (3.3). Another possibility
would be to study qasiminimizers with more general variational integrals
than the p-Dirichlet integral, but we leave this for the interested reader.
6 JUHAKINNUNENANDMATHIASMASSON
Example 3.1. When K > 1, then being a quasiminimizer is not a local
property. Indeed, consider the function u : (0,∞)×(0,1) → R defined by
setting
i−1
x−(i−1) 1
u(x,t) = + ,
k j
j=1
X
when i − 1 < x ≤ i with i = 1,2,... This function is an elliptic quasi-
minimizer with some K > 1 when tested on intervals of finite length by a
criterion given in [GG2]. However, it fails to be a quasiminimizer on the
whole positive axis by the same criterion.
3.2. Metric case. The advantage of the notion of a quasiminimizer is that
it makes sense also in metric spaces and this enables us to develop the
theory of nonlinear parabolicpartial differential equations also in the metric
context.
Definition 3.2. Suppose that Ω ⊂ X is an open set and 0 < T < ∞.
Let 1 < p < ∞, α > 0 and K ≥ 1. We say that a function u ∈
Lp (0,T;N1,p(Ω)) ∩ L2 (Ω ) is a parabolic K-quasiminimizer, if for ev-
loc loc loc T
ery open U ⋐ Ω and for all functions φ∈ Lip(Ω ) such that {φ6= 0} ⊂U,
T T
we have
∂φ
(3.4) α u dν + gpdν ≤ K gp dν.
∂t u u+φ
ZU ZU ZU
A function u ∈ Lp (0,T;N1,p(Ω))∩L2 (Ω ) is a parabolic K-quasisuper-
loc loc loc T
minimizer,if (3.4)holdsforeveryopenU ⋐ Ω andforallnonnegativefunc-
T
tions φ ∈ Lip(Ω ) such that {φ > 0} ⊂ U. A function u is a parabolic K-
T
quasisubminimizer, if −u is a parabolic K-quasisuperminimizer. If K = 1,
thentheparabolicK-quasiminimizerandparabolicK-quasi-superminimizer
are called parabolic minimizer and parabolic superminimizer, respectively.
We proceed by proving characterizations for parabolic quasiminimizers.
The characterizations are proved for parabolic quasisuperminimizers. How-
ever, thereadershouldnotethatLemmas 3.3 through3.6can beformulated
andprovedalsoforK-quasisubminimizers,afterreplacingtheword‘nonneg-
ative’ in the proofs with the word ‘nonpositive’. In particular, this implies
that the corresponding results also hold for parabolic minimizers.
Lemma 3.3. A function u ∈ Lp (0,T;N1,p(Ω))∩L2 (Ω ) is a parabolic
loc loc loc T
K-quasisuperminimizer if and only if for every ν-measurable set E ⋐ Ω
T
and every nonnegative φ ∈ Lip(Ω ) such that {φ > 0} ⊂ E, we have
T
∂φ
α u dν + gpdν ≤ K gp dν.
∂t u u+φ
ZE ZE ZE
Proof. Assume that u is a parabolic K-quasisuperminimizer in Ω and let
T
E ⋐ Ω be a ν-measurable set. Let φ ∈ Lip(Ω) be nonnegative with the
property {φ > 0} ⊂ E. Since E is ν-measurable, and since u + φ ∈
Lp (0,T;N1,p(Ω))∩ L2 (Ω ), by the regularity of µ there exists an open
loc loc loc T
set U ⋐ Ω such that
T
ε
p
g dν < .
u+φ K
ZU\E
7
Moreover, {φ 6= 0} ⊂ E, and since φ is continuous with respect to time, we
have
ν((F \E)∩{∂φ/∂t 6= 0}) = 0.
Since u is a K-quasisuperminimizer, we arrive at
∂φ ∂φ
α u dν + gpdν ≤ α u dν + gpdν
∂t u ∂t u
ZE ZE ZU ZU
p p p
≤K g dν +K g dν ≤ K g dν +ε.
u+φ u+φ u+φ
ZE ZU\E ZE
Thisholdsforeveryε,andsoonedirectionoftheclaimistruebypassingε →
0. The other direction is immediate, since open sets are ν-measurable. (cid:3)
Thefollowing two characterizations areoften usefulinapplications andin
many cases can be taken as the definition of a parabolic K-quasiminimizer.
Lemma 3.4. A function u ∈ Lp (0,T;N1,p(Ω))∩L2 (Ω ) is a K-quasi-
loc loc loc T
superminimizer if and only if for every nonnegative φ ∈ Lip(Ω ) such that
T
{φ 6= 0} ⋐ Ω , we have
T
∂φ
(3.5) α u dν + gpdν ≤ K gp dν.
∂t u u+φ
Z{φ6=0} Z{φ6=0} Z{φ6=0}
Proof. Suppose first that (3.5) holds. Let φ∈ Lip(Ω ) be nonnegative with
T
{φ 6= 0} ⋐ Ω . Let U ⋐ Ω be an open set such that {φ 6= 0} ⊂ U. Since φ
T T
is continuous with respect to time, we have
ν({φ = 0, ∂φ/∂t 6= 0}) = 0,
and consequently
∂φ ∂φ
α u dν + gpdν = α u dν + gpdν + gpdν
∂t u ∂t u u
ZU ZU Z{φ6=0} Z{φ6=0} ZU∩{φ=0}
= K gp dν + gpdν ≤ K gp dν.
u+φ u u+φ
Z{φ6=0} ZU∩{φ=0} ZU
This shows that u is a parabolic K-quasisuperminimizer.
Suppose then that u is a K-quasisuperminimizer. Consider an open set
U ⋐ Ω and a nonnegative function φ ∈ Lip(Ω ) such that {φ 6= 0} ⊂ U.
T T
The set {φ 6= 0} is ν-measurable and {φ 6= 0} ⋐ Ω , and so by Lemma 3.3
T
we have (3.5). This completes the proof. (cid:3)
Lemma 3.5. A function u ∈ Lp (0,T;N1,p(Ω))∩L2 (Ω ) is a K-quasi-
loc loc loc T
superminimizer if and only if for every nonnegative φ ∈ Lip(Ω ) such that
T
suppφ ⊂ Ω , we have
T
∂φ
(3.6) α u dν + gpdν ≤ K gp dν.
∂t u u+φ
Zsuppφ Zsuppφ Zsuppφ
Proof. Suppose u is a K-quasisuperminimizer and let φ ∈ Lip(Ω ) be non-
T
negative such that suppφ⊂ Ω . Because φ is continuous, we have
T
ν({φ = 0,∂φ/∂t 6= 0}) = 0.
8 JUHAKINNUNENANDMATHIASMASSON
Since K ≥ 1, we can write
∂φ
u dν + gpdν
∂t u
Zsuppφ Zsuppφ
∂φ
≤ u dν + gpdν + gpdν
∂t u u
Z{φ6=0} Z{φ6=0} Zsuppφ\{φ6=0}
p p
≤ K g dν + g dν
u+φ u+φ
Z{φ6=0} Zsuppφ\{φ6=0}
p
≤ K g dν.
u+φ
Zsuppφ
Letthen φ ∈Lip(Ω ) besuch that {φ 6= 0} ⋐ Ω . For i = 1,2,..., define
T T
ψ = (φ−i−1) −(φ+i−1) .
i + −
Then for each i we have
ψ ∈ Lip(Ω ), {ψ 6= 0} ⊂ {φ 6= 0}, ν({φ 6= 0}\{ψ 6= 0}) → 0
i T i i
as i→ ∞, and
∂ψ ∂φ
i
= and g = g ,
∂t ∂t ψi φ
in the set {ψ 6= 0}. Moreover, since φ is continuous, we have suppψ ⊂
i i
{|φ| ≥ i−1}. Let ε > 0. By the absolute continuity of the integral and the
above properties, there exists a large enough i such that
∂φ ∂φ
u dν +C gpdν ≤ u dν +C gpdν +ε
∂t 1 u ∂t 1 u
Z{φ6=0} Z{φ6=0} Zsuppψi Zsuppψi
p p
≤ K g dν +ε≤ K g dν +2ε.
u−ψi u−φ
Zsuppψi Z{φ6=0}
Since this is true for any positive ε, by Lemma 3.4 u is a parabolic K-
quasisuperminimizer. (cid:3)
ItturnsoutthataftermollifyingaK-quasisuperminimizerintime, weob-
tain estimates for test functions which do not necessarily have to be smooth
in time. In what follows (·) denotes the standard time mollification
ε
ε
f (x,t) = η (s)f(x,t−s)ds.
ε ε
Z−ε
Lemma 3.6. Let u∈ Lp (0,T;N1,p(Ω))∩L2 (Ω ) be a parabolic K-quasi-
loc loc loc T
superminimizer. Then for every nonnegative φ∈ Lp(0,T;N1,p(Ω))∩L2(Ω )
T
such that {φ6= 0} ⋐Ω , we have
T
∂u
(3.7) −α εφdν + (gp) dν ≤ K (gp ) dν
∂t u ε u(x,t−s)+φ ε
Z{φ6=0} Z{φ6=0} Z{φ6=0}
for every small enough positive ε. Moreover, if u ∈ Lp (0,T;N1,p(Ω)) ∩
loc
L2 (Ω ), then the same inequality also holds for every nonnegative φ ∈
loc T
Lpc(0,T;N01,p(Ω))∩L2(ΩT).
Proof. See Lemma 2.3, Corollary 2.4 and Lemma 2.7 in [MMPP]. (cid:3)
9
4. Parabolic comparison principle
In this section we prove a comparison principle for minimizers, and as a
consequence obtain the uniqueness of parabolic minimizers. We emphasize
that it is essential to have K = 1 in the discussion below. Indeed, we give
an example which shows that the comparison principle and the uniqueness
result do not hold for parabolic K-quasiminimizers when K > 1.
Theorem 4.1. Let u ∈ Lp(0,T;N1,p(Ω))∩L2(Ω ) be a parabolic supermin-
T
imizer and let v ∈ Lp(0,T;N1,p(Ω))∩L2(Ω ) be a parabolic subminimizer,
T
both with the same constant α > 0 in (3.4). Suppose u ≥ v near the parabolic
boundary of Ω , in the sense that for almost every 0 < t < T we have the
T
lateral boundary condition (v(x,t)−u(x,t)) ∈ N1,p(Ω) and also the initial
+ 0
condition
1 h
(4.1) (v−u)2 dν → 0, as h→ 0.
h +
Z0 ZΩ
Then u≥ v ν-almost everywhere in Ω .
T
Proof. Assume that u and v are as in the formulation of the result. Let
t′ ∈ (0,T), and for h > 0 define
t−h, h ≤ t ≤ 2h,
h
1, 2h ≤ t ≤ t′−2h,
χ = 
h t′+h−t, t′−h ≤ t ≤ t′+h,
 2h
0, otherwise.


Choose the test functionφ = (vε −uε)+χh. By the assumptions made on
(u−v) nearthelateralboundaryofΩ ,weseethatforεandhsmallenough
+ T
φ ∈ Lp(0,T;N1,p(Ω))∩L2(Ω ). Therefore, since u is a superminimizer, by
c 0 T
Lemma 3.6 we have
∂u
−α ε(v −u ) χ dν + (gp) dν
∂t ε ε + h u ε
Z{φ6=0} Z{φ6=0}
p
≤ (g ) dν.
u(x,t−s)+(vε−uε)χh ε
Z{φ6=0}
On the other hand, since v is a subminimizer, we may use −φ as a test
function, to obtain
∂v
α ε(v −u ) χ dν + (gp) dν
∂t ε ε + h v ε
Z{φ6=0} Z{φ6=0}
p
≤ (g ) dν.
v(x,t−s)+(uε−vε)χh ε
Z{φ6=0}
By adding the above two inequalities together we have
1 ∂
α (v −u )2 χ dν + (gp) dν + (gp) dν
2∂t ε ε + h u ε v ε
(4.2) Z{φ6=0} Z{φ6=0} Z{φ6=0}
(cid:0) (cid:1)
p p
≤ (g ) dν + (g ) dν.
u(x,t−s)+(vε−uε)χh ε v(x,t−s)−(vε+uε)χh ε
Z{φ6=0} Z{φ6=0}
10 JUHAKINNUNENANDMATHIASMASSON
Next we note that by adding and subtracting, and then using Minkowski’s
inequality, and since χ does not depend on the spatial variable, we have
h
1/p
gp dν ≤ gpdν
Z{φ6=0} u(x,t−s)+(vε−uε)χh !ε  Z{φ6=0} v !
1/p  1/p
p p
+ g dν + g dν
u(x,t−s)−u vε−v
Z{φ6=0} ! Z{φ6=0} !
p
1/p 1/p
+ gp dν + gp (χ −1)pdν .
Z{φ6=0} u−uε ! Z{φ6=0} (vε−uε) h !  
ε
 
Lemma 2.1 implies that
limsup (gp ) dν ≤ gpdν,
ε→0 Z{φ6=0} u(x,t−s)+vε−uε ε Z{ϕ6=0} v
where we have denoted ϕ= (v−u) χ . Similarly, we see that
+ h
limsup (gp ) dν ≤ gpdν.
v(x,t−s)−vε+uε ε u
ε→0 Z{φ6=0} Z{ϕ6=0}
On the left hand side of (4.2) we integrate by parts and use the definition
of χ to obtain
h
1 ∂
α (v −u )2 χ dν
2 ∂t ε ε + h
Z{φ6=0} (cid:18) (cid:19)
α t′+h α 2h
= (v −u )2 dµdt− (v −u )2 dµdt
4h Zt′−h ZΩ ε ε + 2h Zh ZΩ ε ε +
Hence from (4.2) we obtain, after first taking the limit ε → 0 and then
h → 0, and also taking into account initial condition (4.1),
α
(v(x,t′)−u(x,t′))2 dµ+ gpdν + gpdν
4h + u v
ZΩ Z{ϕ6=0} Z{ϕ6=0}
≤ gpdν + gpdν
v u
Z{ϕ6=0} Z{ϕ6=0}
for almost every t′ ∈ (0,T). Since the upper gradient terms cancel each
other, the above implies that for almost every t′ ∈(0,T) we have
(v(x,t′)−u(x,t′)) =0
+
at µ-almost every x ∈ Ω. This completes the proof. (cid:3)
A parabolic minimizer is both a sub- and superminimizer, and so the
comparison principle above immediately implies the following uniqueness
result.
Theorem 4.2. Let u,v ∈ Lp(0,T;N1,p(Ω))∩L2(Ω ) be parabolic minimiz-
T
ers, both with the same constant α > 0 in (3.4). Suppose that for almost