PII. S0161171201004306 http://ijmms.hindawi.com © Hindawi Publishing Corp.
TOPOLOGICAL DEGREE AND APPLICATION TO A PARABOLIC
VARIATIONAL INEQUALITY PROBLEM
A. ADDOU and B. MERMRI
(Received 3 January 2000)
Abstract.We are interested in constructing a topological degree for operators of the form F=L+A+S, whereLis a linear densely defined maximal monotone map,Ais a bounded maximal monotone operators, andSis a bounded demicontinuous map of class(S+)with respect to the domain ofL. By means of this topological degree we prove an existence result that will be applied to give a new formulation of a parabolicvariational inequality problem.
2000 Mathematics Subject Classification. Primary 47H11, 35K30, 47H05.
1. Introduction. The subject to which this paper is devoted, topological degree, is one of the most useful tools to study the existence of solutions of nonlinear problems. This tool has a powerful property known as the invariance under homotopy, which assures under appropriate hypotheses that the solvability of family of problems is invariant under continuous perturbations.
The notion of topological degree was first introduced explicitly by Brouwer (cf. [7]) in 1912 for continuous mappings from a bounded subset ofRntoRn.
In 1934, Leray and Schauder extended this concept to infinite dimensional Banach spaces with mappings of the formf =I−g, whereI is the identity map andg is a compact map (see [7]). The uniqueness of the two degree functions was proved independently by Fuhrer in 1972 and Amann and Weiss [1], respectively. Browder [5, 6] extended the concept of this theory for operators of monotone type from a reflexive Banach spaceXto its dual spaceX∗, mapping of class(S+), perturbation of maximal monotone operators by mappings of class(S+), and other special cases defined on Sobolev spaces. A new and easy construction of the topological degree for mappings of class(S+)from separable Banach space to its dual space, is given by Berkovits and Mustonen [3]. In [4] they constructed a degree function for a class of mappings of the formF=L+S, whereLis a linear densely defined maximal monotone map from the domainD(L)inXtoX∗andSis a bounded demicontinuous map of class(S+)[4].
We construct a topological degree for operators of the formL+A+SfromD(L)⊂X, domain ofL, to 2X∗
a new formulation of the following parabolicvariational inequality problem:
P
findu∈D(L)⊂Xsuch that T
0
∂u(t)
∂t ,v(t)−u(t) +
−∆p(u),v−u+(h,v−u)≥0, ∀v∈,
where
=v∈Lp0,T ,W1,p
0 (Ω)
:v(t)∈K, K=v∈W01,p(Ω):v≥0 a.c. inΩ,
(1.1)
andhis an element ofLp(Q)withQ=Ω×]0,T [. Then we obtain a problem without a set of constraint equivalent to problem(P).
2. Preliminaries. LetXbe a real reflexive Banach space and letX∗stand for its dual space with respect to the continuous pairing···. We may assume, without loss of generality, thatX and X∗ are locally uniformly convex, by virtue of the renorming theorem of Trojanski (cf. [5]). In particular, this implies that the duality mappingJof
XintoX∗given by the following relations:
J(u)X∗= uX, J(u),u= u2X, ∀u∈X, (2.1)
is bijective bicontinuous. The norm convergence inXandX∗is denoted by
/
/
, and the weak convergence by/
. We consider a multi-values mapping (operator)T fromXto 2X∗(i.e., with values subsets ofX∗). With each such map, we associate its graph
G(T )=(u,w)∈X×X∗:w∈T (u). (2.2)
The multi-values mappingT is said to bemonotone(T ∈(M))if for any pair of ele-ments(u1,w1),(u2,w2)inG(T ), we have the inequality
w1−w2,u1−u2≥0. (2.3) T is said to bemaximal monotone(T∈(MM))if it is monotone and maximal in the sense of graph inclusion among monotone multi-values mappings fromXto 2X∗
. An equivalent version of the last clause is that for any(u0,w0)∈X×X∗for which
w0−w,u0−u≥0, (2.4)
for all(u,w)∈G(T ), we have(u0,w0)∈G(T ).By Rockafellar theorem (see [9]) an equivalent statement is that the range ofT+λJ is allX∗for allλ >0.
LetT be a maximal monotone operator fromXto 2X∗
.
(i) A sequence {Tn, n∈N}of maximal monotone operators fromX to 2X∗ is said to begraph-convergent to T if for any(u,w)∈G(T )there exists a sequence
(un,wn)∈G(Tn)such thatun
/
/
uinXandwn/
/
winX∗.there exists a sequence(wn,un)∈G(Ttn)such thatun
/
/
uinXandwn/
/
w inX∗.(iii) A family{Tt, t∈[0,1]}of maximal monotone operators is said to be pointwise graph-continuous if for any sequencetn
/
/
t∈[0,1]and any(u,w)∈G(T ), there exists a sequencewn∈Ttn(u)such thatwn/
/
winX∗.(iv) A family{Tt, t∈[0,1]}of operators fromXto 2X∗is said to be bounded if the range of a bounded subset ofXbyTt is contained in a bounded subset ofX∗ for all
t∈[0,1].
LetT∈(MM)fromX
/
/
2X∗. Then for everyλ >0 and everyu∈X, there exists a unique elementRT
λ(u)belonging toD(T ), domain ofT, which is called the resolvant of indexλofT, such that
0∈JRT λ(u)−u
+λTRT λ(u)
. (2.5)
We denote
Tλ(u)=1λJu−RTλ(u)
, that is,Tλ(u)∈TRλT(u)
. (2.6)
The operatorTλis called the Yosida approximation, it is a monotone continuous map fromXtoX∗, graph-convergent toT asλgoes to zero and verify
Tλ(u)X∗≤ wX∗, ∀u∈X, w∈T (u). (2.7)
The reader is referred to [2] for more details.
Now letT be a linear densely defined monotone map fromD(T )⊂XtoX∗, then a necessary and sufficient condition forT∈(MM)is thatG(T )is a subspace ofX×X∗ and the conjugateT∗ofTis monotone (cf. [7]).
We also need the following classes of mappings of monotone type. A mappingT:
D(T )⊂X
/
/
X∗is called(v) quasimonotone(T∈(QM))if for any sequence{un}inD(L)withun
/
u, we have limsupT (un),un−u ≥0,(vi) pseudomonotone(T∈(PM))if for any sequence{un}inD(L)withun
/
u and limsupT (un),un−u ≤0, we have limT (un),un−u =0, and ifu∈D(T ), thenT (un)
/
T (u),(vii) of class (S+) (T∈(S+))if for any sequence{un}inD(L)withun
/
uand limsupT (un),un−u ≤0, we haveun/
/
u.An example of mapping of class(S+)is the duality mapJ, moreover, it is strictly monotone. SinceJ−1can be identified with the duality map fromX∗toX∗∗, it is also of class(S+).
If we assume that all mappings are demicontinuous and defined in the whole space
X, then(S+)⊂(PM)⊂(QM)and(MON)⊂(PM). It is also important to observe that
(S+)+(QM)is contained in(S+).
3. Construction of a degree function. LetXbe a real reflexive Banach space. We assume thatXand its conjugateX∗are locally uniformly convex. LetAbe a bounded maximal monotone operator fromXto 2X∗
graph ofLis a closed set inX×X∗, Y=D(L)equipped with the graph norm uY= uX+LuX∗, u∈Y , (3.1)
becomes a real reflexive Banach space. We assume thatY and its dual spaceY∗ are also locally uniformly convex.
Letj stand for the natural embedding of Y toX and letj∗ stand for its adjoint fromX∗toY∗. For each open and bounded subsetGofX, let
ᏲG=L+A+S: ¯G∩D(L)
/
/
X∗|Sis a bounded demicontinuous map of classS+with respect toD(L)from ¯GtoX∗, and letᏴG=L+At+S(t): ¯G∩D(L)
/
/
X∗|At(0≤t≤1)is a bounded pointwise graph-continuous homotopy of(MM)operators fromXto 2X∗andS(t)
is a bounded homotopy of classS+with respect toD(L)from ¯GtoX∗. The familyS(t), with 0≤t≤1, is called a homotopy of class(S+)with respect toD(L), if the conditionsun
/
u,Lun/
Lu, tn/
/
t, and limsupS(tn)un,un−u ≤0 implyun/
/
uandS(tn)(un)/
S(t)(u). Note that the classᏴGincludes all affine homotopiesL+(1−t)(A1+S1)+t(A2+S2)with(L+Ai+Si)∈ᏲG, i=1,2. In order to find suitable approximations for mappingsF∈ᏲG, letˆ
L=j∗◦L◦j, (3.2)
which is obviously a bounded linear monotone map fromY toY∗. Similarly, let ˆ
S(t)=j∗◦S(t)◦j:j−1G¯
/
/
Y∗ (3.3) wheneverS(t)is a homotopy from ¯GtoX∗, and for everyλ >0,ˆ
At,λ=j∗◦At,λ◦j, (3.4)
whereAt,λis the Yosida approximation of indexλof the operatorAt. Sincejis con-tinuous fromY toX,j−1(G)¯ =G¯∩D(L)is closed andj−1(G)=G∩D(L)is open inY. It is easy to check that
j−1(G)⊂j−1G¯; ∂j−1(G)⊂j−1(∂G). (3.5)
Note that we have the same notation for closures and boundaries in bothXandY. In what follows, we also need the map M :Y
/
/
Y∗defined by
M(u),v=Lv,J−1(Lu), u,v∈Y , (3.6)
where(·,·)denotes the pairing betweenYandY∗andJ−1is the inverse of the duality mapJ:X
/
/
X∗. In fact, for allu∈Y for which M(u)∈j∗(X∗), we haveJ−1(Lu)∈ D(L∗)and by (3.6),M(u)=j∗L∗J−1(Lu). (3.7)
We need this representation later in proving Lemma 3.4. For each admissible map
F∈ᏲGor homotopyF(t)∈ᏴGand for eachλ >0, we define ˆ
Lemma3.1. Let{Tn, n∈N}andT be a sequence of maximal monotone operators fromXto2X∗
. Then the following properties hold.
(a)IfTnis graph-convergent toT, then for any sequence{un} ⊂Xsuch thatun
/
/
u inX, we haveTn,λ(un)/
/
Tλ(u)inX∗ for everyλ >0. Moreover, if the sequence{Tn}is bounded, we haveTn,λn(un)
/
β∈T (u)for a subsequence{un}and any sequence{λn; λn≥0}withλn/
/
0.(b) If the sequence {Tn} is a pointwise graph-continuous to T, then we have Tn,λn(u)
/
/
T◦(u)for any sequence{λn; λn≥0}withλn/
/
0, whereT◦(u)is the element of minimal norm of the closed convex subsetT (u)ofX∗.Proposition3.2(see [5]). Let{Tn},Tbe a sequence of maximal monotone operators withTnis graph-convergent toT and let{(αn,βn)}be a sequence inG(Tn)such that αn
/
αinX,βn/
βinX∗andlimsupβn,αn ≤ β,α. Then we haveβ∈T (α) andβn,αn/
/
β,α.Proof of Lemma3.1. In this proof, we use the same notation of norm · ofX
andX∗.
(a) We first notice that the sequencesRλTn(un)and{Tn,λ(un)}are bounded. Let (u0,w0)∈T, then there exists(u0n,w0n)∈G(Tn)such that(u0n,w0n)
/
/
(u0,w0). From monotonicity ofTnand relations (2.6),1
λJ
un−RλTn
un−w0n,RλTn
un−u0n ≥0. (3.9)
Thus 1
λJ
un−RλTn
un−w0n,RλTn
un−un+un−u0n ≥0, (3.10) that is,
un−RTn λ
un2≤un−RTnλ
unun−u0n+λw0n+λw0nun−u0n
·un−RλTn
un≤2un−u0n+λw0n.
(3.11)
Hence, the sequencesRTnλ (un)and{Tn,λ(un)}being bounded in reflexive Banach spacesXandX∗, respectively. We can extract a subsequence (still denoted by{u
n}) such thatRTnλ (un)
/
αinXandTn,λ(un)/
βinX∗.Let us identifyαandβ. By monotonicity ofJ, for every integersn,m, 1
λJ
un−RTnλ
un−1λJum−RλTm
um,un−RTnλ
un−um−RTmλ
um ≥0.
(3.12) Thus
Tn,λun,RλTn
un+Tm,λum,RTmλ
um
≤Tn,λun−Tm,λum,un−um+Tn,λun,RTmλ
um+Tm,λum,RTnλ
un.
(3.13)
Let us first fixmand letngo to+∞, next letmgo to+∞, then we have
limsupTn,λun,RTnλ
Applying Proposition 3.2 to (3.14) withαn=RλTn(un),βn=Tn,λ(un), we obtain
β∈T (α), Tn,λ(un),RTnλ
un
/
/
β,α. (3.15)By applying Proposition 3.2 to the operatorJwithαn=(1/λ)(un−RλTn(un))andβn=
(1/λ)J(un−RTnλ (un)), and taking into account thatβn,αn
/
/
β,(1/λ)(u−α),we obtain
β=1
λJ(u−α), un−R
Tn λ
un
/
/
u−α. (3.16)HenceRTnλ (un)
/
/
α. Combining (3.15) and (3.16), we have0∈T (α)+1λJ(α−u), (3.17)
and from the unicity of the solution of (3.17), we conclude that
α=RT
λ(u). (3.18)
HenceRTnλ (un)
/
/
RλT(u). Consequently,Tn,λ(un)/
/
Tλ(u).Now letλn
/
/
0.For(u,w)∈G(T ) there exists a sequence(u0n,wn)∈G(Tn) such that u0n/
/
u0 in X and wn/
/
w0 in X∗. The same estimation as (3.11) yieldsun−RTn λn
un≤2un−u0n+λnwn (3.19) implies that RTnλn(un)
/
/
uasngoes to infinity. Since{Tn,λn(un)}is bounded in reflexive Banach spaceX∗, we can extract a subsequence (still denoted by{un}) such that Tn,λn(un)/
β. Applying Proposition 3.2 to the sequence of operators {Tn} withαn=RλnTn(un)andβn=Tn,λn(un), we obtainβ∈T (u).(b) Letλn
/
/
0 andw∈T (u), then the property pointwise graph-continuous of Tn implies that there exists a sequencewn∈Tn(u)such thatwn/
/
w inX∗. By monotonicity ofTn,
wn−λ1
nJ
u−RTnλn(u),u−RTnλn(u) ≥0. (3.20)
Thus
1
λn
u−RTn
λn(u)≤wn. (3.21)
The relation above implies thatRTnλn(u)
/
/
uandlimsupTn,λn(u)≤ w, ∀w∈T (u). (3.22)
Hence,
limsupTn,λn(u)≤T◦(u). (3.23)
Then there exists a subsequence (still denoted by{un}) such that
Applying Proposition 3.2 to the sequence of operators {Tn}with αn=RTnλn(u)and βn = Tn,λn(u), we obtain β ∈ T (u), which implies that β ≥ T◦(u). Since Tn,λn(u)
/
β, thenliminfTn,λn(u)≥ β. (3.25) Combining (3.23) and (3.25), we have
Tn,λn
un
/
/
T◦(u)= β. (3.26)Now, by definition ofT◦(u)we conclude thatβ=T◦(u). Finally, from (3.24) and (3.26) we have
Tn,λn(u)
/
/
T◦(u). (3.27)Lemma3.3. IfF(t)∈ᏴGandλ >0, thenFˆλ(t)is a bounded homotopy of class(S+)
fromj−1(G)¯ ⊂Y toY∗. In particular, for eachλ >0, Fˆλis a bounded demicontinuous
map of class(S+)fromj−1(G)¯ ⊂Y toY∗.
Proof. AssumeF(t) ∈ᏴG and λ >0. Let {un} ⊂G¯∩D(L) with un
/
u inY , tn
/
/
t, and limsup(Fˆλ(tn)(un),un−u)≤0. Thenun/
uinX,Lun/
LuinX∗and sinceA
tn,λ(u)
/
/
At,λ(u),limsupLun−Lu,un−u+Atn,λun−Atn,λ(u),un−u
+Stnun,un−u+λLun−Lu,J−1Lun−J−1(Lu)≤0. (3.28) SinceL,J−1, andAtn,λare monotone we conclude that
limsupStnun,un−u≤0. (3.29)
By the(S+)-property ofS(t)we obtainun
/
/
uinXandS(tn)(un)/
S(t)(u)in X∗. Then by Lemma 3.1(a),Atn,λun
/
/
At,λ(u). (3.30)Therefore,
limLun−Lu,J−1Lun−J−1(Lu)=0 (3.31) implying, by the(S+)-property ofJ−1, thatLun
/
/
LuinX∗, then the assertion fol-lows.LetF(t)∈ᏴGand let{h(t), 0≤t≤1}be a continuous curve inX∗. We denote
K=u∈j−1G¯|sFˆ
λ(t)(u)+(1−s)Fˆδ(t)(u)=j∗h(t)
for someλ,δ >0,ands,t∈[0,1]. (3.32)
Note thatj(K)⊂G¯implying thatKis bounded inX. The fact thatKis bounded also inY follows from the following.
Lemma3.4. There exists a constantR >0, independent ofλ,δ,s, and t, such that
Proof. Without loss of generality, we may assume thath(t)≡0. Let u∈K be arbitrary. Then for someλ,δ >0 ands,t∈[0,1],
Lu,v+w,v+S(t)(u),v+sλ+(1−s)δL∗J−1(Lu),v=0, ∀v∈D(L),
(3.33) where w=sAt,λ(u)+(1−s)At,δ(u). Observe thatJ−1(Lu)∈D(L∗) since M(u)∈ j∗(X∗). SinceD(L)is dense inX, equation (3.33) holds for allv∈X. Hence we can insertv=J−1(Lu)to get
Lu,J−1(Lu)+w,J−1(Lu)+S(t)(u),J−1(Lu)
+sλ+(1−s)δL∗J−1(Lu),J−1(Lu)=0. (3.34)
Recalling thatL∗is monotone, we obtain
Lu2
X∗≤sAt,λ(u)+(1−s)At,δ(u)+S(t)(u)X∗J−1(Lu)X. (3.35)
SinceJ−1(Lu)
X= LuX∗ and sinceAt andS(t)are bounded homotopies from a
bounded set ¯G⊂Xwe conclude that
LuX∗≤c (3.36)
for some positive constantc independent ofλ,δ,s andt, this completes the proof.
The relationship betweenF(t)∈ᏴG and its approximation ˆFλ(t)is shown by the following.
Lemma3.5. LetΩ⊂G¯be a closed set,F(t)∈ᏴGan admissible homotopy, andh(t) a continuous curve inX∗such that
h(t)∉F(t)Ω∩D(L), ∀t∈[0,1]. (3.37)
Then there existsλ0>0such that
j∗h(t)∉Fˆ
λ(t)j−1(Ω), ∀t∈[0,1],0< λ < λ0. (3.38)
Proof. We may assume thath(t)≡0. We argue by a contradiction. Let us assume that there exist the sequences{λn},{tn}, and{un} ⊂j−1(Ω)such thatλn
/
/
0+, tn/
/
t∈[0,1], andˆ
Lun+Aˆtn,λnun+Sˆtnun+λnMun=0. (3.39)
By Lemma 3.4 the sequence{un}is bounded inY implying thatun
/
uinXand Lun/
LuinX∗with u∈D(L), for a subsequence. Using the fact thatL,Atn,λn, J−1are monotone andAtn,λn(u)/
/
A◦t(u)we get from (3.39),
limsupStnun,un−u=limsup−Lun−Lu,un−u
−Atn,λnun−Atn,λn(u),un−u
SinceS(t)is a homotopy of class(S+), thenun
/
/
uinXandS(tn)(un)/
S(t)(u) inX∗withu∈Ω. By (3.39),ˆ
Lun,v+Aˆtn,λnun,v+Sˆtnun,v+λnMun,v=0 (3.41)
for allv∈Y and n∈N. SinceAtn,λn(un)
/
wt∈At(u)for a subsequence{un} (Lemma 3.1), then asntends to+∞,Lu,v+wt,v+S(t)(u),v=0, ∀v∈D(L). (3.42)
NowD(L)is dense inX,
0∈Lu+At(u)+S(t)(u) (3.43)
withu∈Ω∩D(L), which contradicts our assumption. Hence the proof is complete.
By choosingΩ=∂G,F(t)=F∈ᏲG, andh(t)=h∈X∗as in Lemma 3.5, the
condi-tionh∉(L+A+S)(∂G∩D(L))implies that there existsλ0>0 such that
j∗h∉ˆL+Aˆ
λ+Sˆ+λMj−1(∂G), ∀0< λ < λ0. (3.44)
Recalling (3.5), we also have
j∗h∉Fˆ
λ∂j−1(G), ∀0< λ < λ0. (3.45)
Moreover, by Lemma 3.4 there exists a constantR >0, independent ofλ, such that
j∗h∉Fˆ
λ(u), ∀u∈G,¯ u ≥R, λ>0. (3.46)
DenotingGR(Y )=j−1(G)∩BR(Y ), we therefore have
j∗h∉Fˆ
λ∂G(Y ) (3.47)
for allλwith 0< λ < λ0. Since ˆFλ=ˆL+Aλ+Sˆ+λM is a map of classS+fromj−1(G)¯ ⊂Y toY∗by Lemma 3.3, the value of the unique topological degree (see [5]),
dS+
ˆ
Fλ,GR(Y ),j∗h (3.48)
is well defined for all 0< λ < λ0.
Lemma 3.6. Leth∈X such that h∉(L+A+S)(∂G∩D(L)),then there exists a constantλ >0such that
dS+
ˆ
L+Aˆλ+Sˆ+λM,GR(Y ),j∗h=constant (3.49)
for allλwith0< λ < λ.
Proposition 3.7(see [5]). LetX be a reflexive Banach space,{At, t ∈[0,1]} a graph-continuous family of maximal monotone operators fromXto2X∗ with(0,0)∈ G(A)for allt. Let{un}be a sequence inX,un
/
uand for sequences{tn} ∈[0,1],tn
/
/
t, 0< λn,0< δn, λn/
/
0,andδn/
/
0, letSuppose further that for another sequence{sn}in[0,1], and for
wn=1−snyn+snzn, (3.51)
we have wn
/
w in X∗. If limsupwn,un ≤ w,u, then w ∈ At(u) andwn,un
/
/
w,u.Proof of Lemma3.6. In order to employ the homotopy argument of the degree
dS+, we show the existence ofλ such that
j∗h∉(1−s)Fˆ
λ+sFˆδj−1(∂G) (3.52) for all s ∈[0,1], 0< λ≤λ, 0< δ≤λ. Indeed, we assume the contrary, that is, there exist sequences{un} ⊂j−1(∂G),{sn,s}in[0,1]and two sequences of positive numbersλn, δn
/
/
0 such thatj∗h=Luˆ
n+1−snAˆλnun+snAˆδnun+Sˆun+1−snλn+snδnMun. (3.53)
SinceAis a bounded operator fromXto 2X∗
, then the sequence
wn=1−snAˆλnun+snAˆδnun (3.54)
is bounded inX∗. We may assume thatw
n
/
winX∗andun/
uinX, we havelimsupwn,un−u=limsup−Sun−Lun,un−u
≤limsup−Sun,un−u+limsup−Lun,un−u
≤limsup−Sun,un−u
= −liminfSun,un−u≤0,
(3.55)
that is,
limsupwn,un≤ w,u. (3.56) Proposition 3.7 implies thatwn,un
/
/
w,uandw∈A(u). HencelimsupSun,un−u≤0. (3.57)
The property of class(S+)of the operatorS implies thatun
/
/
u∈j−1(∂G)andS(un)
/
S(u).Applyingvto the two sides of the equality (3.53) and tendingntoinfinity, then we have
h,v = Lu,v+w,v+S(u),v, ∀v∈D(L). (3.58)
SinceD(L)is dense inX,
h=Lu+S(u)+w, (3.59)
wherew∈A(u)andu∈j−1(∂G). Which contradicts the relation (3.52). That is,
j∗h∉(1−s)Fˆ
for allδ,λ,0< δ < λ,0< λ < λ, s∈[0,1]andRis a constant satisfying Lemma 3.4. By the invariance under homotopy property of(S+)-degree, we obtain
dS+
ˆ
Fλ,GR(Y ),j∗h=dS+
ˆ
Fδ,GR(Y ),j∗h=constant (3.61)
for allλ,0< λ < λ
Consequently, it is relevant to define a functionDby
DL+A+S,G,h= lim
λ→0+dS+
ˆ
L+Aˆλ+Sˆ+λM,GR(Y ),j∗h (3.62)
wheneverh∉(L+A+S)(∂G∩D(L))andRis sufficiently large (satisfying Lemma 3.4). Theorem3.8. LetXbe a real reflexive Banach space,La linear maximal monotone densely defined map fromD(L)⊂XtoX∗, Gan open bounded subset inXandᏲ
Gthe class of admissible operators. Then there exists a function topological degreeDdefined
from{(F,G,h):F∈ᏲG, Gan open bounded subset inXandh∉F(∂G∩D(L))}toZ.
Satisfying the following properties:
(D1)IfD(F,G,h)≠0, then there existsu∈G∩D(L)such thath∈F(u).
(D2) (Additivity of domain.) IfG1 andG2 are open disjoint subsets ofG andh∉
F[(G¯\(G1∪G2))∩D(L)], then
D(F,G,h)=DF,G1,h+DF,G2,h. (3.63)
(D3)(Invariance under admissible homotopies.) IfF(t)∈ᏴGandh(t)∉F(t)(∂G∩
D(L))for allt∈[0,1], whereh(t)is a continuous curve inX∗, then
DF(t),G,h(t)=constant, ∀t∈[0,1]. (3.64)
(D4)L+Jis the normalising map, that is,
D(L+J,G,h)=1, wheneverh∈(L+J)G∩D(L). (3.65)
Proof. It is the same as the one given in [4] (properties of the degree).
4. Existence result. LetXbe a real reflexive Banach space,La linear maximal mono-tone map from D(L)⊂X toX∗ with D(L) dense inX, and A a bounded maximal monotone operator fromXto 2X∗
with(0,0)∈G(A)andGan open bounded subset ofX. Let
ᏲG(PM)=F=L+A+S:Sis a bounded demicontinuous
pseudomonotone with respect toD(L)map from ¯GtoX∗,
ᏲG(QM)=F=L+A+S:Sis a bounded demicontinuous
quasimonotone with respect toD(L)map from ¯GtoX∗.
(4.1)
For a givenh∈X∗, we are interested in the solvability of the equation
Lemma4.1. LetGbe a convex open bounded subset inX andF∈ᏲG. ThenF(G¯∩
D(L))is closed inX∗.
Proof. Let{zn}be a sequence in (L+A+S)(G¯∩D(L)) with zn=Lun+wn+ S(un)
/
/
z, un∈G¯∩D(L) andwn∈A(un). Since subsequences{un}, {S(un)}, and{wn}are bounded in reflexive Banach spacesXandX∗, then there exists a sub-sequence (still denoted by{un}), such thatun/
uinX withu∈G, w¯ n/
w inX∗ andS(un)
/
sinX∗. HenceLun/
z−w−s.Since alsoG(L)is weakly closed, we haveu∈D(L)andLu=z−w−s. Consequently,limsupSun,un−u=limsupzn−Lun−wn,un−u
=limsup−Lun−wn,un−u
= −liminfLun−Lu,un−u+wn−w,u¯ n−u≤0, (4.3)
where ¯wis an element ofA(u). SinceSis pseudomonotone, we obtainS(un)
/
S(u)= s and S(un),un
/
/
S(u),u. Therefore, limsupwn,un ≤ w,u, then byProposition 3.2 we havew∈A(u). Hencez=Lu+w+S(u)∈(G¯∩D(L)).
Theorem4.2. LetL+A+S∈ᏲX(PM)(respectively,L+A+S∈ᏲX(QM)) and assume thatSsatisfies the two conditions:
(i) ifLun+wn+S(un)
/
/
hinX∗withwn∈A(un),then the sequence{un}is bounded inX,(ii) there exists R > 0 such that S(u),u> 0 for all u ≥R, Then (L+A+
S)(D(L))=X∗(respectively,(L+A+S)(D(L))is dense inX∗).
Proof. By condition (i) there exist two constantsR ≥Randδ >0 such that
Lu+w+S(u)+λJ(u)−th≥δ, wherew∈A(u), (4.4)
for allu∈D(L),u =R,0≤t≤1,0≤λ< δ/R. Thus we obtain
DL+A+S+λJ,BR(X),h
=DL+A+S+λJ,BR(X),0
(4.5)
whenever 0< λ < δ/R. Let
F(t)(u)=Lu+(1−t)J(u)+tS(u)+A(u)+λJ(u), 0≤t≤1. (4.6)
Since(0,0)∈G(A),then by (ii), we obtain
0∉F(t)(u) (4.7)
for allt∈[0,1]and 0< λ < δ/R. Therefore, by invariance under homotopy, we have for all 0< ε < δ/R,
DL+A+S+λJ,BR(X),0=DL+J,BR(X),0=1. (4.8)
Hence there existsuλ∈D(L) such thath−λJ(uλ)∈Luλ+A(uλ)+S(uλ).Letting λ
/
/
0+, we haveh∈Lu+A(u)+S(u)for someu∈D(L)(Lemma 4.1) (respectively,5. Application to a parabolic variational inequality problem. LetΩbe a bounded open subset ofRnfor somen≥1, andQthe cylinderΩ×]0,T [for a givenT >0, we consider the operators of the form
∂u
∂t +A(u)+B(u), (5.1)
where
B(u)=
|α|≤m
(−1)|α|DαA
αx,t,u,Du,...,Dmu. (5.2)
The supposed functionsAα satisfying the Carathéodory, polynomial growth condi-tions, and monotonicity such that the operator defined from ᐂ=Lp(0,T ,V )with
V=W0m,p(Ω)toᐂ∗=Lp(0,T ,V∗)by
S(u),v =
|α|≤m
QAα
x,t,u,Du,...,DmuDαv, u,v∈ᐂ (5.3)
is bounded continuous pseudomonotone (see [4], for instance). Moreover, we assume that the functionsAαsatisfy the coercivity condition. There existsc >0 andk∈L1(Q)
such that
|α|≤m
Aαx,t,ξξα≥c|ξ|p−k(x,t) (5.4)
for all(x,t)∈Qandξ∈RN.LetAbe a bounded maximal monotone operator from ᐂtoᐂ∗. Indeed, letϕbe a convex lower semi-continuous function from a reflexive Banach spaceXtoR∪{+∞}, then the subdifferential ofϕatuinX, given by
∂ϕ(u)=w∈X∗:ϕ(v)−ϕ(u)≥ w,v−u ∀v∈X (5.5)
is maximal monotone from X to 2X∗
. We assume in what follows that 2≤p <∞. Then, for eachu∈ᐂwithu ∈ᐂ∗which also belongs toC([0,T ],L2(Ω)), the initial conditionu(x,0)=0 inΩmakes sense. Thus the operator∂/∂tinduces a linear map from the subsetD(L)= {v∈ᐂ|v ∈ᐂ∗,v(0)=0}ofᐂtoᐂ∗by
Lu,v =
T
0
u(t),v(t)dt, u∈D(L), v∈ᐂ (5.6)
hereu stands for the generalized derivative ofu, that is, T
0u(t)ϕ(t) dt= −
T
0u(t) ∂ϕ(t)
∂t dt, ∀ϕ∈C0∞(0,T ). (5.7)
It can be shown (see [9]) thatLis a linear densely defined maximal monotone map. Theorem5.1. LetΩbe an open bounded subset inRnandQthe cylinderΩ×]0,T [. Then the equation
h∈Lu+A(u)+S(u) (5.8)
Proof. We use this result to give a new reformulation of a variational inequal-ity problem—obstacle problem. We consider the operator−∆pdefined fromLp(0,T ,
W01,p(Ω))toLp(0,T ,W−1,p (Ω))by
−∆p(u),v=n
i=1
Q|Du| p−2∂u
∂xi
∂v
∂xi, u,v∈L
p0,T ,W1,p
0 (Ω). (5.9) It is a bounded continuous pseudomonotone map, then verifies the hypotheses as-sumed on the operatorBdefined above. Forv∈Lp(Q),let
v+=max(v,0), v−=min(v,0). (5.10)
We verify that
∆p(u),u−=∆pu−,u−. (5.11)
We define
K=v∈W01,p(Ω):v≥0 a.e. inΩ
,
=v∈Lp0,T ,W1,p
0 (Ω):v(t)∈K.
(5.12)
The setis a closed convex cone ofLp(0,T ,W1,p
0 (Ω)). Leth∈Lp(Q)be given. We are interested in the following problem:
P
findu∈D(L)∩such that
Lu,v−u+−∆p(u),v−u+(h,v−u)≥0, ∀v∈,
where(·,·)designates the pairing betweenLp(Q)andLp(Q). The problem(P)is also called obstacle problem. If(P)admits a solutionu, then it is unique. Indeed, we suppose that there exists two solutionsu1andu2, then from the inequality of(P) we obtain
Lu1,u2−u1+−∆pu1,u2−u1+h,u2−u1≥0,
Lu2,u1−u2+−∆pu2,u1−u2+h,u1−u2≥0, (5.13) and by adding (5.13), we obtain
Lu1−Lu2,u1−u2−∆pu1−∆pu2,u1−u2≤0. (5.14) SinceLis monotone and−∆pis strictly monotone, then we haveu1=u2.
We consider the functionϕdefined fromLp(0,T ,W1,p
0 (Ω))toRby :v
/
/
ϕ(v)= (h+,v+), it is continuous onLp(0,T ,W1,p0 (Ω))and convex. Hence the subdifferential of ϕ:∂ϕis a maximal monotone operator fromLp(0,T ,W1,p
0 (Ω))toLp(0,T ,W−1,p(Ω))
(see [7, 9]). Furthermore, we verify that∂ϕis bounded. Theorem5.2. The problem
(P)
findu∈D(L)such that
Lu,v−u+−∆p(u),v−u+ϕ(v)−ϕ(u)+h−,v−u≥0,
∀v∈Lp0,T ,W1,p
Proof. The problem(P)is equivalent to findu∈D(L)such that
h−∈Lu−∆p(u)+∂ϕ(u). (5.15)
Then by Theorem 5.1 and the strict monotonicity of−∆p, we conclude that(P)admits a unique solution. Letube this solution. It suffices to show thatuis an element of to complete the proof. Forv=u+∈, the inequality of(P)becomes
Lu,−u−+−∆p(u),−u−+h−,−u−≥0. (5.16)
SinceLu,−u− ≤0 (see [8]) and(h−,−u−)≤0, we have
−∆p(u),u−≤0. (5.17)
Then by (5.11), we obtain
−∆p(u−),u−≤0. (5.18)
Henceu−
Lp(0,T ,W1,p
0 (Ω))=0, thereforeu∈.
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A. Addou: University Mohammed I, Faculty of Sciences, Department of Mathematics and Computing, Oujda, Morocco
E-mail address:[email protected]
B. Mermri: University Mohammed I, Faculty of Sciences, Department of Mathematics and Computing, Oujda, Morocco
