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

Authors: Shuh-Jye Chern; Po-Chun Huang
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

A nonlinear boundary value problem (BVP) from the modelling of the transport phenomena in the cathode catalyst layer of a one-dimensional half-cell single-phase model for proton exchange membrane (PEM) 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 spaceH1.


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## Body

## 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 scientists (see, 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 in [1, 3], are not in the scope of this paper.In [4], 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. In [6], 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 in [6] 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 Section2, 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 Liu [2, 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 in [6], we describe the derivation here for the reader's convenience.Recall (e.g., see [8]) the species equations are(2.1)0=ɛρDkeff∇2Yk+{0,channelanddiffusionlayers,ɛρSk,catalystlayer,
where Yk is the concentration of kth component gas mixture, and Dkeff is the effective diffusivity of the kth component in the gas mixture, which is given by(2.2)Dkeff={Dk,channel,Dkɛ1.5,porousmedia.
At the cathode, the mass generation source termsSk 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 density at anode and cathode, which represent the reaction rates. Note that the value of jc is negative and ja is positive. The relationships between ja, jc and the species concentration (YH2 and YO2) are given by the Butler-Volmer equations(2.3)ja=(a¯i0ref)a(YH2YH2ref)1/2[exp(αaFRTηa)-exp(-(1-αa)FRTηa)],jc=(a¯i0ref)c(YO2YO2ref)[exp(αcFRTηc)-exp(-(1-αc)FRTηc)],
where a¯ is the active catalyst surface area per unit volume of the catalyst layer, i0ref 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(2.4)0=keff∇2T+{0,channelsanddiffusionlayers,Q,catalystlayerandmembrane,
with (2.5)keff={kg,channels,-2ks+1ɛ/(2ks+kg)+(1-ɛ)/3ks,porousmedia,
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(2.6)Q={ja·ηa+i2σcl,anodecatalystlayer,i2σm,membranejc·ηc+i2σcl,cathodecatalystlayer.
Note that this is a main difference between the 3D model of [2] and the 2D model of [5].The phase potential satisfies(2.7)∇·(σ∇ϕ)={jc,cathodecatalystlayer,0,membrane,ja,anodecatalystlayer,
where ϕ is the phase potential, and σ is the ionic/electric conductivity which depends on T:(2.8)σ(T)=ηckefflcl(0.005139λ-0.00326)exp[1268(1303-1T)].
The current density is given by(2.9)i=-σ∇ϕ.Next, we assume thatT,Φ,Yk depend on one space variable and restrict to the cathode side of the catalyst layer, following [4], 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.From (2.4), the equation for energy becomes(2.10)T′′-k(T)Y+λf(T)(Φ′)2=0,x∈(a,b),
where Φ is the catalyst layer phase potential, T is the energy, and Y is the oxygen mass fraction; f(T)∈Cb1(ℝ) is a regularization of 1/λ·(σ(T)/keff) away from T=0 so that f≥δ1>0 is required, and k(T)∈Cb1(ℝ) is a regularization of(2.11)-ηckeff(a¯i0ref)c(1YO2ref)[exp(αcFRTηc)-exp(-(1-αc)FRTηc)].By (2.7), in the cathode catalyst layer, we have the following equation for the phase potential(2.12)(f(T)Φ′)′+g(T)Y=0,x∈(a,b),
where g(T)∈Cb1(ℝ) is a regularization of(2.13)-ηcjckefflclY=-ηckefflcl(a¯i0ref)c(1YO2ref)[exp(αcFRTηc)-exp(-(1-αc)FRTηc)].
And for the oxygen mass fraction, via (2.2), we obtain the equation in the cathode catalyst layer:(2.14)Y′′-h(T)Y=0,x∈(a,b),
where h(T)∈Cb1(ℝ) is a regularization of(2.15)-jc2FcDkeffY=-1Dkeff·12Fc·(a¯i0ref)c(1YO2ref)[exp(αcFRTηc)-exp(-(1-αc)FRTηc)].
We can assume that h≥δ2>0. The boundary conditions for this 1D model are(2.16)μ1T(a)-μ2T′(a)=1,T′(b)=0,Φ(b)+βf(T(b))·Φ′(b)=0,Φ′(a)=0,α1Y(a)-α2Y′(a)=1,Y′(b)=0,
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 in [6]; therefore we do not repeat here.Now, we formulate a weak form of the boundary value problem (2.10)~(2.16).LetΩ=(a,b) and consider (T,Φ,Y)∈(H1(Ω))3; thus it is a  weak solution of (2.10)~(2.16) if the following equations hold:(2.17)-(μ1T(a)-1μ2)·φ(a)-∫abT′·φ′dx=∫abk(T)Yφdx-λ∫abf(T)(Φ′)2φdx,∀φ∈H1(Ω),-1βΦ(b)·φ(b)-∫abf(T)(Φ′)·φ′dx+∫abg(T)Yφdx=0,∀φ∈H1(Ω),-(α1Y(a)-1α2)·φ(a)-∫abY′·φ′dx=∫abh(T)Yφdx,∀φ∈H1(Ω).
For (2.17), we have the following existence theorem.Theorem 2.1.
There exists at least one solution(T,Φ,Y) of (2.17) in (H1(Ω))3.


## 3. Linear Results

Before we prove Theorem2.1, 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:(3.1)-(α1Y(a)-1α2)·φ(a)-∫abY′·φ′dx=∫abh(T*)Yφdx,∀φ∈H1(Ω),(3.2)-1βΦ(b)·φ(b)-∫abf(T*)(Φ′)·φ′dx+∫abg(T*)Y*φdx=0,∀φ∈H1(Ω),(3.3)-(μ1T(a)-1μ2)·φ(a)-∫abT′·φ′dx=∫abk(T*)Y*φdx-λ∫abf(T*)(Φ*′)2φdx,∀φ∈H1(Ω),
where T*, Y*, Φ*∈H1(Ω). Since the equations for (T,Φ,Y) are decoupled, they can be treated separately.The existence and uniqueness of the solution for the linear generalized Neumann problem (3.1)~(3.3) is guaranteed by the following Lax-Milgram Theorem (see [9]).Theorem 3.1 (A theorem on linear monotone operators).
Let𝒜:X→X* be a linear continuous operator on the real Hilbert space X. Suppose that 𝒜 is strongly monotone, that is, there is a c>0 such that
(3.4)〈𝒜u,u〉≥c∥u∥2∀u∈X,
then for each given b̃∈X*, the operator equation
(3.5)𝒜u=b̃,u∈X,
has a unique solution.Next, we show that the solution for the linear problem has ana priori bound which can be shown to be independent of (T*,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
(3.6)∥Y∥H1(Ω)≤N1,(3.7)∥Φ∥H1(Ω)≤N2∥Y*∥∞,(3.8)∥T∥H1(Ω)≤N3∥Φ*′∥22+N4∥Y*∥∞+N5,
where N1,N2,N3,N4,N5 are positive constants, and they depend on ∥f∥∞,∥g∥∞,∥h∥∞.Proof.
(1∘) Equation (3.6) holds. Since
(3.9)-(α1Y(a)-1α2)·φ(a)-∫abY′·φ′dx=∫abh(T*)Yφdx,∀φ∈H1(Ω),
let φ=Y∈H1(Ω); therefore we have
(3.10)-α1α2Y2(a)+1α2Y(a)=∫ab|Y′|2dx+∫abh(T*)Y2dx≥min(δ2,1)∥Y∥H12.
By (3.10), we can get that(3.11)∥Y∥H12≤1min(δ2,1)(-α1α2Y2(a)+1α2Y(a))≤1min(δ2,1)·14α1α2.
Set N1=((1/min(δ2,1))·(1/4α1α2))1/2, then (3.6) is proved.
(2∘)  Equation (3.7) holds. From   (3.2),(3.12)-1βΦ(b)·φ(b)-∫abf(T*)(Φ′)·φ′dx+∫abg(T*)Y*φdx=0,∀φ∈H1(Ω).
Thus let φ=Φ∈H1(Ω) so that
(3.13)-1βΦ2(b)-∫abf(T*)(Φ′)2dx+∫abg(T*)Y*Φdx=0.
It follows that
(3.14)∫abf(T*)(Φ′)2dx=-1βΦ2(b)+∫abg(T*)Y*Φdx≤∥g∥∞∥Y*∥∞∥Φ∥2·l1/2.
Since 0<δ1≤f, we have
(3.15)δ1∫ab(Φ′)2dx≤∥g∥∞∥Y*∥∞∥Φ∥2·l1/2.
To prove (3.7) we need a lemma (see [10]).
Lemma 3.3.
LetX denote a real Banach space, and let  u∈W1,p([0,τ];X) for some  1≤p≤∞. Then (i)
u∈C([0,τ];X) (after possibly being redefined on a set of measure zero), and(ii)
u(t)=u(s)+∫stu′(x)dxforall0≤s≤t≤τ.
By Lemma3.3, we get(3.16)∥Φ∥H12=∫ab|Φ(x)|2dx+∫ab|Φ′(x)|2dx≤∫ab[∫xb|Φ′(t)|dt+|Φ(b)|]2dx+∫ab|Φ′(x)|2dx.
By (3.15), for (3.16), we arrive at
(3.17)∥Φ∥H12≤2∫ab[∫xb|Φ′(t)|dt]2dx+2∫ab|Φ(b)|2dx+1δ1∥g∥∞∥Y*∥∞∥Φ∥2·l1/2≤(∫ab|Φ′(t)|dt)2·2l+2(Φ(b))2l+1δ1∥g∥∞∥Y*∥∞∥Φ∥2·l1/2.
On the other hand, for (3.2), we take φ=1∈H1(Ω), so we know that
(3.18)Φ(b)=β∫abg(T*)Y*dx.
Substituting (3.18) into (3.17), we arrive at
(3.19)∥Φ∥H12≤2∫ab|Φ′(t)|2dt·l2+2β2(∫abg(T*)Y*dx)2·l+1δ1l1/2∥g∥∞∥Y*∥∞∥Φ∥2≤2δ1l5/2∥g∥∞∥Y*∥∞∥Φ∥H1+1δ1l1/2∥g∥∞∥Y*∥∞∥Φ∥H1+2β2l2∥g∥∞2∥Y*∥∞2.
Set x=∥Φ∥H1, B̃=(2/δ1)l5/2∥g∥∞∥Y*∥∞+(1/δ1)l1/2∥g∥∞∥Y*∥∞,C̃=2β2l2∥g∥∞2∥Y*∥∞2 so that we have
(3.20)x2-B̃x-C̃≤0.
Hence, we get that
(3.21)x≤B̃+B̃2+4C̃2=12(2δ1l5/2+1δ1l1/2+(2δ1l5/2+1δ1l1/2)2+8β2l2)∥g∥∞∥Y*∥∞.
Take N2=(1/2)((2/δ1)l5/2+(1/δ1)l1/2+((2/δ1)l5/2+(1/δ1)l1/2)2+8β2l2)∥g∥∞, and from (3.21), we get (3.7).
(3∘) Equation (3.8) holds. For (3.3),  take φ=T∈H1(Ω); thus we have(3.22)-μ1μ2T2(a)+1μ2T(a)-∫ab(T′)2dx=∫abk(T*)Y*Tdx-λ∫abf(T*)(Φ*′)2Tdx.
Now we introduce the following lemma (see [9]).
Lemma 3.4.
Let G be a bounded region inℝN with  N≥1, and  u∈H1(G), set
(3.23)∥u∥H1=(∫G(u2+Σj=1N(Dju)2)dx)1/2,∥u∥H1*=(∫GΣi=1N(Diu)2dx+∫∂Gu2dσ)1/2,
then the two norms in (3.23) are equivalent on H1(G).
Hence we consider theH1(G) norm with the type(3.24)∥u∥H1*=(∫GΣi=1N(Diu)2dx+∫∂Gu2dσ)1/2,
and estimate by
(3.25)∫ab(T′)2dx+ɛ′T2(b)+ɛ′T2(a)(ɛ′isasmallpositivenumbertobedetermined)=-μ1μ2T2(a)+1μ2T(a)+λ∫abf(T*)(Φ*′)2Tdx-∫abk(T*)Y*Tdx+ɛ′T2(b)+ɛ′T2(a)≤-μ1μ2T2(a)+1μ2T(a)+λ∫abf(T*)(Φ*′)2Tdx+(ɛ∫abT2(x)dx+Cɛ∫ab|k|2|Y*|2dx)+ɛ′T2(b)+ɛ′T2(a)≤-μ1μ2T2(a)+1μ2T(a)+λ(ɛ2∥T∥∞2+12ɛ(∫abf(T*)(Φ*′)2dx)2)+ɛ∥T∥H12+Cɛ∥k∥∞2∥Y*∥∞2·l+ɛ′T2(b)+ɛ′T2(a).
By Lemma 3.3, we have that
(3.26)|T(x)|≤∫ax|T′(t)|dt+|T(a)|,∀x∈[a,b].
Substituting (3.26) into the bound (3.25), we arrive at
(3.27)∫ab(T′)2dx+ɛ′T2(b)+ɛ′T2(a)≤-μ1μ2T2(a)+1μ2T(a)+λ(ɛ2∥T∥∞2+12ɛ(∫abf(T*)(Φ*′)2dx)2)+ɛ∥T∥H12+Cɛ∥k∥∞2∥Y*∥∞2·l+ɛ′(2(∫ab|T′|dt)2+2|T(a)|2)+ɛ′T2(a)(ɛ≪ɛ′)≤-μ1μ2T2(a)+1μ2T(a)+λɛ2c12∥T∥H12+λ2ɛ(∫abf(T*)(Φ*′)2dx)2+ɛ∥T∥H12+Cɛ∥k∥∞2∥Y*∥∞2·l+2ɛ′∫ab(T′(t))2dt·l+2ɛ′T2(a)+ɛ′T2(a).
By (3.27), we have that
(3.28)(1-2ɛ′l)∫ab(T′(t))2dt+ɛ′T2(b)+ɛ′T2(a)≤(-μ1μ2+3ɛ′)T2(a)+1μ2T(a)+λɛ2c12∥T∥H12+λ2ɛ(∫abf(T*)(Φ*′)2dx)2+ɛ∥T∥H12+Cɛ∥k∥∞2∥Y*∥∞2·l,
where c1 is the constant that appeared in the Sobolev inequality (see [11] and note that Ω⊂ℝ)
(3.29)∥u∥∞≤c1∥u∥H1(Ω)
for all u∈H1(Ω).
Hence,(3.30)ɛ′∥T∥H12≤C1(ɛ′∫ab(T′(t))2dt+ɛ′T2(b)+ɛ′T2(a))≤C1((1-2ɛ′l)∫ab(T′(t))2dt+ɛ′T2(b)+ɛ′T2(a))≤C1[λ2ɛ(∫abf(T*)(Φ*′)2dx)2(-μ1μ2+3ɛ′)T2(a)+1μ2T(a)+λɛ2c12∥T∥H12+λ2ɛ(∫abf(T*)(Φ*′)2dx)2+ɛ∥T∥H12+Cɛl∥k∥∞2∥Y*∥∞2(∫abf(T*)(Φ*′)2dx)2],
where C1>0, and ɛ′ is chosen to satisfy ɛ′<min{(1/1+2l),(μ1/3μ2)}.
From (3.30), we arrive at(3.31)(ɛ′-λɛ2c12C1-ɛC1)∥T∥H12≤C1(-μ1μ2+3ɛ′)T2(a)+1μ2C1T(a)+λ2ɛC1(∫abf(T*)(Φ*′)2dx)2+CɛC1∥k∥∞2∥Y*∥∞2·l.
Now we chooseɛ>0 such that ɛ′-(λɛ/2)c12C1-ɛC1>0. Note that, in (3.3), if we choose test function φ=1∈H1(Ω), then(3.32)-(μ1T(a)-1μ2)=∫abk(T*)Y*dx-λ∫abf(T*)(Φ*′)2dx.
Hence, we obtain that T(a) is bounded by a number depending on ∥Y*∥∞, l, ∥Φ*∥H1(Ω), ∥k∥∞ and ∥f∥∞, and is independent of T*. So
(3.33)∥T∥H12≤C1(ɛ′-(λɛ/2)c12C1-ɛC1)[λ2ɛ∥f∥∞2∥Φ*′∥24+Cɛ∥k∥∞2∥Y*∥∞2·l-1-4μ1μ2+12ɛ′μ22],
where -(1/μ22)/4(-μ1/μ2+3ɛ′) is the maximum of (-μ1/μ2+3ɛ′)T2(a)+(1/μ2)T(a). By (3.33), we have that
(3.34)∥T∥H1≤C2[λ2ɛ∥f∥∞∥Φ*′∥22+Cɛl∥k∥∞∥Y*∥∞+(14μ1μ2-12ɛ′μ22)1/2],
where C2=(C1/(ɛ′-(λɛ/2)c12C1-ɛC1))1/2.
Let(3.35)N3=C2λ2ɛ∥f∥∞,N4=C2Cɛl∥k∥∞,N5=C2(14μ1μ2-12ɛ′μ22)1/2,
so (3.8) is proved.


## 4. Proof of Theorem2.1

Now, we show the proof of Theorem2.1.Step 1.
Under the assumptions made in Section3, for each (Tn,Φn,Yn)∈(H1(Ω))3,n∈ℕ∪{0}, we first consider the linear generalized Neumann problem,
(4.1)-(μ1Tn+1(a)-1μ2)·φ(a)-∫abTn+1′·φ′dx=∫abk(Tn)Ynφdx-λ∫abf(Tn)(Φn′)2φdx,-1βΦn+1(b)·φ(b)-∫abf(Tn)(Φn+1′)·φ′dx+∫abg(Tn)Ynφdx=0,-(α1Yn+1(a)-1α2)·φ(a)-∫abYn+1′·φ′dx=∫abh(Tn)Yn+1φdx,
for all φ∈H1(Ω).
Denote byRv≡(Yn+1,Φn+1,Tn+1)=(RYn,RΦn,RTn) the unique solution of problem (4.1), where v=(Yn,Φn,Tn)∈(H1(Ω))3.From Theorem3.2, we know that (4.2)∥Yn+1∥H1(Ω)≤N1≡M1,∥Φn+1∥H1(Ω)≤N2∥Yn∥∞≤N2c1∥Yn∥H1(Ω)≤N2c1M1≡M2,∥Tn+1∥H1(Ω)≤N3∥Φn′∥22+N4∥Yn∥∞+N5≤N3∥Φn∥H12+N4c1∥Yn∥H1(Ω)+N5≤N3M22+N4c1M1+N5≡M3.Now we consider the convex set(4.3)S={(v1,v2,v3)∈(H1(Ω))3:∥v1∥H1≤M1,∥v2∥H1(Ω)≤M2,∥v3∥H1(Ω)≤M3}.
By the estimate (4.2), we know that R maps S into S.Step 2.
We show thatR is continuous on S, that is,
(4.4)limj→∞(∥RTj̃-RT̃∥H1+∥RΦj̃-RΦ̃∥H1+∥RYj̃-RỸ∥H1)=0
if (Tj̃,Φj̃,Yj̃)→(T̃,Φ̃,Ỹ) in S, as j→∞.
Consider the equations(4.5)-(α1RỸj(a)-1α2)φ(a)-∫ab(RỸj)′φ′dx-∫abh(Tj̃)(RỸj)φdx=0,-(α1RỸ(a)-1α2)φ(a)-∫ab(RỸ)′φ′dx-∫abh(T̃)(RỸ)φdx=0,
for all φ∈H1(Ω). The difference (4.5) gives
(4.6)-α1α2(RỸj-RỸ)(a)·φ(a)+∫ab(RỸ-RỸj)′φ′dx+∫ab(h(T̃)(RỸ)-h(Tj̃)(RỸj))φdx=0.
By (4.6), we have
(4.7)-α1α2(RỸj-RỸ)(a)·φ(a)+∫ab(RỸ-RỸj)′φ′dx+∫ab[h(Tj̃)(RỸ-RỸj)+RỸ(h(T̃)-h(Tj̃))]φdx=0.
Set Dj=RỸ-RỸj, and let φ=Dj in (4.7), then
(4.8)∫ab(Dj′)2dx+δ2∫abDj2dx≤∥RỸ∥∞∥h′∥∞∥Dj∥∞∥T̃-Tj̃∥2l1/2.
Since 0<δ2≤h, we know that
(4.9)min(δ2,1)∥Dj∥H12≤c1∥RỸ∥∞∥h′∥∞∥Dj∥H1∥T̃-Tj̃∥H1l1/2.
Since limj→∞∥Tj̃-T̃∥H1=0, hence we have
(4.10)limj→∞∥RỸj-RỸ∥H1=0.
The proof of
(4.11)limj→∞∥RΦj̃-RΦ̃∥H1=0,limj→∞∥RTj̃-RT̃∥H1=0
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 (4.12)𝕃u=Di(aij(x)Dju+bi(x)u)+ci(x)Diu+d(x)u,
whose coefficients aij,bi,ci,d(i,j=1,…,n) are continuous on a domain Ω⊂ℝn.Theorem 4.1.
Letu∈H1(Ω) be a weak solution of the equation 𝕃u=p in Ω where 𝕃 is strictly elliptic on Ω, the coefficients aij,bi(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 L2(Ω). Also, assume that ∂Ω is of class C2 and that there exists a function φ∈H2(Ω) for which u-φ∈H01(Ω). Then one has also u∈H2(Ω) and
(4.13)∥u∥H2(Ω)≤C(∥u∥L2(Ω)+∥p∥L2(Ω)+∥φ∥H2(Ω))forC=C(n,λ̃,K,∂Ω),
where K=max{∥aij,bi∥∞,∥ci,d∥∞}.Remark 4.2.
Theorem4.1 continues to hold for sufficiently smooth ∂Ω with φ∈H2(Ω) if we assume only that the principal coefficients aij are in C0(Ω¯).Step 3.
R(S) is precompact.
By Theorem4.1, one has the following:(4.14)∥Yn+1∥H2(Ω)≤C(∥Yn+1∥L2(Ω)+∥Yn+1φ∥H2(Ω)),
where Yn+1φ∈H2(Ω) is a function for which Yn+1-Yn+1φ∈H01(Ω),
(4.15)∥Φn+1∥H2(Ω)≤C(∥Φn+1∥L2(Ω)+∥g(Tn)Yn∥L2(Ω)+∥Φn+1φ∥H2(Ω)),
where Φn+1φ∈H2(Ω) is a function for which Φn+1-Φn+1φ∈H01(Ω),
(4.16)∥Tn+1∥H2(Ω)≤C(∥Tn+1∥L2(Ω)+∥k(Tn)Yn+λf(Tn)(Φn′)2∥L2(Ω)+∥Tn+1φ∥H2(Ω)),
where Tn+1φ∈H2(Ω) is a function for which Tn+1-Tn+1φ∈H01(Ω).
For (4.14)~(4.16), set(4.17)Yn+1φ(x)=Yn+1(b)-Yn+1(a)b-a(x-a)+Yn+1(a),Φn+1φ(x)=Φn+1(b)-Φn+1(a)b-a(x-a)+Φn+1(a),Tn+1φ(x)=Tn+1(b)-Tn+1(a)b-a(x-a)+Tn+1(a),
for all x∈[a,b].
By (4.17), we obtain that(4.18)|Yn+1φ(x)|=|Yn+1(b)-Yn+1(a)b-a(x-a)+Yn+1(a)|≤∫ab|Yn+1′|dx+|Yn+1(a)|≤(l1/2+c1)M1,
for all x∈[a,b].
From (4.14), (4.18) and by a simple calculation, we obtain(4.19)∥Yn+1∥H2(Ω)≤C(∥Yn+1∥L2(Ω)+∥Yn+1φ∥H2(Ω))≤C(M1+M1l1/2+c12l+4c12l)≡γ1.
Using the same formula, we also obtain that(4.20)∥Φn+1∥H2(Ω)≤C(∥Φn+1∥L2(Ω)+∥g(Tn)Yn∥L2(Ω)+∥Φn+1φ∥H2(Ω))≤C(M2+∥g∥∞M1+M2l1/2+c12l+4c12l)≡γ2,∥Tn+1∥H2(Ω)≤C(∥Tn+1∥L2(Ω)+∥k(Tn)Yn+λf(Tn)(Φn′)2∥L2(Ω)+∥Tn+1φ∥H2(Ω))≤C(M3+∥k∥∞M1+λ∥f∥∞(c14γ24l)1/2+M3(l1/2+c1)2l+4c12l)≡γ3.
Thus, we have
(4.21)∥RYn∥H2(Ω)≤γ1,∥RΦn∥H2(Ω)≤γ2,∥RTn∥H2(Ω)≤γ3,∀n∈ℕ.
So,R maps S into a bounded set in (H2(Ω))3, since H2(Ω) is compactly imbedded in H1(Ω) (see, e.g., [9]); hence, R(S) is precompact in (H1(Ω))3.Hence by Schauder's fixed point theorem,R has a fixed point and there exists a (T,Φ,Y)∈(H1(Ω))3 satisfyng (2.17). Thus, we complete the proof of Theorem 2.1.



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*Source: 701096-2010-06-28.xml*