# An Obstacle Problem for Noncoercive Operators

Authors: Luigi Greco; Gioconda Moscariello; Gabriella Zecca
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

We study the obstacle problem for second order nonlinear equations whose model appears in the stationary diffusion-convection problem. We assume that the growth coefficient of the convection term lies in the Marcinkiewicz spaceweak-LN.


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## Body

## 1. Introduction

LetΩ be a bounded domain in RN with C1-boundary, N>2, and let A:Ω×RN→RN be a Carathèodory function; that is,
(1)x⟶Ax,ξismeasurableforanyξ∈RN;ξ⟶Ax,ξiscontinuousforalmosteveryx∈Ω.
We assume that there exist 0<α<β such that for almost every x∈Ω we have
(2)Ax,ξ⩽βξ+φxwithφ∈L2Ω,(3)αξ-η2⩽Ax,ξ-Ax,η,ξ-ηkkkkkkkkkkkkstrongmonotonicity
for any vectors ξ and η in RN. Moreover, we assume that B:Ω×R→RN is a Carathèodory function verifying the following properties. (i)
There exists a nonnegative functionb:Ω→R+ in the Lorentz space b∈LN,∞(Ω) such that
(4)Bx,s-Bx,t⩽bxs-t,
for almost every  x∈Ω and for any s,t∈R.(ii)
Consider(5)b0x:=Bx,0∈L2Ω.The spaceLN,∞ is also known as the Marcinkiewicz space weak-LN.Letg∈W1,2(Ω) and let ψ:Ω→[-∞,+∞]. We define
(6)Kψ,g=v∈g+W01,2Ω:v⩾ψa.e.inΩ.Definition 1.
GivenF∈L2(Ω,RN), one says that u∈Kψ,g is a solution of the obstacle problem OP(F,ψ,g) if
(7)∫Ω‍Ax,∇u+Bx,u,∇v-udx⩾∫Ω‍F,∇v-udx
for every v∈Kψ,g.For a classical treatment of obstacle problem we refer to [1, 2]. See also [3, 4] and references therein.Under assumptions (2) and (4) the left hand side of (7) is finite by the Sobolev embedding theorem.We point out that assumptions (1)–(5) do not guarantee that the operator
(8)Ax,s,ξ=Ax,ξ+B(x,s)
for any ξ∈RN, s∈R, and almost every x∈Ω is coercive and monotone.The aim of this paper is to establish existence and uniqueness of solutions ofOP(F,ψ,g) in the sense of Definition 1. Our first result is the following.Theorem 2.
Assume that assumptions (3) and (4) are verified, and let
(9)b∈LN,∞(Ω).
Then, there exists at most one solution u∈Kψ,g of problem (7).We also prove the following.Theorem 3.
Let assumptions (1)–(5) be verified and let Kψ,g≠∅. Assume that
(10)distLN,∞b,L∞<α4S2.
Then, for every F∈L2(Ω,RN), problem (7) admits a solution u∈Kψ,g. Here S2 is the Sobolev constant.We remark thatL∞ is not dense in LN,∞. Moreover, condition (10) does not give any smallness control on the norm of b in LN,∞ (see Section 2.1). This fact is very relevant when we have to prove a priori estimates for the solutions of OP(F,ψ,g). Indeed, in order to prove our results we follow a classical approach. First, we construct a coercive and monotone operator. Then we reduce the existence to applying a fixed point theorem.Theorem3 is new also in case of equations. In [5–8], the authors considered operators with a lower order term having the growth coefficient b in spaces in which the bounded functions are dense.A condition similar to (10) has been used in [9] for proving the existence of solutions to linear equations. In that paper, an example shows that, in general, condition (10) cannot be dropped in order to achieve existence of solutions. Regularity results for solutions have been obtained in [10].


## 2. Preliminary Results

### 2.1. Some Functional Spaces

LetΩ be a bounded domain in RN. For a measurable E⊂Ω, we denote by |E| its Lebesgue measure. For a measurable function f:Ω→R we denote by
(11)μf(λ)=x∈Ω:fx>λ,λ⩾0
its distribution function and by
(12)f*(t)=inf⁡λ:μfλ⩽t
its decreasing rearrangement; see [11]. Clearly, f*(t)=0 if t>|Ω|. For 0<p<∞ and 0<q<∞, we consider the quantity
(13)fp,q=∫0∞‍t1/pf*tqdtt1/q
and for q=∞ the obvious modification
(14)fp,∞=sup⁡0<t<∞t1/pf*t.
The Lorentz space Lp,q=Lp,q(Ω) consists of all measurable functions f satisfying ∥f∥p,q<∞. The space Lp,∞ is also known as Marcinkiewicz space Mp or weak-Lp. The quantity ∥∥p,q is equivalent to a norm which makes Lp,q a Banach space; see [11, 12]. For p=q, the space Lp,p coincides with the usual Lebesgue Lp space. Moreover,
(15)1<p1<p2<∞,1⩽q1,q2⩽∞⟹Lp1,q1⊃Lp2,q2,1<p<∞,1⩽q1<q2⩽∞⟹Lp,q1⊂Lp,q2
with continuous injections. In particular, if 1<r<p<∞,
(16)Lp⊂Lp,∞⊂Lr.
The following Hölder-type inequality holds. For 1<pi<∞ and 1⩽qi⩽∞, i=1,…,n, if
(17)∑i=1n‍1pi=1=∑i=1n‍1qi,
then
(18)∫Ω‍∏i=1n‍fi(x)dx⩽∏i=1n‍fipi,qi.
See [9]. An elementary but often useful property is expressed by the equality
(19)fαp,q=fαp,αqα
which holds for α>0.We note the equality(20)χEp,q=pq1/qE1/p
for every measurable E⊂Ω. Here, for q=∞ we assume (p/q)1/q=1.We remark that, for anyp∈]1,∞[, L∞ is not dense in Lp,∞. We consider the distance of a given f∈Lp,∞ to L∞:
(21)distLp,∞f,L∞=inf⁡g∈L∞f-gp,∞.To find a formula for the distance, we consider the truncation operator. Fork>0, we set
(22)Tk(y)=yymin⁡y,k.
Then
(23)distLp,∞f,L∞=lim⁡k→∞f-Tkfp,∞.
Indeed, ∀g∈L∞ and ∀k⩾∥g∥∞, we have, for almost every x∈Ω,
(24)fx-gx⩾fx-Tkfx.
For other comments on the distance to L∞ and some applications, we refer to [13].Example 4.
LetΩ be the unit ball of RN and p∈]1,∞[. The function
(25)f(x)=x-N/p
belongs to Lp,∞. Setting ωN=|Ω|, for k>0 and λ>0, we compute
(26)μf-Tkfλ=ωNλ+k-p,(27)f-Tkf*t=tωN-1/p-k,0<t<ωNk-p0,t⩾ωNk-p.
Hence
(28)f-Tkfp,∞=ωN1/p
does not depend on k.On the contrary, for all1⩽q<∞, starting with the definition of ∥∥p,q, a simple application of Lebesgue dominated convergence theorem shows that L∞ is dense in Lp,q. Hence, for 1⩽q<∞, Lp,q, and in particular the Lebesgue space Lp, is contained in the closure of L∞ in Lp,∞. The closure of L∞ coincides with the closure of C0∞. The elements of the closure can be characterized by the condition of having absolutely continuous norm; see [11, Section 1.3].Fundamental to us will be the Sobolev embedding theorem in Lorentz spaces (see [12]; see also [14, 15]).Theorem 5.
Let one assume that1<p<N, 1⩽q⩽p; then every function g∈W01,1Ω verifying ∇g∈Lp,q actually belongs to Lp*,q, where p*=Np/(N-p), and
(29)gp*,q⩽Sp∇gp,q,
where Sp=c(N)(p/(N-p)).


### 2.2. Monotone Operators

LetX be a reflexive Banach space with dual X*. Let ·,· denote the pairing between X* and X. Let K⊂X be a closed convex set.Definition 6.
A mappingA:K→X* is called monotone if
(30)Au-Av,u-v⩾0∀u,v∈K.
The monotone mapping A is called strictly monotone if
(31)Au-Av,u-v=0impliesu≡v.Definition 7.
A
:
K
→
X
* is called coercive on K if there exists an element φ∈K such that
(32)Au-Aφ,u-φu-φ⟶+∞asu⟶+∞foranyu∈K.The following existence and uniqueness result is contained in [1] (see [1], Cap. III, Theorem 1.7 and Corollary 1.8).Theorem 8.
LetK≠∅ and let A:K→X* be strictly monotone, coercive, and continuous on finite dimensional subspaces. Then, there exists
(33)u∈K:Au,v-u⩾0foranyv∈K.
Such a solution is unique.


### 2.3. The Leray-Schauder Theorem

We will use the well-known Leray-Schauder fixed point theorem in the following form (see [16, Theorem 11.3, page 280]).A continuous mapping between two Banach spaces is called compact if the images of bounded sets are precompact.Theorem 9.
LetF be a compact mapping of a Banach space X into itself, and suppose there exists a constant K such that ∥x∥X<K for all x∈X and t∈[0,1] satisfying x=tF(x). Then, F has a fixed point.


## 2.1. Some Functional Spaces

LetΩ be a bounded domain in RN. For a measurable E⊂Ω, we denote by |E| its Lebesgue measure. For a measurable function f:Ω→R we denote by
(11)μf(λ)=x∈Ω:fx>λ,λ⩾0
its distribution function and by
(12)f*(t)=inf⁡λ:μfλ⩽t
its decreasing rearrangement; see [11]. Clearly, f*(t)=0 if t>|Ω|. For 0<p<∞ and 0<q<∞, we consider the quantity
(13)fp,q=∫0∞‍t1/pf*tqdtt1/q
and for q=∞ the obvious modification
(14)fp,∞=sup⁡0<t<∞t1/pf*t.
The Lorentz space Lp,q=Lp,q(Ω) consists of all measurable functions f satisfying ∥f∥p,q<∞. The space Lp,∞ is also known as Marcinkiewicz space Mp or weak-Lp. The quantity ∥∥p,q is equivalent to a norm which makes Lp,q a Banach space; see [11, 12]. For p=q, the space Lp,p coincides with the usual Lebesgue Lp space. Moreover,
(15)1<p1<p2<∞,1⩽q1,q2⩽∞⟹Lp1,q1⊃Lp2,q2,1<p<∞,1⩽q1<q2⩽∞⟹Lp,q1⊂Lp,q2
with continuous injections. In particular, if 1<r<p<∞,
(16)Lp⊂Lp,∞⊂Lr.
The following Hölder-type inequality holds. For 1<pi<∞ and 1⩽qi⩽∞, i=1,…,n, if
(17)∑i=1n‍1pi=1=∑i=1n‍1qi,
then
(18)∫Ω‍∏i=1n‍fi(x)dx⩽∏i=1n‍fipi,qi.
See [9]. An elementary but often useful property is expressed by the equality
(19)fαp,q=fαp,αqα
which holds for α>0.We note the equality(20)χEp,q=pq1/qE1/p
for every measurable E⊂Ω. Here, for q=∞ we assume (p/q)1/q=1.We remark that, for anyp∈]1,∞[, L∞ is not dense in Lp,∞. We consider the distance of a given f∈Lp,∞ to L∞:
(21)distLp,∞f,L∞=inf⁡g∈L∞f-gp,∞.To find a formula for the distance, we consider the truncation operator. Fork>0, we set
(22)Tk(y)=yymin⁡y,k.
Then
(23)distLp,∞f,L∞=lim⁡k→∞f-Tkfp,∞.
Indeed, ∀g∈L∞ and ∀k⩾∥g∥∞, we have, for almost every x∈Ω,
(24)fx-gx⩾fx-Tkfx.
For other comments on the distance to L∞ and some applications, we refer to [13].Example 4.
LetΩ be the unit ball of RN and p∈]1,∞[. The function
(25)f(x)=x-N/p
belongs to Lp,∞. Setting ωN=|Ω|, for k>0 and λ>0, we compute
(26)μf-Tkfλ=ωNλ+k-p,(27)f-Tkf*t=tωN-1/p-k,0<t<ωNk-p0,t⩾ωNk-p.
Hence
(28)f-Tkfp,∞=ωN1/p
does not depend on k.On the contrary, for all1⩽q<∞, starting with the definition of ∥∥p,q, a simple application of Lebesgue dominated convergence theorem shows that L∞ is dense in Lp,q. Hence, for 1⩽q<∞, Lp,q, and in particular the Lebesgue space Lp, is contained in the closure of L∞ in Lp,∞. The closure of L∞ coincides with the closure of C0∞. The elements of the closure can be characterized by the condition of having absolutely continuous norm; see [11, Section 1.3].Fundamental to us will be the Sobolev embedding theorem in Lorentz spaces (see [12]; see also [14, 15]).Theorem 5.
Let one assume that1<p<N, 1⩽q⩽p; then every function g∈W01,1Ω verifying ∇g∈Lp,q actually belongs to Lp*,q, where p*=Np/(N-p), and
(29)gp*,q⩽Sp∇gp,q,
where Sp=c(N)(p/(N-p)).


## 2.2. Monotone Operators

LetX be a reflexive Banach space with dual X*. Let ·,· denote the pairing between X* and X. Let K⊂X be a closed convex set.Definition 6.
A mappingA:K→X* is called monotone if
(30)Au-Av,u-v⩾0∀u,v∈K.
The monotone mapping A is called strictly monotone if
(31)Au-Av,u-v=0impliesu≡v.Definition 7.
A
:
K
→
X
* is called coercive on K if there exists an element φ∈K such that
(32)Au-Aφ,u-φu-φ⟶+∞asu⟶+∞foranyu∈K.The following existence and uniqueness result is contained in [1] (see [1], Cap. III, Theorem 1.7 and Corollary 1.8).Theorem 8.
LetK≠∅ and let A:K→X* be strictly monotone, coercive, and continuous on finite dimensional subspaces. Then, there exists
(33)u∈K:Au,v-u⩾0foranyv∈K.
Such a solution is unique.


## 2.3. The Leray-Schauder Theorem

We will use the well-known Leray-Schauder fixed point theorem in the following form (see [16, Theorem 11.3, page 280]).A continuous mapping between two Banach spaces is called compact if the images of bounded sets are precompact.Theorem 9.
LetF be a compact mapping of a Banach space X into itself, and suppose there exists a constant K such that ∥x∥X<K for all x∈X and t∈[0,1] satisfying x=tF(x). Then, F has a fixed point.


## 3. Uniqueness of Solutions: Proof of Theorem2

Proof of Theorem2.
Suppose thatu1,u2∈Kψ,g verify (7); that is, suppose that
(34)∫Ω‍Ax,∇u1+Bx,u1,∇v-u1dx⩾∫Ω‍F,∇v-u1dx,(35)∫Ω‍Ax,∇u2+Bx,u2,∇v-u2dx⩾∫Ω‍F,∇v-u2dx∀v∈Kψ,g. We will prove that u=u1-u2≡0 a.e. in Ω. To this aim we use as test functions vɛ=Tɛ(u2-u1)+u1 in (34) and wɛ=Tɛ(u1-u2)+u2 in (35) for a number ɛ>0. Those functions are admissible since vɛ and wɛ belong to g+W01,2(Ω) and vɛ⩾ψ, wɛ⩾ψ a.e. on Ω. Observing that ∇Tɛ(u1-u2)=-∇Tɛ(u2-u1), we obtain
(36)∫Ω‍Ax,∇u1-Ax,∇u2+Bx,u1-Bx,u2,KKK∇Tɛu1-u2dx⩽0.
Now we set
(37)Ωɛ=x∈Ω:u>ɛ.
We have, using (3), (36), and (4),
(38)α∫Ω‍∇Tɛ(u)2dx=α∫Ω∖Ωɛ‍∇(u1-u2)2dx⩽∫Ω‍A(x,∇u1)-A(x,∇u2),∇Tɛ(u1-u2)dx⩽∫Ω‍bxu1-u2∇Tɛu1-u2dx⩽ɛ∫0<u⩽ɛ‍bx∇Tɛudx⩽ɛ∫0<u⩽ɛ‍bx2dx1/2∫Ω‍∇Tɛu2dx1/2.
Then we have(39)α2∇Tɛ(u)22⩽ɛ2∫0<u⩽ɛ‍bx2dx.
Now, let 0<ɛ<η, so that
(40)ɛ2|Ωη|=∫|u|>η‍Tɛ(u)2dx⩽c∫Ω‍∇Tɛ(u)2dx,
where c=c(N). Combining (39) and the last inequality we obtain
(41)α2Ωη⩽c∫0<u⩽ɛ‍bx2dx.
Lettingɛ→0+ we obtain |Ωη|=0, and then, by the arbitrariness of η>0, we can conclude that u(x)=u1(x)-u2(x)≡0 for almost every x∈Ω.


## 4. Existence of Solutions: Proof of Theorem3

AsKψ,g=∅, it is not restrictive to assume g⩾ψ a.e. in Ω. Moreover, let us observe that if assumption (10) holds true then by (23) there exists a positive constant M=M(α,b,N) such that
(42)b-TMbN,∞<α4S2.
Let us fix such a value of M.Here below we denote(43)ϑx=TMbxbx,
where as above TM is the truncation operator at level M.LetA:Ω×RN→RN and B:Ω×R→RN be Carathèodory functions satisfying (1)–(4). Let us consider the operator A:W1,2(Ω)→(W1,2(Ω))* defined by
(44)Au,v=∫Ω‍A(x,∇u)+(1-θ(x))B(x,u),∇vdx,KKKKKKKKKKKKKKKKKKKKKKKKu,v∈W1,2.
The operator A is strictly monotone and coercive on Kψ,g(Ω). In fact for u,v∈Kψ,g(Ω) we have
(45)Au-Av,u-v=∫Ω‍Ax,∇u-Ax,∇v+1-θxBx,ukkkk-Bx,v,∇u-vdx⩾α∫Ω‍∇u-∇v2dx-∫Ω‍bx-TMbxu-v∇u-vdx⩾α∇(u-v)22-S2b-TMbLN,∞∇(u-v)22=α-S2b-TMbLN,∞∇u-v22.Then, by (42), we have α-S2∥b-TMb∥LN,∞⩾α/2>0.The following technical lemma will be useful in the sequel. We shall follow closely the proof of Lemma 4.1 in [7]. We include some details for the sake of completeness.Lemma 10.
Let one assume (1)–(5), F∈L2(Ω,Rn), g∈W1,2(Ω), ψ⩽g, and let 0<t⩽1. Assume that u∈W1,2 is such that u/t∈Kψ,g(Ω) and verifies
(46)∫Ω‍Ax,∇ut+(1-θ(x))Bx,ut,∇v-utdx⩾∫Ω‍F-θ(x)B(x,u),∇v-utdx
for all v∈Kψ,g. Then,
(47)log⁡1+ut-g2*⩽CF2+b02+b2+bN,∞g2*,2+∇g2+φ2,
where C depends only on α, β, and N.Proof.
Takingv=u/t-(u/t-g)/(1+u/t-g)∈Kψ,g as a test function in (46), we obtain
(48)∫Ω‍u/t-g1+u/t-gAx,∇ut+1-θxBx,ut,kkkk∇u/t-g1+u/t-gdx⩽∫Ω‍F-θxBx,u,∇u/t-g1+u/t-gdx.
By assumptions (2)–(5) we have
(49)α∫Ω‍∇u/t-g21+u/t-g2dx⩽2∫Ω‍Ax,∇u/t,∇u/t-g1+u/t-g2dx+2∫Ω‍Ax,∇u/t,∇g1+u/t-g2dx+2∫Ω‍φx∇u/t1+u/t-g2dx+2α∇g22⩽2∫Ω‍Ax,∇u/t,∇u/t-g1+u/t-g2dx+α2∫Ω‍∇u/t-g21+u/t-g2dx+c∇g22+φ22
with c=c(α,β). Note that in the last inequality we used also Young’s inequality.
Noting that(50)ϑxBx,u⩽TMbxu+b0x
and that
(51)1-ϑxBx,ut⩽bx-TMbx|u|t+b0(x)
and combining (48) and (49), by Hölder’s inequality, we have
(52)∫Ω‍∇u/t-g21+u/t-g2dx⩽4α∫Ω‍F∇u/t-g1+u/t-gdx+4α∫Ω‍TMbxut+b0∇u/t-g1+u/t-g2dx+4α∫Ω‍bx-TMbxut+b0x×∇u/t-g1+u/t-g2dx+c∇g22+φ22⩽4α∇u/t-g1+u/t-g2×F2+2b02+b2+bN,∞g2*,2+c∇g22+2αφ22.
Then, by the elementary relationa⩾0, b⩾0, x2⩽ax+b⇒x⩽a+b we obtain
(53)∇u/t-g1+u/t-g2⩽4αF2+2b02+b2+bN,∞g2*,2+c(∇g2+φ2),
with c=c(α,β). This concludes our proof, observing that by Sobolev embedding theorem
(54)log⁡1+ut-g2*⩽C∇u/t-g1+u/t-g2
with C=C(N).Proof of Theorem3.
We will obtain the existence of a solution to problem (7) by applying the Leray-Schauder fixed point theorem stated in Section 2.3 to a suitable compact operator F. Hence, we will now construct such operator.
Letu¯∈W1,2(Ω). Using Theorem 8 we have that the problem
(55)∫Ω‍Ax,∇u+(1-θ(x))Bx,u,∇v-udx⩾∫Ω‍F-θxBx,u¯,∇v-udx,∀v∈Kψ,g,
admits a unique solution u∈Kψ,g, since the operator A:W1,2(Ω)→(W1,2(Ω))* defined by
(56)Au,v=∫Ω‍Ax,∇u+1-θxBx,u,∇vdx,kkkkkkkkkkkkkkkkkkkkkkkkkkkku,v∈W1,2
is strictly monotone and coercive in Kψ,g (see (45)).
Hence we can define an operator(57)F:W1,2⟶Kψ,g⊆W1,2,u=F(u¯).
We claim that such an operator F is compact.
Let us prove the compactness. (The proof thatF is continuous is similar.) To this aim, suppose that (u¯j) is a bounded sequence in W1,2(Ω). Then, up to a subsequence, there exists u¯∈W1,2(Ω) such that u¯j→u¯ in L2(Ω). Denoting uj=F(u¯j) and u=F(u¯) we have that
(58)∫Ω‍Ax,∇uj+(1-θ(x))Bx,uj,∇u-ujdx⩾∫Ω‍F-θ(x)B(x,u¯j),∇u-ujdx,(59)∫Ω‍Ax,∇u+(1-θ(x))Bx,u,∇uj-udx⩾∫Ω‍F-θ(x)B(x,u¯),∇uj-udx.
Hence, adding (59) to (58) and using (3), (50), (51), and (4) we have
(60)α∫Ω‍∇u-uj2dx⩽∫Ω‍A(x,∇u)-A(x,∇uj),∇u-ujdx⩽∫Ω‍bx-TMbxu-uj∇u-ujdx+∫Ω‍TMbxu¯-u¯j∇u-ujdx⩽S2b-TMbN,∞∇u-uj22+Mu¯-u¯j2∇(u-uj)2.
Dividing the last inequality by∇u-uj2, by (10) and (42),
(61)∇u-uj2⩽4M3αu¯-u¯j2.
This implies that F(u¯j)→F(u¯) in W1,2(Ω) so that we can conclude that F is compact.
A fixed point ofF is a solution to problem (7). We will prove that F has a fixed point. In particular, we will find a constant K>1 such that the a priori estimate ∥u∥W1,2<K holds for every u∈W1,2 and t∈[0,1] satisfying u-tF(u)=0.
To this aim lett∈(0,1] and let u∈W1,2 be a solution to the equation u=tF(u). Then, u/t∈Kψ,g and
(62)∫Ω‍Ax,∇ut+1-θxBx,ut,∇v-utdx⩾∫Ω‍F-θxBx,u,∇v-utdx,kkkkkkkkkkkkKKLkklkklkkkkkk∀v∈Kψ,g.
Now our aim is to estimate∇Tku/t-g2.
We usev=u/t-Tku/t-g∈Kψ,g as a test function in (62) obtaining
(63)∫Ω‍Ax,∇ut+1-θxBx,ut,∇Tkut-gdx⩽∫Ω‍F-θxBx,u,∇Tkut-gdx.
Moreover, we observe that by (2) and (3)(64)α∫u/t-g⩽k‍∇ut2dx⩽∫u/t-g⩽k‍Ax,∇ut-Ax,0,∇utdx⩽∫u/t-g⩽k‍Ax,∇ut,∇ut-gdx+β∫u/t-g⩽k‍∇ut∇gdx+∫u/t-g⩽k‍φxβ∇g+∇utdx⩽∫u/t-g⩽k‍Ax,∇ut,∇ut-gdx+α2∫u/t-g⩽k‍∇ut2dx+c∇g22+φ22
with c=c(α,β). This gives, using (63),
(65)α2∫u/t-g⩽k‍∇ut2dx⩽∫Ω‍F-θxBx,u,∇Tkut-gdx+∫Ω‍1-θxBx,ut∇Tkut-gdx+c∇g22+φ22⩽F2∇Tkut-g2+∫Ω‍TMbxu+2b0x∇Tkut-gdx+∫Ω‍bx-TMbxut∇Tkut-gdx+c∇g22+φ22⩽F2∇Tkut-g2+2b02∇Tkut-g2+∫Ω‍bxut∇Tkut-gdx+c∇g22+φ22.
By (65), using Hölder’s inequality and Theorem 5, we obtain
(66)∫Ω‍∇Tkut-g2dx=∫u/t-g⩽k‍∇ut-g2dx⩽2∫u/t-g⩽k‍∇ut2+∇g2dx⩽4α∇Tkut-g2×F2+2b02+kb2+bN,∞g2*,2+c∇g22+φ22.
Hence, we obtain, for every k∈N,
(67)∇Tkut-g2⩽4αF2+2b02+kb2+bN,∞g2*,2+c∇g2+φ2
with c=c(α,β).
Now, let us denote(68)Gkut-g=ut-g-Tkut-g.
And let us set
(69)Ωk=x∈Ω:ut-g>k.
At this point our aim is to estimate ∥∇Gku/t-g∥2.
Let us preliminarily observe that usingv=u/t-Gku/t-g∈Kψ,g as a test function in (62) we obtain
(70)∫Ω‍Ax,∇ut+1-θxBx,ut,∇Gkut-gdx⩽∫Ω‍F-θxBx,u,∇Gkut-gdx.
Using (2) and (3) and arguing as above by (70) we obtain
(71)α4∫Ω‍∇Gkut-g2dx⩽(F2+2b02+bN,∞g2*,2)∇Gkut-g2+∫Ωk‍TMb(x)Gkut-g+k∇Gkut-gdx+∫Ωk‍(b(x)-TMb(x))Gkut-g+k∇Gkut-gdx+c∇g22+φ22⩽∇Gkut-g2∇Gkut-g2F2+2b02+bN,∞g2*,2+kb2KKKKKKKKKKKKK+S2TMbLN,∞(Ωk)∇Gkut-g2KKKKKKKKKKKKK+S2b-TMbLN,∞∇Gkut-g2+c(∇g22+φ22)
with c=c(α,β). Using (42), this leads to the estimate
(72)∇Gkut-g22⩽c∇Gkut-g2×F2+b02+kb2+bN,∞g2*,2+cTMbLN,∞(Ωk)∇Gkut-g22+c(∇g22+φ22)
with c=c(α,β,b,N,Ω).
On the other hand, sinceu∈W1,2 is a solution to u-tF(u)=0, we can apply Lemma 10 to obtain
(73)Ωk⩽Clog⁡(1+k),
where C=Cα,β,N,Ω,b,g,∥b0∥2,∥F∥2,∥φ∥2. Moreover, by (20), we have
(74)TMbLN,∞Ωk⩽MΩk1/N.
Then, combining (73) and (74), we can now fix k=k0, independent of t and such that
(75)TMbLN,∞Ωk0⩽12c.
By (75) and (72), we obtain
(76)∇Gk0ut-g2⩽cF2+b02+k0b2+bN,∞g2*,2+∇g2+φ2
with c=cα,β,b,N,Ω.
Now we are in a position to estimate∇u/t-g2. We obtain, combining (67) and (76),
(77)∇ut-g2⩽∇Tk0ut-g2+∇Gk0ut-g2⩽cF2+b02+k0b2+bN,∞g2*,2KKK+∇g2+φ2g2*,2=K¯.
Hence, for allt∈[0,1] and all u∈W1,2(Ω) solution to u-tFu=0, we have uW1,2(Ω)<K¯+gW1,2=K, with K=Kα,β,N,Ω,b,g,b02,F2,φ2.
SinceF is a compact operator, Theorem 9 implies that F has a fixed point, which is a solution u of (7).



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*Source: 890289-2015-01-15.xml*