Báo cáo hóa học: "APPLICATIONS OF THE POINCARÉ INEQUALITY TO EXTENDED KANTOROVICH METHOD" potx

TO EXTENDED KANTOROVICH METHODDER-CHEN CHANG, TRISTAN NGUYEN, GANG WANG, AND NORMAN M.. WERELEY Received 3 February 2005; Revised 2 March 2005; Accepted 18 April 2005 We apply the Poinca

TO EXTENDED KANTOROVICH METHOD

DER-CHEN CHANG, TRISTAN NGUYEN, GANG WANG,

AND NORMAN M WERELEY

Received 3 February 2005; Revised 2 March 2005; Accepted 18 April 2005

We apply the Poincar´e inequality to study the extended Kantorovich method that wasused to construct a closed-form solution for two coupled partial differential equationswith mixed boundary conditions

Copyright © 2006 Der-Chen Chang et al This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited

The famous Poincar´e inequality can be stated as follows: foru ∈ H1(Ω), then there exists

a universal constantC such that

Hindawi Publishing Corporation

Journal of Inequalities and Applications

DOI 10.1155/JIA/2006/32356

where x=(x1, , x n) with a fixed f ∈ C( ¯ Ω) Then the function v in (1.3) satisfied theboundary value problem

Δv = − f , inΩ

In this paper, we will use the Poincar´e inequality to study the extended Kantorovichmethod, see [6] This method has been used extensively in many engineering problems,for example, readers can consult papers [4,7,8,11,12], and the references therein Let usstart with a model problem, see [8] For a clamped rectangular boxΩ= n

whereΦ is the lateral deflection which satisfies the boundary conditions, η is the flexural

rigidity of the box, and

It is known that (1.9) and (1.11) are called the Galerkin equations of the extended torovich method Now we may first choose

Continue this process until we obtainΦ1···1(x)= f11(x1)f21(x2)··· f n1( x n) and

there-fore completes the first cycle Next, we use f21(x2)··· f n1( x n) as our priori data and find

f12(x1) We continue this process and expect to find a sequence of “approximate tions.” The problem reduces to investigate the convergence of this sequence Therefore, it

solu-is crucial to analyze (1.15) Moreover, from numerical point of view, we know that thissequence converges rapidly (see [1,2]) Hence, it is necessary to give a rigorous mathe-matical proof of this method

2 A convex linear functional onH2(Ω)

Denote

I[φ] =

 Ω

ForΩ⊂ R2, we define the Lagrangian functionL associated to I[φ] as follows:

L :Ω× R × R2× R4−→ R,(x, y; z; X, Y ; U, V , S, W) (U + V )2−2ᏼ(x, y)z, (2.3)whereᏼ(x, y) is a fixed function on Ω which shows up in the integrand of I[φ] With the

above definitions, we have

L

x, y; φ; ∇ φ; D2φ

= | Δφ |2−2ᏼ(x, y)φ(x, y), (2.4)where we have identified

This implies that

L

x, y; φ;∇  φ; D2φ≥ L(x, y; φ;∇ φ; D2φ

+ 2Δφ[ ∇  φ − ∇ φ] (2.12)Therefore,

|Δφ|2−2ᏼ(x, y)φ≥ | Δφ |2−2ᏼ(x, y)φ + 2 Δφ[Δ φ− Δφ]. (2.13)

Lemma 2.1 Suppose either

(1)φ ∈ H2(Ω)∩ C4(Ω) and η∈ C1

c(Ω); or(2)φ ∈ H2(Ω)∩ C3( ¯Ω)∩ C4(Ω) and η ∈ H2(Ω).

Let δI[φ; η] denote the first variation of I at φ in the direction η, that is,

η ∈ H2(Ω), the boundary term vanishes, which proves the lemma 

Lemma 2.2 Let φ ∈ H2(Ω) Then

 φ  H2 ( Ω)≈  Δφ  L2 ( Ω). (2.19)

Proof The function φ ∈ H2(Ω) implies that there exists a sequence{ φ k } ⊂ C ∞ c (Ω) suchthat limk→∞φ k = φ in H2-norm From a well-known result for the Calder ´on-Zygmundoperator (see, Stein [10, page 77]), one has

for all f ∈ C2

c(Rn) and 1< p < ∞ HereC is a constant that depends on n only Applying

this result to eachφ k, we obtain

Lemma 2.3 Let { φ k } be a bounded sequence in H2(Ω) Then there exist φ∈ H2(Ω) and a

subsequence { φ k j } such that

I[φ] ≤lim infI

Proof By a weak compactness theorem for reflexive Banach spaces, and hence for Hilbert

spaces, there exist a subsequence{ φ k j }of{ φ k }andφ in H2(Ω) such that φkj → φ weakly

inH2(Ω) Since

H2(Ω)⊂ H1(Ω)⊂⊂ L2(Ω), (2.25)

by the Sobolev embedding theorem, we have

after passing to yet another subsequence if necessary

Now fix (x, y, φ k j(x, y)) ∈ R2× Rand apply inequality (2.13), we have

Δφk

j 2

−2ᏼ(x, y)φk j(x, y) ≥ | Δφ |2−2ᏼ(x, y)φk j(x, y) + 2 ΔφΔφk j − Δφ (2.27)This implies that

| Δφ |2−2ᏼ(x, y)φ dx d y = I[φ]. (2.29)

Besidesφ k j → φ weakly in H2(Ω) implies that



It follows that when taking limit

I[φ] ≤lim inf

j I

Remark 2.4 The above proof uses the convexity of L(x, y; z; X, Y ; U, V , S, W) when (x, y; z) is fixed We already remarked at the beginning of this section that when (x, y) is fixed, L(x, y; z; X, Y ; U, V , S, W) is convex in the remaining variables, including the z-variable.

That is, we are not required to utilize the full strength of the convexity ofL here.

3 The extended Kantorovich method

Now, we shift our focus to the extended Kantorovich method for finding an approximatesolution to the minimization problem

min

when Ω=[− a, a] ×[− b, b] is a rectangular region in R 2 In the sequel, we will write

φ(x, y) (resp., φ k( x, y)) as f (x)g(y) (resp., f k( x)g k(y)) interchangeably as notated in Kerr

and Alexander [8] More specifically, we will study the extended Kantorovich method forthe casen =2, which has been used extensively in the analysis of stress on rectangularplates Equivalently, we will seek for an approximate solution of the above minimizationproblem in the formφ(x, y) = f (x)g(y) where f ∈ H2([− a, a]) and g ∈ H2([− b, b]).

To phrase this differently, we will search for an approximate solution in the tensorproduct Hilbert spacesH2([− a, a]) ⊗$H2([− b, b]), and all sequences { φ k },{ φ k j }involvedhereinafter reside in this Hilbert space Without loss of generality, we may assume that

Ω=[−1, 1]×[−1, 1] for all subsequent results remain valid for the general case where

Ω=[− a, a] ×[− b, b] by approximate scalings/normalizing of the x and y variables As in

[8], we will treat the special caseᏼ(x, y) = γ, that is, we assume that the load ᏼ(x, y) is

distributed equally on a given rectangular plate

To start the extended Kantorovich scheme, we first choose g0(y) ∈ H2([−1, 1])∩

C ∞ c (−1, 1), and find the minimizerf1(x) ∈ H2([−1, 1]) of the functional:

I

f g0 =

 Ω

where the last equality was obtained via the integration by parts of f f andg0g0 Since

g0has been chosen a priori; we can rewrite the functionalI as

for some positive constantsα and β, independent of g0 Consequently,K(x; z; V ; W) is a

strictly convex function in variablez, V , W when x is fixed In other words, K satisfies K(x; z;V ; W) − K(x; z; V ; W)

for all (x; z; V ; W) and (x; z + z ;V + V ;W + W )∈ R4, with equality at (x; z; V ; W) only

η (x) d dx



η2(x)

Ifη (x) ≡0, thenη (x) =constant which implies thatη (x) ≡0 (since η ∈ C ∞ c (α, β)).

This tells us thatη ≡constant and conclude thatη ≡0 on the interval (α, β).

Ifη (x) ≡0, setU = { x ∈(α, β) : η (x) =0} ThenU is a non-empty open set which

implies that there existx0∈ U and some open setᏻx0ofx0contained inU Then η (ξ) =

0 for allξ ∈ᏻx0⊂ U Thus

d dx

Corollary 3.2 Let J[ f ] be as in ( 3.4 ) Then f1∈ H2([− 1, 1]) is the unique minimizer for J[ f ] if and only if f1solves the following ODE:

Proof Suppose f1 is the unique minimizer Then f1 is a local extremum ofJ[ f ] This

implies thatδJ[ f , η] =0 for allη ∈ H2([−1, 1]) Using the notations in (3.4), we have

H2-norm), implies that

Reversing the roles of f and g, that is, fixing f0and findingg1∈ H2to minimizeI[ f0g]

overg ∈ H2([−1, 1]), we obtain the same conclusion by using the same arguments

Corollary 3.3 Fix f0∈ H2([− 1, 1]) Then g1∈ H2([− 1, 1]) is the unique minimizer for

if and only if g1solves the Euler-Lagrange equation

Remark 3.4 In general when g ∈ H2, that is,g needs not satisfy the zero boundary

con-ditions for function inH2, then the quantity

and equality holds if and only if g ≡ 0.

Proof Integration by parts yields

(i)| g(x) | = λ | Δg(x) |almost everywhere for someλ > 0,

So,g must satisfy the following PDE:

Δg −1

whereg ∈ H2(Ω) But the only solution to this PDE is g≡0 (see, Evans [3, pages 300–

Remark 3.6 If n =1, one can solveg  − λ −1g =0 directly without having to appeal to thetheory of elliptic PDEs

Proposition 3.7 The solutions of ( 3.18 ) and ( 3.24 ) have the same form.

Proof Using eitherLemma 3.5in casen =1 to the above remark, we see that

As observed inRemark 3.4and proved inProposition 3.7, the positivity requirement isnot sufficient The fact that f0,g0∈ H2must be used to conclude this assumption

4 Explicit solution for ( 3.26 )

We now find the explicit solution for (3.26), and hence for (3.18) Let

Thus the homogeneous solution of (3.26) is

f h( x) = c1cosh(ρx) cos(κx) + c2sinh(ρx) cos(κx)

+c cosh(ρx) sin(κx) + c sinh(ρx) sin(κx). (4.3)

It follows that a particular solution of (3.26) is

Thus the solution of (3.18) is

f (x) = c1cosh(ρx) cos(κx) + c2sinh(ρx) cos(κx)

+c3cosh(ρx) sin(κx) + c4sinh(ρx) sin(κx) + c p, (4.5)

wherec p =2 & 1

−1g0(y)d y/  g0 2

L2is a known constant This implies that

f (x) = ρc1sinh(ρx) cos(κx) − κc1cosh(ρx) sin(κx)

+ρc2cosh(ρx) cos(κx) − κc2sinh(ρx) sin(κx)

+ρc3sinh(ρx) sin(κx) + κc3cosh(ρx) cos(κx)

+ρc4cosh(ρx) sin(κx) + κc4sinh(ρx) cos(κx).

(4.6)

Apply the boundary conditions f (1) = f ( −1)= f (1)= f (−1)=0, we get

c1cosh(ρ) cos(κ) + c2sinh(ρ) cos(κ) + c3cosh(ρ) sin(κ) + c4sinh(ρ) sin(κ) = − c p,

c1cosh(ρ) cos(κ) − c2sinh(ρ) cos(κ) − c3cosh(ρ) sin(κ) + c4sinh(ρ) sin(κ) = − c p,

ρ sinh(ρ) sin(κ) + κ cosh(ρ) cos(κ) +c4

ρ cosh(ρ) sin(κ) + κ sinh(ρ) cos(κ) =0,

c1

− ρ sinh(ρ) cos(κ) + κ cosh(ρ) sin(κ) +c2

ρ cosh(ρ) cos(κ) − κ sinh(ρ) sin(κ)

c1cosh(ρ) cos(κ) + c4sinh(ρ) sin(κ) = − c p, (4.8)

c2sinh(ρ) cos(κ) + c3cosh(ρ) sin(κ) =0, (4.9)

ρ cosh(ρ) sin(κ) + κ sinh(ρ) cos(κ) =0. (4.11)

We know, beforehand, that there must be a unique solution Thus (4.9) and (4.23) force

c2= c3=0 We are left to solve forc1andc4from (4.8) and (4.11) But (4.11) tells us that

c1= − c4

ρ cosh(ρ) sin(κ) + κ sinh(ρ) cos(κ)

ρ sinh(ρ) cos(κ) − κ cosh(ρ) sin(κ) . (4.12)

Substituting (4.12) into (4.8), we have

c4= c p

ρ sinh(ρ) cos(κ) − κ cosh(ρ) sin(κ)

ρ sin(κ) cos(κ) + κ sinh(ρ) cosh(ρ) . (4.13)

Plugging (4.13) into (4.12), we have

c1= − c p ρ cosh(ρ) sin(κ) + κ sinh(ρ) cos(κ)

ρ sin(κ) cos(κ) + κ sinh(ρ) cosh(ρ) . (4.14)

Therefore, the solution f1(x) can be written in the form

f1(x) = c p

"K1

K0= ρ sin(κ) cos(κ) + κ sinh(ρ) cosh(ρ),

K1= − ρ cosh(ρ) sin(κ) − κ sinh(ρ) cos(κ),

K2= ρ sinh(ρ) cos(κ) − κ cosh(ρ) sin(κ).

(4.16)

The next step in the extended Kantorovich method is to fix f1(x) just found above and

solve forg1(y) ∈ H2([−1, 1]) from (3.24).Lemma 2.2and the computation above showthat

n−1 

L2/g n−1

L2+g  n−1  2

n−1 

L2/g n−1

L2−g  n−1  2

5 Convergence of the solutions

In order to discuss the convergence of the extended Kantorovich method, let us start withthe following auxiliary lemma

Lemma 5.1 Let φ n( x, y) = f n( x)g n( y) and ψ n( x, y) = f n+1( x)g n( y) Then these two sequences are bounded in H2(Ω).

Proof We will verify the boundedness of { ψ n }for the arguments which is identical forthe sequence{ φ n } Fix an integern ∈Z+and assume thatg nhas been determined fromthe extended Kantorovich scheme whenn ≥1 org nhas been chosen a priori whenn =0

Then f n+1is determined by minimizing

ByCorollary 3.2, if f n+1 is as in (4.19), then f n+1is the unique minimum forJ[ f ] over

H2(Ω) Thus we must have

Now we are in a position to prove the main theorem of this section

Theorem 5.2 There exist subsequences { φ n j } j and { ψ n j } j of { φ n } and { ψ n } which converge

in L2(Ω) to some functions φ,ψ ∈ H2(Ω) Furthermore if

Therefore, the above limits are zero if and only if g0∈ ᐆ.

Proof FromLemma 5.1,{ φ n }and{ ψ n }are bounded inH2(Ω) As a consequence of aweak compactness theorem, there are subsequences{ φ n j }and{ ψ n j }and functionsφ and

From (4.19), we see thatg0∈ ᐆ if and only if f1≡0 Hence ifg0∈ᐆ, the iteration process

of the extended Kantorovich method stops and we haveψ1(x, y) = f1(x)g0(y) ≡0 Nowsupposeg0∈ ᐆ, that is, f1≡0 As in the proof ofLemma 5.1,Corollary 3.2implies that

I[ψ] ≤lim inf

j I

ψ n j :=lim inf

j I

f n j+1g n j (5.11)

In view of (5.10), we must haveJ[ψ] < 0, which implies lim j  ψ n j  L2=  ψ  L2> 0;

oth-erwise, we would have ψ  L2=0 which implies thatJ[ψ] =0 Similarly, limj φ n j  L2=

 φ  L2> 0 Since ψ n j → ψ and φ n j → φ in L2, we also haveψ n j → ψ and φ n j → φ in L1 Thus

Corollary 5.3 Let g0∈ ᐆ and set

Combining with the Poincar´e inequality, it follows that

Corollary 5.4 If g0∈ ᐆ, then there exists a subsequence { f n j g n j } j that converges wisely to a function of the form

Proof Let us observe the expression of φ n( x, y) = f n( x)g n( y) in (4.23) Applying

Corollary 5.3to the constants on the right-hand side of (4.23), we can find convergentsubsequences:

hence Theorem 5.2 and Corollary 5.3 guarantee the convergence of the subsequence

{ c n−1cn−j } Altogether, after replacing all sequences on the right-hand side of (4.23) with

either convergent subsequences, we get

these sequences have convergent subsequences Hence whenx, y are fixed, we conclude

that all derivatives of f n j g n j at (x, y) will converge to that of Θ(x, y) as k → ∞ The proof

Remark 5.5. Corollary 5.4implies that

References

[1] D.-C Chang, G Wang, and N M Wereley, A generalized Kantorovich method and its application

to free in-plane plate vibration problem, Applicable Analysis 80 (2001), no 3-4, 493–523.

[2] , Analysis and applications of extended Kantorovich-Krylov method, Applicable Analysis

82 (2003), no 7, 713–740.

[3] L C Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol 19, American

Mathematical Society, Rhode Island, 1998.

[4] N H Farag and J Pan, Model characteristics of in-plane vibration of rectangular plates, Journal of

Acoustics Society of America 105 (1999), no 6, 3295–3310.

[5] F John, Partial Differential Equations, 3rd ed., Applied Mathematical Sciences, vol 1, Springer,

[8] A D Kerr and H Alexander, An application of the extended Kantorovich method to the stress

analysis of a clamped rectangular plate, Acta Mechanica 6 (1968), 180–196.

[9] E H Lieb and M Loss, Analysis, Graduate Studies in Mathematics, vol 14, American

Mathe-matical Society, Rhode Island, 1997.

[10] E M Stein, Singular Integrals and Differentiability Properties of Functions, Princeton

Mathemat-ical Series, no 30, Princeton University Press, New Jersey, 1970.

[11] G Wang, N M Wereley, and D.-C Chang, Analysis of sandwich plates with viscoelastic damping

using two-dimensional plate modes, AIAA Journal 41 (2003), no 5, 924–932.

[12] , Analysis of bending vibration of rectangular plates using two-dimensional plate modes,

AIAA Journal of Aircraft 42 (2005), no 2, 542–550.

Der-Chen Chang: Department of Mathematics, Georgetown University, Washington,

DC 20057-0001, USA

E-mail address:chang@math.georgetown.edu

Tristan Nguyen: Department of Defense, Fort Meade, MD 20755, USA

E-mail address:tristan@afterlife.ncsc.mil

Gang Wang: Smart Structures Laboratory, Alfred Gessow Rotorcraft Center,

Department of Aerospace Engineering, University of Maryland,

College Park, MD 20742, USA

E-mail address:gwang@eng.umd.edu

Norman M Wereley: Smart Structures Laboratory, Alfred Gessow Rotorcraft Center,

Department of Aerospace Engineering, University of Maryland,

College Park, MD 20742, USA

E-mail address:wereley@eng.umd.edu

That is, we are not required to utilize the full strength of the convexity of< i>L here.

3 The extended Kantorovich. .. directly without having to appeal to thetheory of elliptic PDEs

Proposition 3.7 The solutions of ( 3.18 ) and ( 3.24 ) have the same form.

Proof Using eitherLemma 3.5in... class="page_container" data-page="16">

5 Convergence of the solutions

In order to discuss the convergence of the extended Kantorovich method, let us start withthe following auxiliary lemma

Lemma