On the Existence of a Weak Solution of a Half-Cell Model for PEM Fuel Cells

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Volume 2010, Article ID 701096,16pages doi:10.1155/2010/701096

Research Article

On the Existence of a Weak Solution of a Half-Cell

Model for PEM Fuel Cells

Shuh-Jye Chern

1

_{and Po-Chun Huang}

2

1_{Department of Mathematics, National Tsing-Hua University, Hsin-Chu 30013, Taiwan}

2_{Department of Applied Mathematics, National Chiao-Tung University, Hsin-Chu 30010, Taiwan}

Correspondence should be addressed to Po-Chun Huang,semihaha@math.nctu.edu.tw

Received 16 December 2009; Accepted 4 May 2010 Academic Editor: Katica R.Stevanovic Hedrih

Copyrightq 2010 S.-J. Chern and P.-C. Huang. 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.

A nonlinear boundary value problemBVP from the modelling of the transport phenomena in the cathode catalyst layer of a one-dimensional half-cell single-phase model for proton exchange membranePEM fuel cells, derived from the 3D model of Zhou and Liu 2000, 2001, is studied. It is a BVP for a system of three coupled ordinary diﬀerential equations of second order. Schauder’s fixed point theorem is applied to show the existence of a solution in the Sobolev space H1_.

1. Introduction

The modelling of fuel cells has been an attractive topic in the field of electrochemical theory. In the last decade, models for proton exchange membrane PEM fuel cells have been formulated by many scientistssee, e.g., 1. Among these models, some complicated

systems of partial diﬀerential equations PDEs were constructed from principles of fluid mechanics, electrostatics, and heat transfers; however, most of them were solved by numerical simulations only. We are interested in the mathematical analysis of the system of diﬀerential equations and the discussion is restricted on the transport phenomenon of a single-phase model given by 2. The more complicated two-phase models, like those

mentioned in1,3, are not in the scope of this paper.

In4, by reducing space variables to one dimension and making several assumptions,

a system of PDEs in 5 was simplified to a boundary value problem BVP for a linear

system of decoupled ordinary diﬀerential equations ODEs, and an exact solution was constructed. In6, a 1D half-cell model reduced from 5 is considered; that model is a BVP

for a nonlinear system of three ODEs of second order which are no longer decoupled and it seems to be hard to find an exact solution. By Schaefer’s fixed point theorem, the study in6

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is able to show the existence of a solution in the space of continuously twice diﬀerentiable functions. In this paper, motivated by 4,6, we will derive a 1D half-cell model from the

3D model of 2; it is still a BVP for a nonlinear system of three ODEs of second order;

however, the nonlinearity is diﬀerent from that of 6 and an alternative strategy will be

applied; namely, a weak formulation of the BVP will be considered. In this weak formulation, the function space is replaced by the Sobolev space H1 _{and an iteration process associated}

with Schauder’s fixed point theorem will be adopted. The result of this paper indicates a direction of attacking the complicated system of PDEs for the modelling of PEM fuel cells.

Now, we briefly describe the contents of this paper. In Section 2, the governed equations and boundary conditions in the cathode catalyst layer for the 1D half-cell model of PEM fuel cells are derived. In Section 3, the weak form of a linear generalized Neumann problem is described. Existence and uniqueness of the generalized Neumann problem is guaranteed by the Lax-Milgram theorem and it will be shown that the solution for the linear problem has an a priori bound. In Section 4, Schauder’s fixed point theorem is applied to prove the existence of an H1_{solution for the nonlinear system of ODEs.}

2. The Model

In this section, we will reduce a 3D model of Zhou and Liu2,7 to a 1D half-cell model. This

3D model was a modification of the 2D model given by Gurau et al.5, so the derivation of

the 1D model is quite the same with what we did in6, we describe the derivation here for

the reader’s convenience.

Recalle.g., see 8 the species equations are

0 ερD_keﬀ∇2Yk

⎧ ⎨ ⎩

0, channel and diﬀusion layers,

ερSk, catalyst layer,

2.1

where Yk is the concentration of kth component gas mixture, and D_keﬀ is the eﬀective

diﬀusivity of the kth component in the gas mixture, which is given by

Deﬀ_k ⎧ ⎨ ⎩ Dk, channel, Dkε1.5, porous media. 2.2

At the cathode, the mass generation source terms Sk for oxygen, water, and protons are

jc/2Fc, −jc/2Fc, and jc/Fc, respectively. At the anode, the source terms for hydrogen

molecules and protons are −ja/2Fc and ja/Fc, where ja and jc are the transfer current

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is negative and jais positive. The relationships between ja, jcand the species concentration

YH2and YO2 are given by the Butler-Volmer equations

ja airef₀ a YH2 Y_Href 2 1/2 exp αa_F RT ηa − exp −1 − αaF RT ηa , jc airef₀ c YO2 Y_Oref 2 exp αc_F RTηc − exp −1 − αcF RT ηc , 2.3

where a is the active catalyst surface area per unit volume of the catalyst layer, iref 0 is the

exchange current density under the reference conditions, T is the absolute temperature,

R is the universal gas constant, αa and αc are symmetric factors, and ηa and ηc are the

corresponding overpotentials. The energy equation is

0 keﬀ∇2T

⎧ ⎨ ⎩

0, channels and diﬀusion layers,

Q, catalyst layer and membrane, 2.4

with keﬀ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ kg, channels, −2ks 1 ε/2ks kg  1 − ε/3ks , porous media, 2.5

where kg is the thermal conductivity of the gas while ks is the thermal conductivity of the

solid matrix of the porous media. The heat generation rates Q in diﬀerent regions are given by Q ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ja· ηa i 2 σcl

, anode catalyst layer, i2

σm, membrane

jc· ηc

i2

σcl, cathode catalyst layer.

2.6

Note that this is a main diﬀerence between the 3D model of 2 and the 2D model of 5.

The phase potential satisfies

∇ ·σ∇φ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

jc, cathode catalyst layer,

0, membrane,

ja, anode catalyst layer,

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where φ is the phase potential, and σ is the ionic/electric conductivity which depends on T: σT  ηc keﬀlcl0.005139λ − 0.00326 exp 1268 1 303− 1 T . 2.8

The current density is given by

i −σ∇φ. 2.9

Next, we assume that T,Φ, Ykdepend on one space variable and restrict to the cathode

side of the catalyst layer, following4, only one species the oxygen i.e., k 1, and let

Y1 Y. For simplicity, the derivative with respect to x is denoted by  d/dx.

From2.4, the equation for energy becomes

T− kTY λfTΦ2  0, x ∈ a, b, 2.10

where Φ is the catalyst layer phase potential, T is the energy, and Y is the oxygen mass fraction; fT ∈ C1

bR is a regularization of 1/λ · σT/keﬀ away from T 0 so that

f≥ δ1> 0 is required, and kT ∈ C1_bR is a regularization of

−ηc k_eﬀ airef₀ c 1 Y_Oref 2 exp αc_F RT ηc − exp −1 − αcF RT ηc . 2.11

By2.7, in the cathode catalyst layer, we have the following equation for the phase

potential fTΦ gTY 0, x ∈ a, b, 2.12 where gT ∈ C1 bR is a regularization of − ηcjc keﬀlclY − ηc keﬀlcl airef₀ c 1 Yref O2 exp αc_F RTηc − exp −1 − αcF RT ηc . 2.13

And for the oxygen mass fraction, via2.2, we obtain the equation in the cathode catalyst

layer: Y− hTY 0, x ∈ a, b, 2.14 where hT ∈ C1 bR is a regularization of − jc 2FcDeﬀ_k Y −1 D_keﬀ· 1 2Fc · airef₀ c 1 Y_Oref 2 exp αc_F RTηc − exp −1 − αcF RT ηc . 2.15

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We can assume that h≥ δ2> 0. The boundary conditions for this 1D model are μ1Ta − μ2Ta 1, Tb 0,

Φb βfTb · Φ_{b 0, Φ}_{a 0,} α1Ya − α2Ya 1, Yb 0,

2.16

where μ1, μ2, α1,α2, β > 0. It is convenient to let l b − a in the following discussions.

Note that the derivation of these boundary conditions can be found in6; therefore

we do not repeat here.

Now, we formulate a weak form of the boundary value problem2.10∼2.16.

LetΩ a, b and consider T, Φ, Y ∈ H1_Ω3_{; thus it is a weak solution of}_2.10_∼

2.16 if the following equations hold:

− _μ 1Ta − 1 μ2 · ϕa − _b a T· ϕdx _b a kTYϕdx − λ _b a fTΦ2ϕdx, ∀ϕ ∈ H1Ω, −1 βΦb · ϕb − _b a fTΦ· ϕdx _b a gTYϕdx 0, ∀ϕ ∈ H1Ω, − α1Ya − 1 α2 · ϕa − b a Y· ϕdx b a hTYϕdx, ∀ϕ ∈ H1Ω. 2.17 For2.17, we have the following existence theorem.

Theorem 2.1. There exists at least one solution T, Φ, Y of 2.17 in H1_Ω3_.

3. Linear Results

Before we prove Theorem 2.1, some linear results should be proved and we still use the notation T, Φ, Y for the solution of the following weak linear generalized Neumann problem: − α1Ya − 1 α2 · ϕa − _b a Y· ϕdx _b a hT∗Yϕdx, ∀ϕ ∈ H1Ω, 3.1 −1 βΦb · ϕb − b a fT∗Φ· ϕdx b a gT∗Y∗ϕdx 0, ∀ϕ ∈ H1Ω, 3.2 − _μ 1Ta − 1 μ2 · ϕa − _b a T· ϕdx _b a kT∗Y∗ϕdx− λ _b a fT∗Φ∗ 2 ϕdx, ∀ϕ ∈ H1Ω, 3.3 where T∗, Y∗,Φ∗∈ H1Ω. Since the equations for T, Φ, Y are decoupled, they can be treated

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The existence and uniqueness of the solution for the linear generalized Neumann problem3.1∼3.3 is guaranteed by the following Lax-Milgram Theorem see 9.

Theorem 3.1 A theorem on linear monotone operators. Let A : X → X∗ _{be a linear}

continuous operator on the real Hilbert space X. Suppose thatA is strongly monotone, that is, there is a c > 0 such that

Au, u ≥ cu2 _{∀u ∈ X,} _3.4 then for each given b∈ X∗, the operator equation

Au b, u ∈ X, 3.5

has a unique solution.

Next, we show that the solution for the linear problem has an a priori bound which can be shown to be independent ofT∗,Φ∗, Y∗ so that a domain for the iteration process exists.

Theorem 3.2. Suppose that Y, Φ, T is a weak solution for 3.1∼3.3, then one has

YH1_Ω≤ N₁, 3.6

ΦH1_Ω≤ N2Y∗∞, 3.7

TH1_Ω≤ N3Φ_∗₂2 N4Y∗∞ N5, 3.8 where N1, N2, N3, N4, N5are positive constants, and they depend onf_∞,g_∞,h∞.

Proof. 1◦ Equation 3.6 holds. Since

− α1Ya − 1 α2 · ϕa − _b a Y· ϕdx _b a hT_∗Yϕdx, ∀ϕ ∈ H1_Ω, _3.9

let ϕ Y ∈ H1_{Ω; therefore we have}

−α1 α2 Y2a 1 α2 Ya  b a _Y2 dx b a hT∗Y2dx≥ minδ2, 1Y2_H1. 3.10

By3.10, we can get that

Y2 H1 ≤ 1 minδ2, 1 −α1 α2 Y2a 1 α2 Ya ≤ _minδ1 2, 1· 1 4α1α2 . 3.11

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2◦_Equation_3.7_{holds. From}_3.2_, −1 βΦb · ϕb − _b a fT∗Φ· ϕdx _b a gT∗Y∗ϕdx 0, ∀ϕ ∈ H1Ω. 3.12

Thus let ϕ Φ ∈ H1Ω so that

−1 βΦ 2_{b −} b a fT∗Φ2dx b a gT∗Y∗Φdx 0. 3.13 It follows that b a fT∗Φ2dx −_β1Φ2b b a gT∗Y∗Φdx ≤g_∞Y∗∞Φ2· l1/2. 3.14 Since 0 < δ1 ≤ f, we have δ1 b a Φ2 dx≤g_∞Y∗∞Φ2· l1/2. 3.15

To prove3.7 we need a lemma see 10.

Lemma 3.3. Let X denote a real Banach space, and let u ∈ W1,p_{0, τ; X for some 1 ≤ p ≤ ∞.}

Then

i u ∈ C0, τ; X (after possibly being redefined on a set of measure zero), and ii ut us t

suxdx for all 0 ≤ s ≤ t ≤ τ. ByLemma 3.3, we get Φ2 H1 b a |Φx|2_dx b a Φ_x2 dx≤ b a b x Φ_t_dt_|Φb| 2 dx b a Φ_x2 dx. 3.16 By3.15, for 3.16, we arrive at Φ2 H1≤ 2 b a b x Φ_t_dt 2 dx 2 b a |Φb|2_dx 1 δ1 _g ∞Y∗∞Φ2· l1/2 ≤ b aΦ _t_dt 2 · 2l 2Φb2_l 1 δ1 g_∞Y∗∞Φ2· l1/2. 3.17

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On the other hand, for3.2, we take ϕ 1 ∈ H1_{Ω, so we know that}

Φb β b

a

gT∗Y∗dx. 3.18

Substituting3.18 into 3.17, we arrive at

Φ2 H1 ≤ 2 b a Φ_t2 dt· l2 2β2 b a gT∗Y∗dx 2 · l  1 δ1 l1/2g_∞Y∗∞Φ2 ≤ 2 δ1 l5/2_g ∞Y∗∞ΦH1 1 δ1 l1/2_g ∞Y∗∞ΦH1 2β2l2g2_∞Y∗2_∞. 3.19 Set x Φ_H1, B 2/δ₁l5/2g_∞Y_∗_∞ 1/δ₁l1/2g_∞Y_∗_∞, C 2β2l2g2_∞Y_∗2_∞so that we have x2− Bx − C≤ 0. 3.20

Hence, we get that

x≤ B  B2₄_C 2 1 2 ⎛ ⎝ 2 δ1 l5/2 1 δ1 l1/2 2 δ1 l5/2 1 δ1 l1/2 2  8β2_l2 ⎞ ⎠g_∞Y∗∞. 3.21 Take N2 1/22/δ1l5/2 1/δ1l1/2 2/δ1l5/2 1/δ1l1/22 8β2l2g_∞, and from 3.21, we get 3.7.

3◦_Equation_3.8_{holds. For}_3.3_{, take ϕ T ∈ H}1_{Ω; thus we have}

−μ1 μ2T 2_a 1 μ2Ta − b a T2dx b a kT∗Y∗Tdx− λ b a fT∗Φ∗2Tdx. 3.22

Now we introduce the following lemmasee 9.

Lemma 3.4. Let G be a bounded region in RN_{with N}_{≥ 1, and u ∈ H}1_{G, set}

uH1 G u2 ΣN_j₁Dju 2 dx 1/2 , u∗_H1 G ΣN i1Diu2dx ∂G u2dσ 1/2 , 3.23

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Hence we consider the H1_{G norm with the type} u∗ H1 G ΣN i1Diu2dx ∂G u2dσ 1/2 , 3.24 and estimate by _b a

T2dx εT2b εT2aεis a small positive number to be determined

−μ1 μ2 T2a 1 μ2 Ta λ _b a fT∗Φ∗ 2 Tdx− _b a kT∗Y∗Tdx εT2b εT2a ≤ −μ1 μ2 T2a 1 μ2 Ta λ _b a fT∗Φ∗ 2 Tdx ε _b a T2xdx Cε _b a |k|2_|Y ∗|2dx  ε_T2_{b ε}_T2_a ≤ −μ1 μ2 T2a 1 μ2 Ta λ ⎛ ⎝ ε 2T 2 ∞_2ε1 b a fT∗Φ∗ 2 dx 2⎞ ⎠ εT2 H1  Cεk2∞Y∗∞2 · l εT2b εT2a. 3.25

ByLemma 3.3, we have that

|Tx| ≤ _x

a

Ttdt |Ta|, ∀x ∈ a, b. 3.26

Substituting3.26 into the bound 3.25, we arrive at

b a T2dx εT2b εT2a ≤ −μ1 μ2T 2_a 1 μ2Ta λ ⎛ ⎝ ε 2T 2 ∞_2ε1 b a fT∗Φ∗2dx 2⎞ ⎠ εT2 H1  Cεk2_∞Y∗2∞· l ε ⎛ ⎝2b a _T_dt 2  2|Ta|2 ⎞ ⎠ ε_T2_a _ε_ε ≤ −μ1 μ2 T2a 1 μ2 Ta λε 2 c 2 1T2H1 λ 2ε b a fT∗Φ∗2dx 2  εT2 H1  Cεk2∞Y∗2∞· l 2ε b a Tt2dt· l 2εT2a εT2a. 3.27

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By3.27, we have that 1− 2εl b a Tt2dt εT2b εT2a ≤−μ1 μ2  3ε _T2_a 1 μ2Ta λε 2 c 2 1T2H1 λ 2ε b a fT∗Φ∗2dx 2  εT2 H1 Cεk2_∞Y∗2∞· l, 3.28

where c1is the constant that appeared in the Sobolev inequalitysee 11 and note that Ω ⊂ R

u_∞≤ c1uH1_Ω 3.29 for all u∈ H1_Ω. Hence, εT2_H1≤ C1 ε _b a Tt2dt εT2b εT2a ≤ C1 1− 2εl b a Tt2dt εT2b εT2a ≤ C1 ⎡ ⎣−μ1 μ2  3ε _T2_a 1 μ2Ta λε 2 c 2 1T2H1 λ 2ε b a fT∗Φ∗2dx 2  εT2 H1 Cεlk2_∞Y∗2∞ ⎤ ⎦, 3.30

where C1> 0, and εis chosen to satisfy ε< min{1/1 2l, μ1/3μ2}.

From3.30, we arrive at ε−λε 2 c 2 1C1− εC1 T2 H1 ≤ C1 −μ1 μ2  3ε _T2_a 1 μ2 C1Ta λ 2εC1 b a fT_∗Φ_∗2dx 2  CεC1k2∞Y∗2∞· l. 3.31 Now we choose ε > 0 such that ε− λε/2c2

1C1− εC1 > 0. Note that, in3.3, if we

choose test function ϕ 1 ∈ H1_{Ω, then}

− _μ 1Ta − 1 μ2 _b a kT∗Y∗dx− λ _b a fT∗Φ∗ 2 dx. 3.32

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Hence, we obtain that Ta is bounded by a number depending on Y∗∞, l,Φ∗H1_Ω,k_∞

andf_∞, and is independent of T_∗. So

T2 H1≤ C1 ε− λε/2c₁2C1− εC1 λ 2εf 2 ∞Φ∗42 Cεk 2 ∞Y∗2∞· l − 1 −4μ1μ2 12εμ2₂ , 3.33 where−1/μ2

2/4−μ1/μ23ε is the maximum of −μ1/μ23εT2a1/μ2Ta. By 3.33,

we have that TH1 ≤ C2 ⎡ ⎣ λ 2εf∞Φ ∗22 Cεlk_∞Y∗∞ 1 4μ1μ2− 12εμ2₂ 1/2⎤ ⎦, 3.34 where C2 C1/ε− λε/2c2₁C1− εC11/2. Let N3 C2 λ 2εf∞, N4 C2 Cεlk∞, N5  C2 1 4μ1μ2− 12εμ22 1/2 , 3.35 so3.8 is proved.

4. Proof of

Theorem 2.1

Now, we show the proof ofTheorem 2.1.

Step 1. Under the assumptions made inSection 3, for eachTn,Φn, Yn ∈ H1Ω3, n ∈ N ∪

{0}, we first consider the linear generalized Neumann problem, − _μ 1Tn1a − 1 μ2 · ϕa − _b a T_n₁· ϕdx _b a kTnYnϕdx− λ _b a fTn Φ n 2 ϕdx, −1 βΦn1b · ϕb − b a fTn Φ n1 · ϕ_dxb a gTnYnϕdx 0, − α1Yn1a − 1 α2 · ϕa − b a Y_n₁· ϕdx b a hTnYn1ϕdx, 4.1 for all ϕ∈ H1_Ω.

Denote by Rv ≡ Yn1,Φn1, Tn1 RYn, RΦn, RTn the unique solution of problem

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FromTheorem 3.2, we know that Yn1H1_Ω≤ N1≡ M1, Φn1H1_Ω≤ N2Yn_∞≤ N2c1YnH1_Ω≤ N2c1M1≡ M2, Tn1H1_Ω≤ N3Φn 2 2 N4Yn∞ N5 ≤ N3Φn2H1 N4c1YnH1_Ω N₅ ≤ N3M22 N4c1M1 N5≡ M3. 4.2

Now we consider the convex set

S " v1,v2, v3 ∈ H1Ω3:v1H1≤ M1,v2H1_Ω≤ M2,v3H1_Ω≤ M3 # . 4.3

By the estimate4.2, we know that R maps S into S.

Step 2. We show that R is continuous on S, that is,

lim j→ ∞ _R_T j− R T H1 R$Φj− R Φ H1 R$Yj− R Y H1 0 _4.4 if Tj, $Φj, $Yj → T, Φ, Y in S, as j → ∞.

Consider the equations

− α1R Yja − 1 α2 ϕa − _b a R Yj ϕdx− _b a hTj R Yj ϕdx 0, − α1R Ya − 1 α2 ϕa − _b a R Yϕdx− _b a hTR Yϕdx 0, 4.5

for all ϕ∈ H1_{Ω. The diﬀerence}_4.5_gives

−α1 α2 R Yj− R Y a · ϕa  _b a R Y − R Yj ϕdx _b a hTR Y− hTj R Yj ϕdx 0. 4.6 By4.6, we have −α1 α2 R Yj− R Y a · ϕa  b a R Y− R Yj ϕdx b a % hTj R Y− R Yj  R YhT− hTj & ϕdx 0. 4.7

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Set Dj R Y − R Yj, and let ϕ Djin4.7, then _b a D_j2dx δ2 _b a D_j2dx≤R Y ∞h ∞Dj_∞ T − Tj 2l 1/2_. _4.8

Since 0 < δ2 ≤ h, we know that

minδ2,1Dj2_H1≤ c1R Y ∞h ∞Dj_H1 T − Tj H1l 1/2_. _4.9

Since limj→ ∞ Tj− T_H1 0, hence we have

lim j→ ∞ R Yj− R Y H1 0. 4.10 The proof of lim j→ ∞ R$Φj− R Φ H1 0, _jlim_{→ ∞} R Tj− R T H1 0 4.11

is similar and is omitted. Hence, R is continuous on S.

Before the next step, we first state a regularity theorem, see 12. Consider the

operator Lu Di aij_xD ju bixu  ci_xD iu dxu, 4.12

whose coeﬃcients aij_{, b}i_{, c}i_{, d}_{i, j 1, . . . , n are continuous on a domain Ω ⊂ R}n_.

Theorem 4.1. Let u ∈ H1_{Ω be a weak solution of the equation Lu p in Ω where L is strictly}

elliptic onΩ, the coeﬃcients aij_{, b}i _{i, j 1, . . . ., n are uniformly Lipschitz continuous on Ω, the}

coeﬃcients ci _{i 1, . . . , n, d are essentially bounded on Ω, and the function p is in L}2_{Ω. Also,} assume that ∂Ω is of class C2_{and that there exists a function ϕ}_{∈ H}2_{Ω for which u − ϕ ∈ H}1

0Ω. Then one has also u∈ H2_{Ω and}

uH2_Ω≤ C uL2_Ωp_L2_Ωϕ_H2_Ω for C Cn, λ, K, ∂Ω, 4.13 where K max{aij_{, b}i ∞,ci, d∞}.

Remark 4.2. Theorem 4.1continues to hold for suﬃciently smooth ∂Ω with ϕ ∈ H2Ω if we

assume only that the principal coeﬃcients aij_{are in C}0_Ω. Step 3. RS is precompact.

ByTheorem 4.1, one has the following:

Yn1H2_Ω≤ C Yn1L2_ΩY ϕ n1 H2_Ω , 4.14

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where Y_nϕ₁∈ H2_{Ω is a function for which Y} n1− Ynϕ1∈ H01Ω, Φn1H2_Ω≤ C Φn1L2_ΩgTnYn_L2_ΩΦ ϕ n1_H2_Ω , 4.15

whereΦϕ_n₁∈ H2_{Ω is a function for which Φ}

n1− Φϕn1∈ H01Ω, Tn1H2_Ω≤ C Tn1L2_ΩkTnYn λfTn Φ n 2 L2_Ω T_nϕ₁ H2_Ω , 4.16

where T_nϕ₁∈ H2_{Ω is a function for which T}

n1− Tnϕ1∈ H01Ω. For4.14∼4.16, set Y_nϕ₁x  Yn1b − Yn1a b− a x − a Yn1a, Φϕ_n₁x  Φn1b − Φn1a b− a x − a Φn1a, T_nϕ₁x  Tn1b − Tn1a b− a x − a Tn1a, 4.17

for all x∈ a, b.

By4.17, we obtain that Y_nϕ₁x Yn1b − Yn1a b− a x − a Yn1a ≤ b a _Y n1dx |Yn1a| ≤ l1/2 c1 M1, 4.18

for all x∈ a, b.

From4.14, 4.18 and by a simple calculation, we obtain

Yn1H2_Ω≤ C Yn1L2_ΩY ϕ n1_H2_Ω ≤ C ⎛ ⎝M1 M1 l1/2_c2 1l 4c₁2 l ⎞ ⎠ ≡ γ1. 4.19

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Using the same formula, we also obtain that Φn1H2_Ω≤ C Φn1L2_ΩgTnYn_L2_ΩΦ ϕ n1_H2_Ω ≤ C ⎛ ⎝M2g_∞M1 M2 l1/2_c2 1l 4c2 1 l ⎞ ⎠ ≡ γ2, Tn1H2_Ω≤ C Tn1L2_ΩkTnYn λfTn Φ n 2 L2_Ω T_nϕ₁ H2_Ω ≤ C ⎛ ⎝M3 k_∞M1 λf_∞ c4₁γ₂4l1/2 M3 l1/2_c₁2_l 4c 2 1 l ⎞ ⎠ ≡ γ3. 4.20 Thus, we have RYnH2_Ω≤ γ1, RΦnH2_Ω≤ γ2, RTnH2_Ω≤ γ3, ∀n ∈ N. 4.21

So, R maps S into a bounded set inH2_Ω3_{, since H}2_{Ω is compactly imbedded in} H1_{Ω see, e.g.,}₉_{; hence, RS is precompact in H}1_Ω3_.

Hence by Schauder’s fixed point theorem, R has a fixed point and there exists a T, Φ, Y ∈ H1_Ω3_satisfyng_2.17_{. Thus, we complete the proof of}_{Theorem 2.1}_.

References

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