**APPLICATIONS OF THE POINCARÉ INEQUALITY** **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 was used to construct a closed-form solution for two coupled partial diﬀerential equations with mixed boundary conditions.

Copyright © 2006 Der-Chen Chang et al. 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*⊂*R* ^{n}*be a Lipschitz domain inR

*. Consider the Dirichlet space*

^{n}*H*

_{0}

^{1}(Ω) which is the collection of all functions in the Sobolev space

*L*

^{2}

_{1}(Ω) such that

*H*_{0}^{1}(Ω)*=*

*u**∈**L*^{2}(Ω) :*u**|**∂Ω**=*0,*u**L*^{2}+
*n*
*k**=*1

*∂u*

*∂x**k*

*L*^{2}

*<**∞*

*.* (1.1)

The famous Poincar´e inequality can be stated as follows: for*u**∈**H*0^{1}(Ω), then there exists
a universal constant*C*such that

Ω*u*^{2}**(x)dx***≤**C*
*n*
*k**=*1

Ω

*∂u*

*∂x*_{k}

^{2}*dx.* (1.2)

One of the applications of this inequality is to solve the modified version of the Dirichlet
problem (see, John [5, page 97]): find a*v**∈**H*0^{1}(Ω) such that

(u,*v)**=*

Ω

_{n}

*k**=*1

*∂u*

*∂x**k*

*∂v*

*∂x**k* *dx**=*

Ω*u(x)f***(x)dx,** (1.3)

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 32356, Pages1–21 DOI10.1155/JIA/2006/32356

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

*Δv**= −**f*, inΩ

*v**=*0, on*∂Ω.* (1.4)

In this paper, we will use the Poincar´e inequality to study the extended Kantorovich
method, 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 us
start with a model problem, see [8]. For a clamped rectangular boxΩ*=*_{n}

*k**=*1[*−**a** _{k}*,a

*], subjected to a lateral distributed load,ᏼ*

_{k}**(x)**

*=*ᏼ(x1,

*. . .,x*

*), the principle of virtual dis- placements yields*

_{n}*n*
*=*1

_{a}_{}

*−**a*

*η**∇*^{4}Φ*−*ᏼ^{}*δΦDx**=*0, (1.5)

whereΦis the lateral deflection which satisfies the boundary conditions,*η*is the flexural
rigidity of the box, and

*∇*^{4}*=*
*n*
*k**=*1

*∂*^{4}

*∂x*^{4}* _{k}*+

^{}

*j**=**k*

2 *∂*^{4}

*∂x*^{2}_{j}*∂x*^{2}_{k}*.* (1.6)

Since the domainΩis a rectangular box, it is natural to assume the deflection in the form
Φ**(x)***=*Φ*k*1*···**k**n***(x)***=*

*n*
*=*1

*f*_{k}_{}^{}*x*_{}^{}, (1.7)

it follows that when *f**k*2(x2)*···**f**k**n*(x*n*) is prescribed a priori, (1.5) can be rewritten as
_{a}_{1}

*−**a*1

_{n}

*=*2

_{a}_{}

*−**a*

*η**∇*^{4}Φ*k*1*···**k**n**−*ᏼ^{}*f*_{k}_{}^{}*x*_{}^{}*dx*_{}*δ f*_{k}_{1}(x1)dx1*=*0. (1.8)

Equation (1.8) is satisfied when
*n*
*=*2

_{a}_{}

*−**a*

*η**∇*^{4}Φ*k*1*···**k**n**−*ᏼ^{}*f*_{k}_{}^{}*x*_{}^{}*dx*_{}*=*0. (1.9)

Similarly, when^{}^{n}_{}_{=}_{1,}_{=}_{m}*f**k*(x) is prescribed a priori, (1.5) can be rewritten as
_{a}_{m}

*−**a**m*

_{n}

*=*1,*=**m*

_{a}_{}

*−**a*

*η**∇*^{4}Φ*k*1*···**k**n**−*ᏼ^{}*f**k*

*x*

*dx* *δ f**k**m*(x*m*)dx*m**=*0. (1.10)

It is satisfied when

*n*
*=*1,*=**m*

_{a}_{}

*−**a*

*η**∇*^{4}Φ*k*1*···**k**n**−*ᏼ^{}*f*_{k}_{}^{}*x*_{}^{}*dx*_{}*=*0. (1.11)

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

*f*20(x2)*···**f**n0*(x*n*)*=*
*n*
*=*2

*c*

*x*^{2}_{}*a*^{2}_{}* ^{−}*1

2

*.* (1.12)

ThenΦ10*···*0**(x)***=**f*11(x1)*f*20(x2)*···**f**n0*(x*n*) satisfies the boundary conditions
Φ10* _{···}*0

*=*0,

*∂Φ*10

*0*

_{···}*∂x*_{}* ^{=}*0 at

*x*

_{}

_{= ±}*a*

*,*

_{}*x*1

*∈*

*−**a*1,a1

, (1.13)

for*=*2,*. . .,n. Now (1.9) becomes*
*n*

*=*2

*c*_{}_{a}_{}

*−**a*

*∇*^{4}Φ10*···*0*−*ᏼ
*η*

*x*^{2}_{}*a*^{2}_{}* ^{−}*1

2

*dx*_{}*=*0, (1.14)

which yields

*C*4

*d*^{4}*f*11

*dx*^{4} +*C*2

*d*^{2}*f*11

*dx*^{2} +*C*0*f*11*=**B.* (1.15)

After solving the above ODE, we can use *f*11(x1)^{}^{n}_{}_{=}_{3}*f** _{0}*(x

*) as a priori data and plug it into (1.10) to find*

_{}*f*21(x2). Then we obtain the function

Φ110*···*0**(x)***=* *f*11

*x*1

*f*21

*x*2

*f*30

*x*3

*···**f**n0*
*x**n*

*.* (1.16)

Continue this process until we obtainΦ1*···*1**(x)***=* *f*11(x1)*f*21(x2)*···**f** _{n1}*(x

*) and there- fore completes the first cycle. Next, we use*

_{n}*f*21(x2)

*···*

*f*

*n1*(x

*n*) as our priori data and find

*f*12(x1). We continue this process and expect to find a sequence of “approximate solu- tions.” The problem reduces to investigate the convergence of this sequence. Therefore, it is crucial to analyze (1.15). Moreover, from numerical point of view, we know that this sequence converges rapidly (see [1,2]). Hence, it is necessary to give a rigorous mathe- matical proof of this method.

**2. A convex linear functional on***H*0^{2}(Ω)
Denote

*I[φ]**=*

Ω

*|**Δφ**|*^{2}*−*2ᏼ(x)φ(x)^{}*dx* (2.1)

forΩ*⊂*R^{n}**a bounded Lipschitz domain. Here x***=*(x1,. . .,x*n*). As usual, denote

*D*^{2}*φ**=*

⎡

⎢⎢

⎢⎣

*∂*^{2}*φ*

*∂x*^{2}

*∂*^{2}*φ*

*∂x ∂y*

*∂*^{2}*φ*

*∂y ∂x*

*∂*^{2}*φ*

*∂y*^{2}

⎤

⎥⎥

⎥⎦*.* (2.2)

ForΩ*⊂*R^{2}, we define the Lagrangian function*L*associated to*I[φ] as follows:*

*L*:Ω*×*R*×*R^{2}*×*R^{4}*−→*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*Ωwhich shows up in the integrand of*I[φ]. With the*
above definitions, we have

*L*^{}*x,y;φ;**∇**φ;D*^{2}*φ*^{}*= |**Δφ**|*^{2}*−*2ᏼ(x,*y)φ(x,y),* (2.4)
where we have identified

*z**←→**φ(x,y),* *X**←→**∂φ*

*∂x*, *Y**←→**∂φ*

*∂y*,
*U**←→**∂*^{2}*φ*

*∂x*^{2}, *V**←→**∂*^{2}*φ*

*∂y*^{2}, *S**←→* *∂*^{2}*φ*

*∂y ∂x*, *W**←→* *∂*^{2}*φ*

*∂x ∂y.*

(2.5)

We also set*H*_{0}^{2}(Ω) to be the class of all square integrable functions such that
*H*_{0}^{2}(Ω)*=*

*ψ**∈**L*^{2}(Ω) : ^{}

*|***k***|≤*2

*∂*^{k}*ψ*

*∂x*^{k}

*L*^{2}

*<**∞*,*ψ**|**∂Ω**=*0, *∇**ψ**|**∂Ω**=***0**

*.* (2.6)

Fix (x,*y)**∈*Ω. We know that

*∇**L(x,y;z;X,Y*;U,V,*S,W)**=*

*−*2ᏼ(x,*y)* 0 0 2(U+*V*) 2(U+*V*) 0 0^{}^{T}*.*
(2.7)
Because the convexity of the function*L*in the remaining variables, then for all (*z;**X,*^{} *Y*^{};*U,*^{}
*V*,*S,*^{}*W)*^{} *∈*R*×*R^{2}*×*R^{4}, one has

*L* *x,y;z;**X,*^{} *Y*^{};*U*^{},*V*^{},*S,*^{}*W*^{}^{!}*≥**L(x,y;z;X,Y*;*U,V*,S,W)*−*2ᏼ(x,*y)*^{}*z**−**z*^{}

+ 2(U+*V*)^{}*U*^{}_{−}*U*^{}+ *V*_{−}*V*^{}*.* (2.8)
In particular, one has, with*z**=**φ(x,* *y),*

*L* *x,y;φ;*^{}*∇**φ;D* ^{2}*φ*^{}^{!}*≥**L*^{}*x,y;φ;**∇**φ;D*^{2}*φ*^{}+ 2*Δφ*^{}*∇**φ**− ∇**φ*^{}

*−*2ᏼ(x,*y)(φ*^{}*−**φ).* (2.9)

This implies that

Δ*φ*^{}^{}^{2}*−*2ᏼ(x,*y)φ*^{}*≥**Δφ*^{}^{2}*−*2ᏼ(x,*y)φ*+ 2Δφ^{}Δ*φ*^{}*−**Δφ*^{}*−*2ᏼ(x,*y)* *φ**−**φ*^{}*.* (2.10)
If instead we fix (x,*y;z)**∈*Ω*×*R, then

*L* *x,y;z;**X,*^{} *Y*^{};*U,*^{} *V*^{},*S,*^{}*W*^{}^{!}*≥**L(x,y;z;X,Y* ;U,V,S,W)

+ 2(U+*V*) *U**−**U*^{}+ *V**−**V*^{}*.* (2.11)

This implies that

*L*^{}*x,y;φ;*^{}*∇**φ;D* ^{2}*φ*^{}^{}*≥**L(x,y;φ;*^{}*∇**φ;D*^{2}*φ*^{}+ 2*Δφ*[*∇**φ**− ∇**φ]* (2.12)
Therefore,

*|*Δ*φ*^{}*|*^{2}*−*2ᏼ(x,*y)φ*^{}*≥ |**Δφ**|*^{2}*−*2ᏼ(x,*y)φ*^{}+ 2Δφ[Δ*φ*^{}*−**Δφ].* (2.13)
*Lemma 2.1. Suppose either*

(1)*φ**∈**H*_{0}^{2}(Ω)*∩**C*^{4}(Ω) and*η**∈**C*_{c}^{1}(Ω); or
(2)*φ**∈**H*_{0}^{2}(Ω)*∩**C*^{3}( ¯Ω)*∩**C*^{4}(Ω*) andη**∈**H*_{0}^{2}(Ω*).*

*LetδI[φ;η] denote the first variation ofIatφin the directionη, that is,*

*δI[φ;η]**=*lim

*ε**→*0

*I*[φ+*εη]**−**I*[φ]

*ε* *.* (2.14)

*Then*

*δI[φ;η]**=*2

Ω

Δ^{2}*φ**−*ᏼ(x,*y)*^{}*η dx d y.* (2.15)
*Proof. We know that*

*I*[φ+*εη]**−**I*[φ]*=*2ε

Ω[ΔφΔη*−*ᏼ*η]dx d y*+*ε*^{2}

Ω(Δη)^{2}*dx d y.* (2.16)
Hence,

*εI*[φ;η]*=*2

Ω[*ΔφΔη**−*ᏼ*η]dx d y.* (2.17)
If either assumption (1) or (2) holds, we can apply Green’s formula to a Lipschitz domain
Ωto obtain

Ω(*ΔφΔη*)dx d y*=*

Ω*η*^{}Δ^{2}*φ*^{}*dx d y*+

*∂Ω*

"*∂η*

*∂nΔφ**−**η* *∂*

*∂nΔφ*^{#}*dx d y,* (2.18)
where*∂/∂n* is the derivative in the direction normal to*∂Ω. Since eitherη**∈**C*_{c}^{1}(Ω) or
*η**∈**H*_{0}^{2}(Ω), the boundary term vanishes, which proves the lemma.

*Lemma 2.2. Letφ*_{∈}*H*_{0}^{2}(Ω). Then

*φ**H*0^{2}(Ω)*≈ **Δφ**L*^{2}(Ω)*.* (2.19)
*Proof. The functionφ**∈**H*0^{2}(Ω) implies that there exists a sequence*{**φ**k**} ⊂**C*^{∞}* _{c}* (Ω) such
that lim

*k*

*→∞*

*φ*

_{k}*=*

*φ*in

*H*

_{0}

^{2}-norm. From a well-known result for the Calder ´on-Zygmund operator (see, Stein [10, page 77]), one has

*∂*^{2}*f*

*∂x**j**∂x*

*L*^{p}*≤**C**Δ f**L** ^{p}*,

*j,*

*=*1,

*. . .,n*(2.20)

for all *f* *∈**C*_{c}^{2}(R* ^{n}*) and 1

*< p <*

*∞*. Here

*C*is a constant that depends on

*n*only. Applying this result to each

*φ*

*k*, we obtain

*∂*^{2}*φ**k*

*∂x*^{2}

*L*^{2}(Ω),^{}_{}*∂*^{2}*φ**k*

*∂x ∂y*

*L*^{2}(Ω),^{}_{}*∂*^{2}*φ**k*

*∂y*^{2}

*L*^{2}(Ω)*≤**C*^{}*Δφ**k*

*L*^{2}(Ω)*.* (2.21)
Taking the limit, we conclude that

*∂*^{2}*φ*

*∂x*^{2}

*L*^{2}(Ω),^{}_{} *∂*^{2}*φ*

*∂x ∂y*

*L*^{2}(Ω),^{}_{}*∂*^{2}*φ*

*∂y*^{2}

*L*^{2}(Ω)*≤**C**Δφ**L*^{2}(Ω)*.* (2.22)
Applying Poincar´e inequality twice to the function*φ**∈**H*_{0}^{2}(Ω), we have

*φ**L*^{2}(Ω)*≤**C*1*∇**φ**L*^{2}(Ω)

*≤**C*2

*∂*^{2}*φ*

*∂x*^{2}

*L*^{2}(Ω)+^{}_{} *∂*^{2}*φ*

*∂x ∂y*

*L*^{2}(Ω)+^{}_{}*∂*^{2}*φ*

*∂y*^{2}

*L*^{2}(Ω)

*≤**C**Δφ**L*^{2}(Ω)*.*

(2.23)

Hence,*φ**L*^{2}(Ω)*≤**C**Δφ**L*^{2}(Ω). The reverse inequality is trivial. The proof of this lemma

is therefore complete.

*Lemma 2.3. Let**{**φ*_{k}*}**be a bounded sequence inH*_{0}^{2}(Ω). Then there exist*φ**∈**H*_{0}^{2}(Ω) and a
*subsequence**{**φ*_{k}_{j}*}**such that*

*I[φ]**≤*lim inf*I*^{}*φ*_{k}_{j}^{}*.* (2.24)

*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**}*and*φ*in*H*0^{2}(Ω) such that*φ**k**j**→**φ*weakly
in*H*_{0}^{2}(Ω). Since

*H*_{0}^{2}(Ω)*⊂**H*_{0}^{1}(Ω)*⊂⊂**L*^{2}(Ω), (2.25)
by the Sobolev embedding theorem, we have

*φ*_{k}_{j}*−→**φ* in*L*^{2}(Ω) (2.26)

after passing to yet another subsequence if necessary.

Now fix (x,*y,φ**k**j*(x,*y))**∈*R^{2}*×*Rand apply inequality (2.13), we have

*Δφ**k**j*^{2}*−*2ᏼ(x,*y)φ*_{k}* _{j}*(x,

*y)*

*≥ |*

*Δφ*

*|*

^{2}

*−*2ᏼ(x,

*y)φ*

_{k}*(x,*

_{j}*y) + 2Δφ*

^{}

*Δφ*

*k*

*j*

*−*

*Δφ*

^{}

*.*(2.27) This implies that

*I*^{}*φ*_{k}_{j}^{}*≥*

Ω

*|**Δφ**|*^{2}*−*2ᏼ(x,*y)φ*_{k}_{j}^{}*dx d y*+ 2

Ω*Δφ**·*

*Δφ**k**j**−**Δφ*^{}*dx d y.* (2.28)
But*φ**k**j**→**φ*in*L*^{2}(Ω), hence

Ω

*|**Δφ**|*^{2}*−*2ᏼ(x,*y)φ*_{k}_{j}^{}*dx d y**−→*

Ω

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

Besides*φ**k**j**→**φ*weakly in*H*_{0}^{2}(Ω) implies that

Ω*Δφ**·*

*Δφ**k**j**−**Δφ*^{}*dx d y**−→*0. (2.30)
It follows that when taking limit

*I[φ]**≤*lim inf

*j* *I*^{}*φ**k**j*

*.* (2.31)

This completes the proof of the lemma.

*Remark 2.4. The above proof uses the convexity ofL(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 of*L*here.

**3. The extended Kantorovich method**

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

min

*φ**∈**H*0^{2}(Ω)*I[φ]* (3.1)

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)) asf*(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 for
the case*n**=*2, which has been used extensively in the analysis of stress on rectangular
plates. Equivalently, we will seek for an approximate solution of the above minimization
problem in the form*φ(x,y)**=* *f*(x)g(y) where *f* *∈**H*0^{2}([*−**a,a]) andg**∈**H*0^{2}([*−**b,b]).*

To phrase this diﬀerently, we will search for an approximate solution in the tensor
product Hilbert spaces*H*_{0}^{2}([*−**a,a])**⊗*$*H*_{0}^{2}([*−**b,b]), and all sequences**{**φ**k**}*,*{**φ**k**j**}*involved
hereinafter 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 thex*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 *g*0(y)*∈**H*_{0}^{2}([*−*1, 1])*∩*
*C*^{∞}* _{c}* (

*−*1, 1), and find the minimizer

*f*1(x)

*∈*

*H*0

^{2}([

*−*1, 1]) of the functional:

*I*^{}*f g*0

*=*

Ω

Δ^{}*f g*0^{2}*−*2γ f(x)g0(y)^{}*dx d y*

*=*

Ω

*g*_{0}^{2}(*f** ^{}*)

^{2}+ 2

*f f*

^{}*g*0

*g*

_{0}

*+*

^{}*f*

^{2}

^{}

*g*

_{0}

^{}^{}

^{2}

*−*2γ f g0

*dx d y*

*=*
1

*−*1(*f** ^{}*)

^{2}

*dx*1

*−*1*g*_{0}^{2}*d y*+ 2
1

*−*1

*g*_{0}^{}^{}^{2}*d y*
1

*−*1(*f** ^{}*)

^{2}

*dx*+

1

*−*1

*g*_{0}^{}^{}^{2}*d y*
1

*−*1*f*^{2}*dx**−*2γ
1

*−*1*g*0*d y*
1

*−*1*f dx,*

(3.2)

where the last equality was obtained via the integration by parts of *f f** ^{}*and

*g*0

*g*

_{0}

*. Since*

^{}*g*0has been chosen a priori; we can rewrite the functional

*I*as

*J[f*]*=**g*0^{2}

*L*^{2}

1

*−*1(*f** ^{}*)

^{2}

*dx*+ 2

^{}

*g*0

^{}^{2}

*L*^{2}

1

*−*1(*f** ^{}*)

^{2}

*dx*+

^{}

*g*

_{0}

^{}^{}

^{2}

*2*

_{L}1

*−*1*f*^{2}*dx**−*2γ
1

*−*1*g*0(y)d y
1

*−*1*f dx*

(3.3)

for all *f* *∈**H*0^{2}([*−*1, 1]). Now we may rewrite (3.3) in the following form:

*J[f*]*=*
_{1}

*−*1

*C*1(*f** ^{}*)

^{2}+

*C*2(

*f*

*)*

^{}^{2}+

*C*3

*f*

^{2}+

*C*4

*f*

^{}

*dx*

*≡*
1

*−*1*K(x,f*,*f** ^{}*,

*f*

*)dx*

^{}(3.4)

with*K*:R*×*R*×*R*×*R*→*Rgiven by

(x;z;V;W)*−→**C*1*W*^{2}+*C*2*V*^{2}+*C*3*z*^{2}+*C*4*z,* (3.5)
where

*C*1*=**g*0^{2}

*L*^{2}, *C*2*=**g*_{0}^{}^{}^{2}* _{L}*2,

*C*3

*=*

*g*

_{0}

^{}^{}

^{2}

*2,*

_{L}*C*4

*= −*2γ

_{1}

*−*1*g*0(y)d y. (3.6)
As long as*g*0*≡*0, as we have implicitly assumed, the Poincar´e inequality implies that

0*< C*1*≤**αC*2*≤**βC*3 (3.7)

for some positive constants*α*and*β, independent ofg*0. Consequently,*K*(x;z;V;W) is a
strictly convex function in variable*z,V*,*W*when*x*is fixed. In other words,*K*satisfies

*K*(x;*z;**V*^{};*W)*^{} *−**K*(x;z;V;W)

*≥**∂K*

*∂z*(x;z;V;W)(*z**−**z) +∂K*

*∂V*(x;z;V;*W)(V**−**V*) + *∂K*

*∂W*(x;z;V;W)(*W**−**W)*
(3.8)
for all (x;*z;V*;W) and (x;*z;V*^{};*W) in*^{} R^{4}, and the inequality becomes equality at (x;z;V;
*W) only if**z**=**z, orV*^{}*=**V*, or*W*^{}*=**W.*

*Proposition 3.1. Let*ᏸ:R*×*R*×*R*×*R*→*R*be a* *C*^{∞}*function satisfying the following*
*convexity condition:*

ᏸ(x;z+*z** ^{}*;V+

*V*

*;W+*

^{}*W*

*)*

^{}*−*ᏸ(x;z;V;W)

*≥**∂ᏸ*

*∂z*(x;z;V;W)z* ^{}*+

*∂ᏸ*

*∂V*(x;z;V;W)V* ^{}*+

*∂ᏸ*

*∂W*(x;z;V;W)W* ^{}* (3.9)

*for all (x;z;V*;W) and (x;z+*z** ^{}*;V+

*V*

*;W+*

^{}*W*

*)*

^{}*∈*R

^{4}

*, with equality at (x;z;V*;W) only

*ifz*

^{}*=*

*0, orV*

^{}*=*

*0, orW*

^{}*=*

*0. Also, let*

*J[f*]*=*
_{β}

*α*ᏸ^{}*x,f*(x),*f** ^{}*(x),

*f*

*(x)*

^{}^{}

*dx,*

*∀*

*f*

*∈*

*H*

_{0}

^{2}(α,

*β).*(3.10)

*Then*

*J[f* +*η]**−**J[f*]*≥**δJ[f*,*η],* *∀**η**∈**C*^{∞}* _{c}* (α,β) (3.11)

*and equality holds only ifη*

*≡*

*0. HereδJ*[

*f*,η] is the first variation of

*Jatf*

*in the directionη.*

*Proof. Condition (3.9) means that at eachx,*
ᏸ(x;*f*+*η;f** ^{}*+

*η*

*;*

^{}*f*

*+*

^{}*η*

*)*

^{}*−*ᏸ(x;

*f*;

*f*

*;*

^{}*f*

*)*

^{}*≥**∂*ᏸ

*∂z*(x;*f*;*f** ^{}*;

*f*

*)η(x) +*

^{}*∂*ᏸ

*∂V*(x;*f*;*f** ^{}*;

*f*

*)η*

^{}*(x) +*

^{}*∂*ᏸ

*∂W*(x;*f*;*f** ^{}*;

*f*

*)η*

^{}*(x) (3.12) for all*

^{}*η*

*∈*

*C*

^{∞}*(α,β) with equality only if*

_{c}*η(x)*

*=*0, or

*η*

*(x)*

^{}*=*0, or

*η*

*(x)*

^{}*=*0. Equivalently, the equality holds in (3.12) at

*x*only if

*η(x)η*

*(x)*

^{}*=*0 or

*η*

*(x)*

^{}*=*0. In other words,

*η** ^{}*(x)

*d*

*dx*

*η*^{2}(x)^{}*=*0. (3.13)

Integrating (3.12) gives
*J*[*f*+*η]**−**J[f*]*≥*

_{β}

*α*

"*∂ᏸ*

*∂zη*+*∂ᏸ*

*∂Vη** ^{}*+

*∂ᏸ*

*∂Wη*^{}

#

*dx**=**δJ[f*,η]. (3.14)
Now suppose there exists*η**∈**C*_{c}* ^{∞}*(α,β) such that (3.14) is an equality. Sinceᏸis a smooth
function, this equality forces (3.12) to be a pointwise equality, which implies, in view of
(3.13), that

*η** ^{}*(x)

*d*

*dx*

*η*^{2}(x)^{}*=*0, *∀**x.* (3.15)

If*η** ^{}*(x)

*≡*0, then

*η*

*(x)*

^{}*=*constant which implies that

*η*

*(x)*

^{}*≡*0 (since

*η*

*∈*

*C*

^{∞}*(α,β)).*

_{c}This tells us that*η**≡*constant and conclude that*η**≡*0 on the interval (α,β).

If*η** ^{}*(x)

*≡*0, set

*U*

*= {*

*x*

*∈*(α,β) :

*η*

*(x)*

^{}*=*0

*}*. Then

*U*is a non-empty open set which implies that there exist

*x*0

*∈*

*U*and some open setᏻ

*x*0of

*x*0contained in

*U*. Then

*η*

*(ξ)*

^{}*=*0 for all

*ξ*

*∈*ᏻ

*x*0

*⊂*

*U. Thus*

*d*
*dx*

*η*^{2}^{}*=*0 onᏻ*x*0*.* (3.16)

Hence,*η(ξ)**≡*constant onᏻ*x*0. But this creates a contradiction because*η** ^{}*(ξ)

*≡*0 onᏻ

*x*0. Therefore,

*J*[*f*+*η]**−**J[f*]*=**δJ[f*,η] (3.17)
only if*η(x)**≡*0, as desired. This completes the proof of the proposition.

*Corollary 3.2. LetJ[f] be as in (3.4). Then* *f*1*∈**H*_{0}^{2}([*−**1, 1]) is the unique minimizer for*
*J[f] if and only if* *f*1*solves the following ODE:*

*g*0^{2}

*L*^{2}

*d*^{4}*f*

*dx*^{4} * ^{−}*2

^{}

*g*

_{0}

^{}^{}

^{2}

*2*

_{L}*d*^{2}*f*

*dx*^{2} +^{}*g*_{0}^{}^{}^{2}* _{L}*2

*f*

*=*

*γ*

_{1}

*−*1*g*0*d y.* (3.18)
*Proof. Suppose* *f*1 is the unique minimizer. Then *f*1 is a local extremum of*J[f*]. This
implies that*δJ[f*,*η]**=*0 for all*η**∈**H*_{0}^{2}([*−*1, 1]). Using the notations in (3.4), we have

0*=**δJ[f*,η]

*=*
1

*−*1

"*∂K*

*∂zη*+*∂K*

*∂Vη** ^{}*+

*∂K*

*∂Wη*^{}

#
*dx*

*=*
1

*−*1

"*∂K*

*∂z* ^{−}*d*
*dx*

*∂K*

*∂V*

+ *d*^{2}
*dx*^{2}

*∂K*

*∂W*

#
*η(x)dx*

(3.19)

for all*η**∈**H*0^{2}([*−*1, 1]). This implies that

*∂K*

*∂z* ^{−}*d*
*dx*

*∂K*

*∂V*

+ *d*^{2}
*dx*^{2}

*∂K*

*∂W*

*=*0, (3.20)

which is the Euler-Lagrange equation (3.18). This also follows fromLemma 2.1directly.

Conversely, assume *f*1solves (3.18). Then the above argument shows that*δJ[f*,η]*=*0
for all*η**∈**H*0^{2}([*−*1, 1]). Since*K*satisfies condition (3.9) inProposition 3.1, we conclude
that

*J*^{}*f*1+*η*^{}*−**J*^{}*f*1

*≥**δJ*^{}*f*1,η^{}, *∀**η**∈**C*^{∞}_{c}^{}[*−*1, 1]^{}*.* (3.21)
This tells us that*J[f*1+*η]**≥**J*[*f*1] for all*η**∈**C*_{c}* ^{∞}*([

*−*1, 1]) and

*J*[

*f*1+

*η]> J[f*1] if

*η*

*≡*0.

Observe that*J*:*H*_{0}^{1}([*−*1, 1])*→*Ras given in (3.4) is a continuous linear functional in
the*H*_{0}^{2}-norm. This fact, combined with the density of*C*^{∞}* _{c}* ([

*−*1, 1]) in

*H*

_{0}

^{2}([

*−*1, 1]) (in the

*H*0

^{2}-norm), implies that

*J*^{}*f*1+*η*^{}*≥**J*^{}*f*1

, *∀**η**∈**C*^{∞}_{c}^{}[*−*1, 1]^{}*.* (3.22)
This means that for all*ϕ**∈**H*_{0}^{2}([*−*1, 1]), we have*J[ϕ]**≥**J*[*f*1] and if*ϕ**≡* *f*1(almost ev-
erywhere), then *ϕ**−**f*1*≡*0 and hence, *J[ϕ]> J[f*1]. Thus *f*1 is the unique minimum

for*J.*

Reversing the roles of *f* and*g*, that is, fixing *f*0and finding*g*1*∈**H*0^{2}to minimize*I*[*f*0*g]*

over*g**∈**H*_{0}^{2}([*−*1, 1]), we obtain the same conclusion by using the same arguments.

*Corollary 3.3. Fix* *f*0*∈**H*_{0}^{2}([*−**1, 1]). Theng*1*∈**H*_{0}^{2}([*−**1, 1]) is the unique minimizer for*
*J[g*]*=**I*^{}*f*0*g*^{}_{=}^{}*f*0^{2}

*L*^{2}

1

*−*1(g* ^{}*)

^{2}

*d y*+ 2

^{}

*f*

_{0}

^{}^{}

^{2}

*2*

_{L}1

*−*1(g* ^{}*)

^{2}

*d y*+

^{}

*f*

_{0}

^{}^{}

^{2}

*2*

_{L}1

*−*1*g*^{2}*d y**−*2γ^{}*f*0

*L*^{1}

1

*−*1*g d y*

(3.23)

*if and only ifg*1*solves the Euler-Lagrange equation*
*f*0^{2}

*L*^{2}

*d*^{4}*g*

*d y*^{4}* ^{−}*2

^{}

*f*

_{0}

^{}^{}

^{2}

*2*

_{L}*d*^{2}*g*

*d y*^{2}+^{}*f*_{0}^{}^{}^{2}* _{L}*2

*g*

*=*2γ

_{1}

*−*1*f*0(x)dx. (3.24)
Now we search for the solution*f*1*∈**H*_{0}^{2}([*−*1, 1]) in (3.18), that is,

*g*0^{2}

*L*^{2}

*d*^{4}*f*

*dx*^{4} * ^{−}*2

^{}

*g*

_{0}

^{}^{}

^{2}

*2*

_{L}*d*^{2}*f*

*dx*^{2} +^{}*g*_{0}^{}^{}^{2}* _{L}*2

*f*

*=*2γ 1

*−*1*g*0(y)d y. (3.25)
Rewrite the above ODE in the following form:

*g*0^{2}

*L*^{2}

⎡

⎣

*D**−**g*^{}^{2}_{L}^{2}
*g*0^{2}

*L*^{2}

2

+^{}*g*^{}^{2}_{L}^{2}
*g*0^{2}

*L*^{2}

*−**g*^{}^{4}_{L}^{2}
*g*0^{4}

*L*^{2}

⎤

⎦*f* *=*2γ
1

*−*1*g*0(y)d y, (3.26)
where*D**=**d*^{2}*/dx*^{2}.

*Remark 3.4. In general wheng**∈**H*^{2}, that is,*g*needs not satisfy the zero boundary con-
ditions for function in*H*0^{2}, then the quantity

*g*_{0}^{}^{}^{2}* _{L}*2

*g*0^{2}

*L*^{2}

*−**g*_{0}^{}^{}^{4}* _{L}*2

*g*0^{4}

*L*^{2}

(3.27)
can take on any values. However, if*g**∈**H*0^{2}and*g*0*≡*0, as proved below, this quantity is
always positive.

*Lemma 3.5. Let*Ω*be a Lipschitz domain in*R^{n}*,n**≥**1. Letg**∈**H*_{0}^{2}(Ω) be arbitrary. Then

*∇**g*^{2}_{L}^{2}*≤ **g**L*^{2}*· **Δg**L*^{2}, (3.28)
*and equality holds if and only ifg*_{≡}*0.*

*Proof. Integration by parts yields*

*∇**g*^{2}_{L}^{2}*=*

Ω*∇**g**· ∇**g dx**= −*

Ω*gΔg dx +*^{}

*∂Ω**g∂g*

*∂ndσ**= −*

Ω*gΔg dx.* (3.29)
By the Cauchy-Schwartz inequality, we have

*∇**g*^{2}_{L}^{2}*≤ **g**L*^{2}*· **Δg**L*^{2}, (3.30)
and the equality holds if and only if (see Lieb-Loss [9])

(i)*|**g(x)**| =**λ**|**Δg***(x)***|*almost everywhere for some*λ >*0,
(ii)*g***(x)Δg(x)***=**e*^{iθ}*|**g***(x)***| · |**Δg(x)**|*.

Since*g*is real-valued, (i) and (ii) imply

*g(x)Δg***(x)***=**λ*^{}*Δg***(x)**^{}^{2}*.* (3.31)

So,*g*must satisfy the following PDE:

*Δg**−*1

*λg**=*0, (3.32)

where*g**∈**H*_{0}^{2}(Ω). But the only solution to this PDE is*g**≡*0 (see, Evans [3, pages 300–

302]). This completes the proof of the lemma.

*Remark 3.6. Ifn**=*1, one can solve*g*^{}*−**λ*^{−}^{1}*g**=*0 directly without having to appeal to the
theory of elliptic PDEs.

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

*Proof. Using either*Lemma 3.5in case*n**=*1 to the above remark, we see that
*g*^{}^{2}_{L}^{2}

*g*0^{2}

*L*^{2}

*−**g*^{}^{4}_{L}^{2}
*g*0^{4}

*L*^{2}

*>*0 if*g*0*≡*0. (3.33)

Hence the characteristic polynomial associated to (3.26) has two pairs of complex con-
jugate roots as long as*g*0*≡*0. Apply the same arguments to the ODE in (3.24) and the

proposition is proved.

*Remark 3.8. The statement in*Proposition 3.7was claimed in [8] without verification.

Indeed the authors stated therein that the solutions of (3.18) and (3.24) are of the same form because of the positivity of the coeﬃcients on the left-hand side of (3.18) and (3.24).

As observed inRemark 3.4and proved inProposition 3.7, the positivity requirement is
not suﬃcient. The fact that *f*0,g0*∈**H*_{0}^{2}must 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
*r**=**g*^{}*L*^{2}

*g*0

*L*^{2}

, *t**=**g*^{}*L*^{2}

*g*0

*L*^{2}

,
*ρ**=*

%
*t*+*r*^{2}

2 , *κ**=*

%
*t**−**r*^{2}

2 *.*

(4.1)

Then from Proposition 3.7and its proof, the 4 roots of the characteristic polynomial associated to ODE (3.26) are

*ρ*+*iκ,* *ρ**−**iκ,* *−**ρ**−**iκ,* *−**ρ*+*iκ.* (4.2)

Thus the homogeneous solution of (3.26) is

*f**h*(x)*=**c*1cosh(ρx) cos(κx) +*c*2sinh(ρx) cos(κx)

+*c*3cosh(ρx) sin(κx) +*c*4sinh(ρx) sin(κx). (4.3)