General Comparison Principle for Variational Hemivariational Inequalities

Volume 2009, Article ID 184348,29pages doi:10.1155/2009/184348

Research Article

General Comparison Principle for

Variational-Hemivariational Inequalities

Siegfried Carl and Patrick Winkert

Department of Mathematics, Martin-Luther-University Halle-Wittenberg, 06099 Halle, Germany

Correspondence should be addressed to Patrick Winkert,patrick@uni.winkert.de

Received 13 March 2009; Accepted 18 June 2009

Recommended by Vy Khoi Le

We study quasilinear elliptic variational-hemivariational inequalities involving general Leray-Lions operators. The novelty of this paper is to provide existence and comparison results whereby only a local growth condition on Clarke’s generalized gradient is required. Based on these results, in the second part the theory is extended to discontinuous variational-hemivariational inequalities.

Copyrightq2009 S. Carl and P. Winkert. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetΩ ⊂ RN_{,N ≥ 1, be a bounded domain with Lipschitz boundary ∂Ω. ByW}1,p_Ωand W₀1,pΩ,1 < p < ∞, we denote the usual Sobolev spaces with their dual spacesW1,p_Ω∗

and W−1,q_{Ω, respectively, where qis the H ¨older conjugate satisfying 1/p1/q 1. We}

consider the following elliptic variational-hemivariational inequality. Findu∈Ksuch that

AuFu, v−u

Ωj 0

1·, u;v−udx

∂Ωj 0 2

·, γu;γv−γudσ≥0, ∀v∈K, 1.1

where j0

kx, s;r, k 1,2 denotes the generalized directional derivative of the locally

Lipschitz functionss→jkx, satsin the directionrgiven by

j_k0x, s;r lim sup

y→s,t↓0

jk

x, ytr−jk

x, y

cf.1, Chapter 2. We denote byK a closed convex subset ofW1,p_{Ω, andAis a}

second-order quasilinear differential operator in divergence form of Leray-Lions type given by

Aux −

N

i1

∂

∂xiaix, ux,∇ux. 1.3

The operator F stands for the Nemytskij operator associated with some Carath´eodory functionf:Ω×R×RN _{→ Rdefined by}

Fux fx, ux,∇ux. 1.4

Furthermore, we denote the trace operator byγ :W1,p_{Ω → L}p_{∂Ωwhich is known to be}

linear, bounded, and even compact.

The aim of this paper is to establish the method of sub- and supersolutions for problem

1.1. We prove the existence of solutions between a given pair of sub-supersolution assuming only a local growth condition of Clarke’s generalized gradient, which extends results recently obtained by Carl in2. To complete our findings, we also give the proof for the existence of extremal solutions of problem1.1for a fixed ordered pair of sub- and supersolutions in case

Ahas the form

Aux −

N

i1

∂

∂xiaix,∇ux.

1.5

In the second part we consider1.1with a discontinuous Nemytskij operatorF involved, which extends results in 3 and partly of4. Let us consider next some special cases of problem1.1, where we supposeA−Δp.

1IfKW1,p_Ωandj

kare smooth, problem1.1reduces to

−ΔpuFu, v

Ωj

1·, uv dx

∂Ω

j₂·, γuγv dσ0, ∀v∈W1,pΩ, 1.6

which is equivalent to the weak formulation of the nonlinear boundary value problem

−ΔpuFu j1u 0 inΩ,

∂u ∂ν j

2

γu0 on∂Ω,

1.7

2For f ∈ V₀∗, K ⊂ W₀1,pΩ and j2 0, 1.1 corresponds to the variational-hemivariational inequality given by

−Δpuf, v−u

Ωj 0

1·, u;v−udx≥0, ∀v∈K, 1.8

which has been discussed in detail in6.

3IfK⊂W₀1,pΩandjk0, then1.1is a classical variational inequality of the form

u∈K:−ΔpuFu, v−u ≥0, ∀v∈K, 1.9

whose method of sub- and supersolution has been developed in7, Chapter 5.

4LetKW₀1,pΩorKW1,p_Ωandj

knot necessarily smooth. Then problem1.1

is a hemivariational inequality, which contains forKW₀1,pΩas a special case the following Dirichlet problem for the elliptic inclusion:

−ΔpuFu ∂j1·, u0 inΩ,

u0 on∂Ω,

1.10

and forKW1,p_{Ωthe elliptic inclusion}

−ΔpuFu ∂j1·, u0 inΩ,

∂u

∂ν ∂j2·, u0 on∂Ω,

1.11

where the multivalued functions s → ∂jkx, s, k 1,2 stand for Clarke’s

generalized gradient of the locally Lipschitz functions → jkx, s, k 1,2 given

by

∂jkx, s

ξ∈R:j_k0x, s;r≥ξr,∀r∈R. 1.12

Problems of the form1.10and1.11have been studied in5,8, respectively.

Existence results for variational-hemivariational inequalities with or without the method of sub- and supersolutions have been obtained under different structure and regularity conditions on the nonlinear functions by various authors. For example, we refer to9–16. In case thatKis the whole spaceW₀1,pΩorW1,p_{Ω, respectively, problem1.1reduces to}

a hemivariational inequality which has been treated in17–25.

Comparison principles for general elliptic operators A, including the negative p -Laplacian−Δp, Clarke’s generalized gradients → ∂jx, s, satisfying a one-sided growth

condition in the form

for allξi ∈∂jx, si, i1,2, for a.a.x∈Ω, and for alls1, s2withs1 < s2,can be found in7. Inspired by results recently obtained in8,26, we prove the existence ofextremalsolutions for the variational-hemivariational inequality1.1within a sector of an ordered pair of sub-and supersolutionsu, uwithout assuming a one-sided growth condition on Clarke’s gradient of the form1.13.

2. Notation of Sub- and Supersolution

For functionsu, v:Ω → Rwe use the notationu∧vminu, v, u∨vmaxu, v, K∧K {u∧v:u, v ∈K}, K∨K {u∨v :u, v∈K}, andu∧K {u} ∧K, u∨K {u} ∨Kand introduce the following definitions.

Definition 2.1. A function u ∈ W1,p_{Ωis said to be a subsolution of 1.1if the following}

holds:

1Fu∈Lq_Ω;

2AuFu, w−u_Ωj₁0·, u;w−udx_∂Ωj0₂·, γu;γw−γudσ≥0,∀w∈u∧K.

Definition 2.2. A functionu ∈W1,p_{Ωis said to be a supersolution of1.1if the following}

holds:

1Fu∈Lq_Ω;

2AuFu, w−u_Ωj0₁·, u;w−udx_∂Ωj0₂·, γu;γw−γudσ≥0,∀w∈u∨K.

In order to prove our main results, we additionally suppose the following assump-tions:

u∨K⊂K, u∧K⊂K. 2.1

3. Preliminaries and Hypotheses

Let 1< p <∞,1/p1/q1, and assume for the coefficientsai:Ω×R×RN → R, i1, . . . , N

the following conditions.

A1Eachaix, s, ξsatisfies Carath´eodory conditions, that is, is measurable inx∈Ωfor

alls, ξ∈R×RN _{and continuous ins, ξfor a.e.x ∈Ω. Furthermore, a constant} c0>0 and a functionk0∈LqΩexist so that

|aix, s, ξ| ≤k0x c0 |s|p−1|ξ|p−1

3.1

for a.e.x ∈Ωand for alls, ξ∈R×RN_{, where|ξ|denotes the Euclidian norm of}

A2The coefficientsaisatisfy a monotonicity condition with respect toξin the form

N

i1

aix, s, ξ−ai

x, s, ξξi−ξ_i

>0 3.2

for a.e.x∈Ω, for alls∈R, and for allξ, ξ∈RN_{withξ /ξ.}

A3A constantc1>0 and a functionk1∈L1Ωexist such that

N

i1

aix, s, ξξi≥c1|ξ|p−k1x 3.3

for a.e.x∈Ω, for alls∈R,and for allξ∈RN_.

ConditionA1implies thatA : W1,p_{Ω → W}1,p_Ω∗ _{is bounded continuous and along}

with A2; it holds that A is pseudomonotone. Due to A1 the operator A generates a mapping fromW1,p_{Ωinto its dual space defined by}

Au, ϕ

Ω

N

i1

aix, u,∇u ∂ϕ

∂xidx,

3.4

where·,·stands for the duality pairing betweenW1,p_ΩandW1,p_Ω∗_{, and assumption}

A3is a coercivity type condition.

Letu, ube an ordered pair of sub- and supersolutions of problem1.1. We impose the following hypotheses onjkand the nonlinearityfin problem1.1.

j1x→j1x, sandx→j2x, sare measurable inΩand∂Ω, respectively, for alls∈R.

j2s→j1x, sands→j2x, sare locally Lipschitz continuous inRfor a.a.x∈Ωand for a.a.x∈∂Ω, respectively.

j3There are functionsL1 ∈ LqΩandL2 ∈Lq∂Ωsuch that for alls ∈ux, ux the following local growth conditions hold:

η∈∂j1x, s:η≤L1x, for a.a. x∈Ω,

ξ∈∂j2x, s:|ξ| ≤L2x, for a.a. x∈∂Ω.

3.5

F1 ix→fx, s, ξis measurable inΩfor alls, ξ∈R×RN_.

ii s, ξ→fx, s, ξis continuous inR×RNfor a.a.x∈Ω.

iiiThere exist a constantc2 >0 and a functionk3∈LqΩsuch that

_{fx, s, ξ≤k3x c2|ξ|}p−1 _3.6

Note that the associated Nemytskij operator F defined byFux fx, ux,∇ux is continuous and bounded fromu, u⊂W1,p_ΩtoLq_{Ω cf.27. We recall that the normed}

spaceLp_{Ωis equipped with the natural partial ordering of functions defined byu ≤ vif}

and only ifv−u∈LpΩ, whereLpΩis the set of all nonnegative functions ofLpΩ. Based on an approach in 8, the main idea in our considerations is to modify the functionsjk. First we set fork1,2

αkx:min

ξ:ξ∈∂jk

x, ux, βkx:max

ξ:ξ∈∂jkx, ux

. 3.7

By means of3.7we introduce the mappingsj1 :Ω×R → Randj2 :∂Ω×R → Rdefined by

jkx, s

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

jk

x, uxαkx

s−ux, ifs < ux,

jkx, s, ifux≤s≤ux,

jkx, ux βkxs−ux, ifs > ux.

3.8

The following lemma provides some properties of the functionsj1andj2.

Lemma 3.1. Let the assumptions in (j1)–(j3) be satisfied. Then the modified functionsj1:Ω×R → R

andj2:∂Ω×R → Rhave the following qualities.

j1x→j1x, sandx→j2x, sare measurable inΩand∂Ω, respectively, for alls∈R, and

s→j1x, sands→j2x, sare locally Lipschitz continuous inRfor a.a.x∈Ωand for

a.a.x∈∂Ω, respectively.

j2Let ∂jkx, sbe Clarke’s generalized gradient of s → jkx, s. Then for all s ∈ Rthe

following estimates hold true:

η∈∂j1x, s:η≤L1x, for a.a. x∈Ω,

ξ∈∂j2x, s:|ξ| ≤L2x, for a.a. x∈∂Ω.

3.9

j3Clarke’s generalized gradients ofs→j1x, sands→j2x, sare given by

∂jkx, s

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

αkx, ifs < ux,

∂jk

x, ux, ifsux,

∂jkx, s, ifux< s < ux,

∂jkx, ux, ifsux,

βkx, ifs > ux,

3.10

and the inclusions∂jkx, ux⊂∂jkx, uxand∂jkx, ux⊂∂jkx, uxare valid

Proof. With a view to the assumptionsj1–j3and the definition ofjk in3.8, one verifies

the lemma in few steps.

With the aid ofLemma 3.1, we introduce the integral functionalsJ1andJ2defined on

Lp_ΩandLp_{∂Ω, respectively, given by}

J1u

Ω

j1x, uxdx, u∈LpΩ, J2v

∂Ω

j2x, vxdσ, v∈Lp∂Ω. 3.11

Due to the propertiesj1–j2and Lebourg’s mean value theoremsee1, Chapter 2, the functionalsJ1 :LpΩ → RandJ2:Lp∂Ω → Rare well defined and Lipschitz continuous on bounded sets ofLp_ΩandLp_{∂Ω, respectively. This implies among others that Clarke’s}

generalized gradients∂J1:LpΩ → 2L

q_Ω

and∂J2:Lp∂Ω → 2L

q_∂Ω

are well defined, too. Furthermore, by means of Aubin-Clarke’s theoremsee1, foru∈Lp_{Ωandv∈ L}p_∂Ω

we get

η∈∂J1u ⇒η∈LqΩ with ηx∈∂j1x, uxfor a.a. x∈Ω,

ξ∈∂J2v ⇒ξ∈Lq∂Ω with ξx∈∂j2x, vxfor a.a. x∈∂Ω.

3.12

An important tool in our considerations is the following surjectivity result for multivalued pseudomonotone mappings perturbed by maximal monotone operators in reflexive Banach spaces.

Theorem 3.2. LetXbe a real reflexive Banach space with the dual spaceX∗,Φ:X → 2X∗a maximal

monotone operator, andu0 ∈domΦ. LetA:X → 2X

∗

be a pseudomonotone operator, and assume that eitherAu0 is quasibounded orΦu0is strongly quasibounded. Assume further thatA:X → 2X

∗

isu0-coercive, that is, there exists a real-valued functionc:R → Rwithcr → ∞asr → ∞

such that for allu, u∗∈graphAone hasu∗, u−u0 ≥cuXuX. ThenA Φis surjective,

that is,rangeA Φ X∗.

The proof of the theorem can be found, for example, in 28, Theorem 2.12. The notationAu0andΦu0stand forAu0u:Au0uandΦu0u: Φu0u, respectively. Note

that any bounded operator is, in particular, also quasibounded and strongly quasibounded. For more details we refer to 28. The next proposition provides a sufficient condition to prove the pseudomonotonicity of multivalued operators and plays an important part in our argumentations. The proof is presented, for example, in28, Chapter 2.

Proposition 3.3. Let X be a reflexive Banach space, and assume thatA : X → 2X∗ _{satisfies the}

following conditions:

ifor eachu∈Xone has thatAuis a nonempty, closed, and convex subset ofX∗;

iiA:X → 2X∗_{is bounded;}

iiiifun uinXandu∗n u∗inX∗withu∗n∈Aunand iflim supu∗n, un−u ≤0, then u∗∈Auandu∗n, un → u∗, u.

We denote byi∗ : Lq_{Ω → W}1,p_Ω∗ _andγ∗ _{: L}q_{∂Ω → W}1,p_Ω∗ _{the adjoint}

operators of the imbeddingi : W1,p_{Ω → L}p_{Ωand the trace operator γ : W}1,p_{Ω →} Lp_{∂Ω, respectively, given by}

i∗η, ϕ

Ωηϕ dx, ∀ϕ∈W

1,p_{Ω, γ}∗_{ξ, ϕ}

∂Ωξγϕ dσ, ∀ϕ∈W

1,p_{Ω. 3.13}

Next, we introduce the following multivalued operators:

Φ1u: i∗◦∂J1◦iu, Φ2u:

γ∗◦∂J2◦γ

u, 3.14

where i, i∗, γ, γ∗ are defined as mentioned above. The operators Φk, k 1,2, have the

following propertiessee, e.g.,5, Lemmas 3.1 and 3.2.

Lemma 3.4. The multivalued operators Φ1 : W1,pΩ → 2W

1,p_Ω∗

and Φ2 : W1,pΩ →

2W1,p_Ω∗

are bounded and pseudomonotone.

Letb:Ω×R → Rbe the cutofffunction related to the given ordered pairu, uof sub-and supersolutions defined by

bx, s

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

s−uxp−1, ifs > ux,

0, ifux≤s≤ux,

−ux−sp−1, ifs < ux.

3.15

Clearly, the mappingbis a Carath´eodory function satisfying the growth condition

|bx, s| ≤k4x c3|s|p−1 3.16

for a.e. x ∈ Ω, for all s ∈ R, where k4 ∈ LqΩ and c3 > 0. Furthermore, elementary calculations show the following estimate:

Ωbx, uxuxdx≥c4u

p

Lp_Ω−c5, ∀u∈LpΩ, 3.17

wherec4andc5are some positive constants. Due to3.16the associated Nemytskij operator

B:Lp_{Ω → L}q_{Ωdefined by}

Bux bx, ux 3.18

is bounded and continuous. Since the embedding i : W1,p_{Ω → L}p_{Ω is compact, the}

Foru∈W1,p_{Ω, we define the truncation operatorT with respect to the functionsu}

andugiven by

Tux

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

ux, ifux> ux,

ux, ifux≤ux≤ux,

ux, ifux< ux.

3.19

The mappingT is continuous and bounded fromW1,p_ΩintoW1,p_{Ωwhich follows from}

the fact that the functions min·,·and max·,·are continuous fromW1,p_{Ωto itself and that} Tcan be represented asTumaxu, u minu, u−ucf.29. LetF◦T be the composition of the Nemytskij operatorFandTgiven by

F◦Tux fx, Tux,∇Tux. 3.20

Due to hypothesis F1iii, the mapping F ◦ T : W1,p_{Ω → L}q_{Ω is bounded and}

continuous. We setF : i∗◦F◦T :W1,p_{Ω → W}1,p_Ω∗_{, and consider the multivalued}

operator

AATuFλB Φ1 Φ2:W1,pΩ−→2W

1,p_Ω∗

, 3.21

whereλis a constant specified later, and the operatorATis given by

ATu, ϕ− N

i1

Ωaix, Tu,∇u

∂ϕ

∂xidx.

3.22

We are going to prove the following properties for the operatorA.

Lemma 3.5. The operatorA:W1,p_{Ω → 2}W1,p_Ω∗

is bounded, pseudomonotone, and coercive for

λsufficiently large.

Proof. The boundedness ofAfollows directly from the boundedness of the specific operators

AT, F, B, Φ1, and Φ2. As seen above, the operator B is completely continuous and thus

pseudomonotone. The elliptic operatorAT Fis pseudomonotone because of hypotheses

A1, A2, and F1, and in view of Lemma 3.4 the operators Φ1 and Φ2 are bounded and pseudomonotone as well. Since pseudomonotonicity is invariant under addition, we conclude thatA : W1,p_{Ω → 2}W1,p_Ω∗

is bounded and pseudomonotone. To prove the coercivity ofA, we have to find the existence of a real-valued functionc:R → Rsatisfying

lim

s→∞cs ∞, 3.23

such that for allu∈W1,p_Ωandu∗_{∈Authe following holds}

u∗, u−u0 ≥c uW1,p_Ω

for someu0∈K. Letu∗∈Au; that is,u∗is of the form

u∗ ATFλB

u i∗ηγ∗ξ, 3.25

where η ∈ Lq_{Ω with ηx ∈ ∂j}

1x, ux for a.a. x ∈ Ω and ξ ∈ Lq∂Ω with ξx ∈

∂j2x, uxfor a.a.x∈∂Ω. ApplyingA1,A3,F1iii,3.17, andj2, the trace operator

γ:W1,pΩ → Lp∂Ωand Young’s inequality yield

u∗, u−u0

ATFλB

u i∗ηγ∗ξ, u−u0

Ω

N

i1

aix, Tu,∇u∂u−∂u0

∂xi dx

Ω

f·, Tu,∇Tuu−u0 λbx, uu−u0

dx

Ω

ηu−u0

dx

∂Ωξγu−u0dσ

≥c1∇up_Lp_Ω− k1L1_Ω−d₁up_L−p_Ω1 −d2∇up_L−p1_Ω−d3−ε∇up_Lp_Ω−cεu

p Lp_Ω

−d5uLp_Ω−d₆∇up_L−p1_Ω−d7λc4up_Lp_Ω−λc5−d8−d9up_L−p_Ω1

−d10uLp_Ω−d₁₁−d₁₂u_Lp_∂Ω−d₁₃

c1−ε∇up_Lp_Ω λc4−cεup_Lp_Ω−d14∇up_L−p_Ω1 −d15up_L−p_Ω1

−d16uLp_Ω−d17,

3.26

wheredjare some positive constants. Choosingε < c1andλsuch thatλ > cε/c4yields the estimate

u∗, u−u0 ≥d18up_W1,p_Ω−d19u

p−1

W1,p_Ω−d20uW1,p_Ω−d21. 3.27

4. Main Results

Theorem 4.1. Let hypotheses (A1)–(A3), (j1)–(j3), and (F1) be satisfied, and assume the existence of sub- and supersolutionsuandu, respectively, satisfyingu≤uand2.1. Then, there exists a solution of 1.1in the order intervalu, u.

Proof. LetIK : W1,pΩ → R∪ {∞}be the indicator function corresponding to the closed

convex set_{K /}∅given by

IKu

⎧ ⎨ ⎩

0, ifu∈K,

∞, ifu /∈K,

4.1

which is known to be proper, convex, and lower semicontinuous. The variational-hemivariational inequality1.1can be rewritten as follows. Findu∈Ksuch that

AuFu, v−uIKv−IKu

Ωj 0

1·, u;v−udx

∂Ω j0

2

·, γu;γv−γudσ≥0 4.2

for allv ∈ W1,p_{Ω. By using the operatorsA}

T,F, B and the functionsj1,j2 introduced in

Section 3, we consider the following auxiliary problem. Findu∈Ksuch that

ATuFu λBu, v−u

IKv−IKu

Ω

j₁0·, u;v−udx

∂Ω

j₂0·, γu;γv−γudσ≥0

4.3

for allv∈W1,p_{Ω. Consider now the multivalued operator}

A∂IK:W1,pΩ−→2W 1,p_Ω∗

, 4.4

where A is as in 3.21, and ∂IK : W1,pΩ → 2W 1,p_Ω∗

is the subdifferential of the indicator function IK which is known to be a maximal monotone operator cf. 28, page

20. Lemma 3.5 provides that A is bounded, pseudomonotone, and coercive. Applying

Theorem 3.2proves the surjectivity ofA∂IK meaning that rangeA∂IK W1,pΩ∗.

Since 0∈W1,p_Ω∗_{, there exists a solutionu∈Kof the inclusion}

Au ∂IKu0. 4.5

This implies the existence ofη∗∈Φ1u, ξ∗∈Φ2u, andθ∗∈∂IKusuch that

ATuFu λBu η∗ξ∗θ∗0, in W1,pΩ

∗

where it holds in view of3.12and3.14that

η∗i∗η, ξ∗γ∗ξ 4.7

with

η∈LqΩ, ηx∈∂j1x, ux as well as ξ∈Lq∂Ω, ξx∈∂j2

x, γux. 4.8

Due to the Definition of Clarke’s generalized gradient∂jk·, u, k1,2, one gets

η∗, ϕ

Ωηxϕxdx≤

Ω

j₁0x, ux;ϕxdx, ∀ϕ∈W1,pΩ,

ξ∗, ϕ

∂Ωξxγϕxdσ≤

∂Ω

j₂0x, γux;γϕxdσ, ∀ϕ∈W1,pΩ.

4.9

Moreover, we have the following estimate:

θ∗, v−u ≤IKv−IKu, ∀v∈W1,pΩ. 4.10

From4.6we conclude

ATuFu λBu η∗ξ∗θ∗, ϕ

0, ∀ϕ∈W1,pΩ. 4.11

Using the estimates in4.9and4.10to the equation above whereϕis replaced byv−u, yields for allv∈W1,p_Ω

0ATFu λBu η∗ξ∗θ∗, v−u

≤ATuFu λBu, v−u

IKv−IKu

Ω

j₁0·, u;v−udx

∂Ω

j₂0·, γu;γv−γudσ.

4.12

Hence, we obtain a solution u of the auxiliary problem 4.3 which is equivalent to the problem. Findu∈Ksuch that

ATuFu λBu, v−u

Ω

j₁0·, u;v−udx

∂Ω

j₂0·, γu;γv−γudσ≥0, ∀v∈K.

In the next step we have to show that any solution u of 4.13 belongs to u, u. By

Definition 2.2and by choosingwu∨uu u−u∈u∨K, we obtain

AuFu,u−u

Ωj 0 1

·, u;u−udx

∂Ωj 0 2

·, γu;γu−udσ≥0, 4.14

and selectingvu∧uu−u−u ∈Kin4.13provides

ATuFu λBu,−u−u

Ω

j₁0·, u;−u−udx

∂Ω

j₂0·, γu;−γu−udσ≥0.

4.15

Adding these inequalities yields

N

i1

Ωaix, u,∇u−aix, Tu,∇u

∂u−u

∂xi dx

ΩFu−F◦Tuu−u _dx

Ω j 0

1·, u; 1 j10·, u;−1

u−udx

∂Ω j 0 2

·, γu; 1j₂0·, γu;−1γu−udσ

≥λ

ΩBuu−u _dx.

4.16

Let us analyze the specific integrals in 4.16. By using A2 and the definition of the truncation operator, we obtain

Ωaix, u,∇u−aix, Tu,∇u

∂u−u ∂xi dx≤0,

ΩFu−F◦Tuu−u

_dx0.

4.17

Furthermore, we consider the third integral of 4.16in caseu > u; otherwise it would be zero. Applying1.12and3.8proves

j₁0x, ux;−1

lim sup

s→ux,t↓0

j1x, s−t−j1x, s

t

lim sup

s→ux,t↓0

j1x, ux β1xs−t−ux−j1x, ux−β1xs−ux

t

lim sup

s→ux,t↓0

−β1xt

t

−β1x.

Proposition 2.1.2 in1along with3.7shows

j₁0x, ux; 1 maxξ:ξ∈∂j1x, ux

β1x. 4.19

In view of4.18and4.19we obtain

Ω j 0

1·, u; 1 j10·, u;−1

u−udx

Ω

β1x−β1x

u−udx0, 4.20

and analog to this calculation

∂Ω j 0 2

·, γu; 1j₂0·, γu;−1γu−udσ0. 4.21

Due to4.17,4.20, and4.21, we immediately realize that the left-hand side in4.16is nonpositive. Thus, we have

0≥λ

ΩBuu−u _dx

λ

Ωb·, uu−u _dx

λ

{x:ux>ux}u−u

p dx

λ

Ω

u−updx

≥0,

4.22

which impliesu−u 0 and hence,u≤u. The proof foru≤uis done in a similar way. So far we have shown that any solution of the inclusion4.5 which is a solution of4.3as well belongs to the intervalu, u. The latter impliesATuAu,Bu 0 andF◦Tu Fu,

and thus from4.5it follows

AuFu i∗ηγ∗ξ, v−u≥0, ∀v∈K, 4.23

where ηx ∈ ∂j1x, ux ⊂ ∂j1x, uxand ξx ∈ ∂j2x, γux ⊂ ∂j2x, γux, which proves that u ∈ u, u is also a solution of our original problem1.1. This completes the proof of the theorem.

LetSdenote the set of all solutions of1.1within the order intervalu, u. In addition, we will assume thatKhas lattice structure, that is,Kfulfills

We are going to show thatSpossesses the smallest and the greatest element with respect to the given partial ordering.

Theorem 4.2. Let the hypothesis ofTheorem 4.1be satisfied. Then the solution setSis compact.

Proof. First, we are going to show thatSis bounded inW1,p_{Ω. Letu ∈ Sbe a solution of}

4.2, and notice thatSisLp_{Ω-bounded because ofu≤ u≤u. This impliesγu≤ γu≤γu,}

and thus,uis also bounded inLp_{∂Ω. Choosing a fixedvu}

0∈Kin4.2delivers

AuFu, u0−u

Ωj 0

1·, u;u0−udx

∂Ωj 0 2

·, γu;γu0−γu

dσ≥0. 4.25

UsingA1,j3,F1iii, Proposition 2.1.2 in1, and Young’s inequality yields

Au, u ≤

Ω

N

i1

|aix, u,∇u|

∂u0

∂xi

dx

Ω

_{fx, u,∇u|u0−u|dx}

Ωmax

ηu0−u:η∈∂j1x, u

dx

∂Ωmax

ξu0−u:ξ∈∂j2x, u

dσ

≤

Ω

N

i1

k0c0|u|p−1c0|∇u|p−1

|∇u0|dx

Ω k3c2|∇u|

p−1_|

u0−u|dx

ΩL1|u0−u|dx

∂ΩL2

_{γu0−γudσ}

≤e1e2u_Lpp−_Ω1 e3∇u_Lp−p_Ω1 e4e5uLp_Ωe₆∇up−1

Lp_Ωε∇u

p Lp_Ω

cεu_Lpp_Ωe7e8uLp_Ωe9e10uLp_∂Ω

≤ε∇up_Lp_Ωe11∇up_L−p_Ω1 e12∇uLp_Ωe13,

4.26

where the left-hand side fulfills the estimate

Au, u ≥c1∇u_Lpp_Ω−k1. 4.27

Thus, one has

c1−ε∇up_Lp_Ω≤e11∇up_L−p_Ω1 e13, 4.28

where the choiceε < c1proves that∇uLp_Ωis bounded. Hence, we obtain the boundedness

ofuin W1,p_{Ω. Let u}

convergent subsequence, not relabelled, which yields along with the compact imbeddingi:

W1,p_{Ω → L}p_{Ωand the compactness of the trace operatorγ:W}1,p_{Ω → L}p_∂Ω

un u inW1,pΩ,

un−→u inLpΩand a.e.pointwise in Ω,

γun−→γu inLp∂Ωand a.e.pointwise in ∂Ω.

4.29

Asunsolves4.2, in particular, forvu∈K, we obtain

Aun, un−u ≤ Fun, u−un

Ωj 0

1·, un;u−undx

∂Ωj 0 2

·, γun;γu−γun

dσ. 4.30

Sinces, r→j_k0x, s;r, k1,2,is upper semicontinuous and due to Fatou’s Lemma, we get from4.30

lim sup

n→ ∞ Aun, un−u ≤lim supn→ ∞ Fun, u−un →0

Ωlim supn→ ∞ j 0

1·, un;u−un

≤j0 1·,u,00

dx

∂Ωlim supn→ ∞ j 0

2·, γun;γu−γun

≤j0

2·,γu,γ00

dσ≤0.

4.31

The elliptic operatorAsatisfies theS-property, which due to4.31and4.29implies

un−→u inW1,pΩ. 4.32

Replacingubyunin1.1yields the following inequality:

AunFun, v−un

Ωj 0

1·, un;v−undx

∂Ωj 0 2

·, γun;γv−γun

dσ≥0, ∀v∈K.

4.33

Passing to the limes superior in4.33and using Fatou’s Lemma, the strong convergence of

uninW1,pΩ, and the upper semicontinuity ofs, r → j_k0x, s;r, k1,2, we have

AuFu, v−u

Ωj 0

1·, u;v−udx

∂Ωj 0 2

·, γu;γv−γudσ≥0, ∀v∈K. 4.34

In order to prove the existence of extremal elements of the solution setS, we drop the

u-dependence of the operatorA. Then, our assumptions read as follows.

A1Eachaix, ξsatisfies Carath´eodory conditions, that is, is measurable inx ∈Ωfor

allξ ∈RN_{and continuous inξfor a.e.x∈Ω. Furthermore, a constantc}

0 >0 and a functionk0∈LqΩexist so that

|aix, ξ| ≤k0x |ξ|p−1 4.35

for a.e.x∈Ωand for allξ∈RN_{, where|ξ|denotes the Euclidian norm of the vector} ξ.

A2The coefficientsaisatisfy a monotonicity condition with respect toξin the form

N

i1

aix, ξ−ai

x, ξξi−ξi

>0 4.36

for a.e.x∈Ω, and for allξ, ξ∈RN_{withξ /ξ.}

A3A constantc1>0 and a functionk1∈L1Ωexist such that

N

i1

aix, ξξi≥c1|ξ|p−k1x 4.37

for a.e.x∈Ω, and for allξ∈RN_.

Then the operatorA:W1,pΩ → W1,pΩ∗acts in the following way:

Au, ϕ

Ω

N

i1

aix,∇u ∂ϕ

∂xidx.

4.38

Let us recall the definition of a directed set.

Definition 4.3. Let P,≤ be a partially ordered set. A subset C of P is said to be upward

directed if for each pairx, y ∈ Cthere is az ∈ Csuch thatx ≤ zandy ≤ z. Similarly,Cis downward directed if for each pairx, y∈ Cthere is aw∈ Csuch thatw≤xandw≤y. IfC is both upward and downward directed, it is called directed.

Theorem 4.4. Let hypotheses (A1)–(A3) and (j1)–(j3) be fulfilled, and assume that (F1) and4.24 are valid. Then the solution setSof problem1.1is a directed set.

problem. Let the operatorB be given basically as in3.15–3.18with the following slight change:

bx, s

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

s−uxp−1, ifs > ux,

0, ifux≤s≤ux,

−u0x−sp−1, ifs < u0x.

4.39

We introduce truncation operatorsTjrelated touj and modify the truncation operatorT as

follows. Forj 1,2, we define

Tjux

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

ux, ifux> ux,

ux, ifujx≤ux≤ux,

ujx, ifux< ujx,

Tux

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

ux, ifux> ux,

ux, ifu0x≤ux≤ux,

u0x, ifux< u0x,

4.40

and we set

Gux fx, Tux,∇Tux−

2

j1

_{fx, Tux,∇Tux−f}

x, Tjux,∇Tjux 4.41

as well as

F:i∗◦G:W1,pΩ−→ W1,pΩ∗. 4.42

Moreover, we define

αk,jx:min

ξ:ξ∈∂jk

x, ujx

, βkx:max

ξ:ξ∈∂jkx, ux

,

αk,0x:

⎧ ⎨ ⎩

αk,1x, ifx∈ {u1≥u2},

αk,2x, ifx∈ {u2> u1}

4.43

fork, j1,2,and introduce the functionsj1:Ω×R → Randj2:∂Ω×R → Rdefined by

jkx, s

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

jkx, u0x αk,0xs−u0x, if s < u0x,

jkx, s, if u0x≤s≤ux,

jkx, ux βkxs−ux, if s > ux.

Furthermore, we define the functionsh1,j :Ω×R → Randh2,j :∂Ω×R → Rforj0,1,2 as

follows:

hk,0x, s

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

αk,0x, if s≤u0x,

αk,0x

βkx−αk,0x

ux−u0x s−u0x,

if u0x< s < ux,

βkx, if s≥ux,

4.45

and forj1,2

hk,jx, s

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

αk,jx, if s≤ujx,

αk,jx

αk,0x−αk,jx u0x−ujx

s−ujx

, if ujx< s < u0x,

hk,0x, s, if s≥u0x,

4.46

wherek1,2.Note that fork2 we understand the functions above being defined on∂Ω. Apparently, the mappingsx, s→hk,jx, sare Carath´eodory functions which are piecewise

linear with respect tos. Let us introduce the Nemytskij operatorsH1 :LpΩ → LqΩand

H2:Lp∂Ω → Lq∂Ωdefined by

H1ux 2

j1

_{h1,jx, ux−h1,0x, ux,}

H2ux 2

j1

_h2,j

x, γux−h2,0

x, γux.

4.47

Due to the compact imbeddingW1,p_{Ω → L}p_{Ωand the compactness of the trace operator} γ : W1,p_{Ω → L}p_{∂Ω, the operators H}

1 i∗ ◦H1 ◦ i : W1,pΩ → W1,pΩ∗ and

H2 γ∗◦H2 ◦γ : W1,pΩ → W1,pΩ∗ are bounded and completely continuous and thus pseudomonotone. Now, we consider the following auxiliary variational-hemivariational inequality. Findu∈Ksuch that

AuFu λBu, v−u

Ω

j₁0·, u;v−udx−H1u, v−u

∂Ω

j₂0·, γu;γv−γudσ−H2γu, γv−γu

≥0

4.48

for allv ∈ K. The construction of the auxiliary problem 4.48including the functionsHk

upward directed, we have to verify that a solutionuof4.48satisfiesul≤u≤u, l1,2. By

assumptionul∈ S, that is,ulsolves

ul∈K:AulFul, v−ul

Ωj 0

1·, ul;v−uldx

∂Ωj 0 2

·, γul;γv−γul

dσ≥0 4.49

for allv∈K. Selectingvu∧ulul−ul−u ∈Kin the inequality above yields

AulFul,−ul−u

Ωj 0 1

·, ul;−ul−u

dx

∂Ωj 0 2

·, γul;−γul−u

dσ≥0.

4.50

Taking the special test functionvu∨ulu ul−u∈Kin4.48, we get

AuFu λBu,ul−u

Ω

j₁0·, u;ul−u

dx−H1,ul−u

∂Ω

j₂0·, γu;γul−u

dσ−H2γu, γul−u

≥0.

4.51

Adding4.50and4.51yields

Ω

N

i1

aix,∇u−aix,∇ul∂ul−u

∂xi dx

Ω

⎛

⎝_{fx, Tu,∇Tu−fx, ul,∇ul−}2

j1

_{fx, Tu,∇Tu−f}

x, Tju,∇Tju

⎞

⎠_ul−udx

Ω

⎛ ⎝_j0

1·, u; 1 j10·, ul;−1− 2

j1

_{h1,jx, u−h1,0x, u}⎞⎠_ul−udx

∂Ω

⎛ ⎝j0

2

·, γu; 1j0 2

·, γul;−1

−2 j1

_h2,j

x, γu−h2,0

x, γu

⎞

⎠_γul−udσ

≥ −λ

ΩBuul−u _dx.

4.52

The conditionA2implies directly

Ω

N

i1

aix,∇u−aix,∇ul∂ul−u

∂xi dx≤

and the second integral can be estimated to obtain

Ω

⎛

⎝_{fx, Tu,∇Tu−fx, ul,∇ul−}2

j1

_{fx, Tu,∇Tu−f}

x, Tju,∇Tju

⎞

⎠_ul−udx

≤

Ω

fx, Tu,∇Tu−fx, ul,∇ul−fx, Tu,∇Tu−fx, Tlu,∇Tluul−udx

{x∈Ω:ulx>ux}

fx, Tu,∇Tu−fx, ul,∇ul−fx, Tu,∇Tu−fx, ul,∇ulul−udx

≤0.

4.54

In order to investigate the third integral, we make use of some auxiliary calculation. In view of4.44we have forulx> ux

_j0

1x, ux; 1 lim sup

s→ux,t↓0

j1x, st−j1x, s

t

lim sup

s→ux,t↓0

j1x, u0x α1,0xst−u0x−j1x, u0x−α1,0xs−u0x

t

lim sup

s→ux,t↓0

α1,0xt

t

α1,0x.

4.55

Applying Proposition 2.1.2 in1and3.7results in

j₁0x, ulx;−1 max

−ξ:ξ∈∂j1x, ulx

−minξ:ξ∈∂j1x, ulx

−α1,lx.

4.56

Furthermore, we have in caseulx> ux

h1,lx, ux α1,lx,

h1,0x, ux α1,0x.

Thus, we get

Ω

⎛ ⎝_j0

1·, u; 1 j10·, ul;−1− 2

j1

_{h1,jx, u−h1,0x, u}⎞⎠_ul−udx

≤

Ω

j₁0·, u; 1 j₁0·, ul;−1− |h1,lx, u−h1,0x, u|

ul−udx

{x∈Ω:ulx>ux}

α1,0x−α1,lx− |α1,lx−α1,0x|ul−udx

≤0.

4.58

The same result can be proven for the boundary integral meaning

∂Ω

⎛ ⎝_j0

2

·, γu; 1j₂0·, γul;−1

−2 j1

_h2,j

x, γu−h2,0

x, γu

⎞

⎠_{γul−udσ≤0. 4.59}

Applying4.53–4.59to4.52yields

0≥ −λ

ΩBuul−u _dx

−λ

{x∈Ω:ulx>ux}

−u0−up−1ul−udx

≥λ

Ω

ul−u

p dx

≥0,

4.60

and hence,ul−u 0 meaning thatul ≤ uforl 1,2. This provesu0 max{u1, u2} ≤u. The proof for u ≤ ucan be shown in a similar way. More precisely, we obtain a solution

u ∈ Kof 4.48satisfyingu ≤ u0 ≤ u ≤ uwhich impliesFu f·, u,∇u,Bu 0 and

H1u H2γu 0. The same arguments as at the end of the proof ofTheorem 4.1apply, which shows thatuis in fact a solution of problem 1.1belonging to the intervalu0, u. Thus, the solution setS is upward directed. Analogously, one proves that S is downward directed.

Theorems4.2and4.4allow us to formulate the next theorem about the existence of extremal solutions.

Proof. SinceS ⊂W1,p_ΩandW1,p_{Ωare separable,Sis also separable; that is, there exists a}

countable, dense subsetZ{zn:n∈N}ofS. We construct an increasing sequenceun⊂ S

as follows. Letu1 z1and selectun1∈ Ssuch that

maxzn, un≤un1≤u. 4.61

By Theorem 4.4, the elementun1 exists because S is upward directed. Moreover, we can

choose byTheorem 4.2a convergent subsequence denoted again byunwith un → u in W1,p_Ωandu

nx → uxa.e.inΩ. Sinceunis increasing, the entire sequence converges

inW1,p_{Ωand further,usupu}

n. One sees at once thatZ⊂u, uwhich follows from

maxz1, . . . , zn≤un1 ≤u, ∀n, 4.62

and the fact thatu, uis closed inW1,p_Ωimplies

S ⊂Z⊂!u, u"!u, u". 4.63

Therefore, asu ∈ S, we conclude thatuis the greatest element in S. The existence of the smallest solution of1.1inu, ucan be proven in a similar way.

Remark 4.6. IfAdepends ons, we have to require additional assumptions. For example, ifA

satisfies insa monotonicity condition, the existence of extremal solutions can be shown, too. In caseKW1,p_{Ω, a Lipschitz condition with respect tosis sufficient for proving extremal}

solutions. For more details we refer to7.

5. Generalization to Discontinuous Nemytskij Operators

In this section, we will extend our problem in1.1to include discontinuous nonlinearitiesf

of the formf:Ω×R×R×RN → R. We consider again the elliptic variational-hemivariational inequality

AuFu, v−u

Ωj 0

1·, u;v−udx

∂Ωj 0 2

·, γu;γv−γudσ≥0, ∀v∈K, 5.1

where all denotations ofSection 1are valid. Here,Fdenotes the Nemytskij operator given by

Fux fx, ux, ux,∇ux, 5.2

where we will allow f to depend discontinuously on its third argument. The aim of this section is to deal with discontinuous Nemytskij operators F : u, u ⊂ W1,p_{Ω → L}q_Ω

Definition 5.1. A functionu∈W1,p_{Ωis called a subsolution of5.1if the following holds:}

1Fu∈Lq_Ω;

2AuFu, w−u_Ωj0

1·, u;w−udx

∂Ωj

0

2·, γu;γw−γudσ≥0,∀w∈u∧K.

Definition 5.2. A functionu∈W1,pΩis called a supersolution of5.1if the following holds:

1Fu∈Lq_Ω;

2AuFu, w−u_Ωj₁0·, u;w−udx_∂Ωj0₂·, γu;γw−γudσ≥0,∀w∈u∨K.

The conditions for Clarke’s generalized gradient s → ∂jkx, s and the functions jk, k1,2,are the same as inj1–j3. We only change the propertyF1to the following.

F2 ix → fx, r, ux, ξ is measurable for all r ∈ R, for all ξ ∈ RN_{, and for all}

measurable functionsu:Ω → R.

ii r, ξ→fx, r, s, ξis continuous inR×RN_{for alls∈Rand for a.a.x∈Ω.}

iiis→fx, r, s, ξis decreasing for allr, ξ∈R×RN_{and for a.a.x∈Ω.}

ivThere exist a constantc2 >0 and a functionk2∈LqΩsuch that

_{fx, r, s, ξ≤k2x c0|ξ|}p−1 _5.3

for a.e.x∈Ω, for allξ∈RN_{, and for allr, s∈ux, ux.}

By31the mappingx→fx, ux, ux,∇uxis measurable foru∈W1,p_{Ω; however, the}

associated Nemytskij operatorF :W1,p_Ω⊂Lp_{Ω → L}q_{Ωis not necessarily continuous.}

An important tool in extending the previous result to discontinuous Nemytskij operators is the next fixed point result. The proof of this lemma can be found in30, Theorem 1.1.1.

Lemma 5.3. LetP be a subset of an ordered normed space,G:P → P an increasing mapping, and

GP {Gx|x∈P}.

1IfGPhas a lower bound inP and the increasing sequences ofGPconverge weakly in

P, thenGhas the least fixed pointx∗, andx∗min{x|Gx≤x}.

2IfGPhas an upper bound inPand the decreasing sequences ofGPconverge weakly in

P, thenGhas the greatest fixed pointx∗, andx∗max{x|x≤Gx}.

Our main result of this section is the following theorem.

Theorem 5.4. Assume that hypotheses (A1)–(A3), (j1)–(j3), (F2), and4.24are valid, and letuand

ube sub- and supersolutions of 5.1satisfyingu≤uand2.1. Then there exist extremal solutions

u∗and u_∗of 5.1withu≤u∗≤u∗≤u.

Proof. We consider the following auxiliary problem:

u∈K:AuFzu, v−u

Ωj 0

1·, u;v−udx

∂Ωj 0 2

·, γu;γv−γudσ≥0, ∀v∈K,

whereFzux fx, ux, zx,∇ux, and we define the setH : {z ∈ W1,pΩ : z ∈

u, u, andzis a supersolution of5.1satisfyingz∧K ⊂K}. OnHwe introduce the fixed point operatorL : H → K byz → u∗ : Lz, that is, for a given supersolutionz ∈ H, the elementLz is the greatest solution of5.4in u, z, and thus, it holdsu ≤ Lz ≤ zfor all

z∈H. This impliesL:H → u, u∩K. Because of4.24,Lzis also a supersolution of5.4

satisfying

ALzFzLz, w−Lz

Ωj 0

1·, Lz;w−Lzdx

∂Ωj 0 2

·, γLz;γw−γLzdσ≥0 5.5

for allw∈Lz∨K. By the monotonicity off with respect to its third argument,Lz≤z, and using the representationwLz v−Lzfor anyv∈Kwe obtain

0≤ALzFzLz,v−Lz

Ωj 0 1

·, Lz;v−Lzdx

∂Ωj 0 2

·, γLz;γv−Lzdσ

≤ ALzFLzLz,v−Lz

Ωj 0 1

·, Lz;v−Lzdx

∂Ω j20

·, γLz;γv−Lzdσ

5.6

for allv∈K. Consequently,Lzis a supersolution of5.1. This showsL:H → H. Letv1, v2∈H, and assume thatv1≤v2. Then we have the following.

Lv1∈

!

u, v1

"

is the greatest solution of

AuFv1u, v−u

Ωj 0

1·, u;v−udx

∂Ωj 0 2

·, γu;γv−γudσ≥0, ∀v∈K.

5.7

Lv2∈

!

u, v2

"

is the greatest solution of

AuFv2u, v−u

Ωj 0

1·, u;v−udx

∂Ωj 0 2

·, γu;γv−γudσ≥0, ∀v∈K. 5.8

Sincev1 ≤ v2, it follows thatLv1 ≤ v2,and due to4.24,Lv1 is also a subsolution of5.7, that is,5.7holds, in particular, forv∈Lv1∧K, that is,

0≥ALv1Fv1Lv1,Lv1−v

₋ Ωj

0 1

·, Lv1;−Lv1−v

dx − ∂Ω j0 2

·, γLv1;−γLv1−v

dσ

for allv∈K. Using the monotonicity offwith respect to its third argumentsyields

0≥ ALv1Fv1Lv1,Lv1−v

₋ Ωj

0 1

·, Lv1;−Lv1−v

dx

−

∂Ωj 0 2

·, γLv1;−γLv1−v

dσ

≥ ALv1Fv2Lv1,Lv1−v

₋ Ωj

0 1

·, Lv1;−Lv1−v

dx

−

∂Ωj 0 2

·, γLv1;−γLv1−v

dσ

5.10

for allv∈K. Hence,Lv1is a subsolution of5.8. ByTheorem 4.5, we know that there exists the greatest solution of5.8inLv1, v2. ButLv2 is the greatest solution of5.8inu, v2 ⊇

Lv1, v2and therefore,Lv1≤Lv2. This shows thatLis increasing.

In the last step we have to prove that any decreasing sequence of LH converges weakly inH. Letun Lzn⊂LH⊂Hbe a decreasing sequence. Thenunxuxa.e. x ∈Ωfor someu∈ u, u. The boundedness ofun inW1,pΩcan be shown similarly as in

Section 4. Thus the compact imbeddingi:W1,pΩ → LpΩalong with the monotony ofun

as well as the compactness of the trace operatorγ:W1,p_{Ω → L}p_∂Ωimplies

un u inW1,pΩ,

un−→u inLpΩand a.e.pointwise in Ω,

γun−→γu inLp∂Ωand a.e.pointwise in ∂Ω.

5.11

Sinceun ∈ K, it followsu ∈K. From5.4withureplaced byunandvbyu, and using the

fact thats, r→j0

kx, s;r, k1,2,is upper semicontinuous, we obtain by applying Fatou’s

Lemma

lim sup

n→ ∞ Aun, un−u ≤lim supn→ ∞ Fznun, u−unlim supn→ ∞

Ωj 0

1·, un;u−undx

lim sup

n→ ∞

∂Ωj 0 2

·, γun;γu−γun

dσ

≤lim sup

n→ ∞

Fznun, u−un

→0

Ωlim supn→ ∞

j₁0·, un;u−un

≤j0 1·,u;00

dx

∂Ωlim supn→ ∞ j 0

2·, γun;γu−γun

≤j0

2·,γu;γ00

dσ

≤0.