ON SOME VARIATIONAL PROBLEMS SET ON DOMAINS
TENDING TO INFINITY
M. CHIPOT, A. MOJSIC, AND P. ROY
ABSTRACT: Let Ω` = `ω1 × ω2 where ω1 ⊂ Rp and ω2 ⊂ Rn−p are
assumed to be open and bounded. We consider the following minimization
problem:
EΩ` (u` ) :=
min EΩ` (u)
u∈W01,q (Ω` )
´
0
where EΩ` (u) = Ω` F (∇u) − f u, F is a convex function and f ∈ Lq (ω2 ).
We are interested in studying the asymptotic behavior of the solution u` as
` tends to infinity.
1. Introduction
For 0 < p < n, let Ω` = `ω1 × ω2 ⊂ Rn where ω1 ⊂ Rp , ω2 ⊂ Rn−p is
assumed to be open and bounded. ω1 is also assumed to be star shaped
with respect to the origin. We will refer Ω` as a cylindrical domain. Points
in Ω` will be denoted by X = (X1 , X2 ) where X1 = (x1 , ..., xp ) ∈ `ω1 ,
0
X2 = (xp+1 , ..., xn ) ∈ ω2 . Let W −1,q (ω) denotes the dual space of the usual
[12] Sobolev space W01,q (ω). For q > 1, let
0
f = f (X2 ) ∈ Lq (ω2 ),
where
1
q
+
1
q0
= 1.
Definition (Uniform convexity of power q-type): We say that a function
G : Rn → R is a “uniformly convex function of power q type” if there exists
a constant α = α(q) such that ∀ξ, η ∈ Rn
ξ+η
) + α|ξ − η|q ≤ G(ξ) + G(η).
2
There is a large amount of literature available on the study of such class of
functions. We refer to [15] and the reference there. For q ≥ 2, the function
G(x) = |x|q belongs to such a class.
Let F : Rn → R be a “uniformly convex function of power q-type”,
satisfying the following growth condition:
(1.1)
(1.2)
2G(
λ|ξ|q ≤ F (ξ) ≤ Λ|ξ|q , ∀ξ ∈ Rn ,
for some λ, Λ > 0.
Date: February 25, 2015.
Key words and phrases. Cylinders, Variational Problem, Asymptotic analysis.
1
2
M. CHIPOT, A. MOJSIC, AND PROSENJIT ROY
We consider the following minimization problem:
(1.3)
EΩ` (u` ) =
where
(1.4)
min
u∈W01,q (Ω` )
EΩ` (u)
ˆ
F (∇u) − f u.
EΩ` (u) :=
Ω`
We refer to [11] for the proof of existence, uniqueness of such u` . In this
article we are mainly interested in studying the asymptotic behavior of u`
as ` tends to infinity. Consider now the following minimization problem
defined on the cross section ω2 of Ω` :
(1.5)
Eω2 (u∞ ) =
where ∀u ∈ W01,q (ω2 ),
min
u∈W01,q (ω2 )
Eω2 (u)
ˆ
F (0, ∇X2 u) − f u.
Eω2 (u) :=
ω2
In [7], the authors considered the same problem for the particular case of
F (ξ) = Aξ · ξ,
where
A11 A12
A :=
At12 A22
is n × n positive definite matrix and “ · ” denotes usual Euclidean scalar
product. A11 , A12 and A22 are p × p, p × (n − p) and (n − p) × (n − p)
matrices respectively with bounded coefficients. At12 denotes the transpose
of the matrix A12 . It is easy to see that, in this case F satisfies (1.2) with
q = 2. This paper can be considered as the principal incentive for our current
work. In the case above the unique minimizer u` additionally satisfies the
following Euler-Lagrange equation:
− div (A∇u` ) = f in Ω` ,
u` = 0 on ∂Ω` ,
where div and ∇ denotes the divergence operator and the gradient in the
X variable. We recall their main result. Let divX2 and ∇X2 denotes the
divergence operator and the gradient in the X2 variable. For x > 0, through
out this paper [x] will the greatest integer less than or equal to x.
Theorem 1.1. [Chipot-Yeressian] There exists some constants A, B > 0,
such that
ˆ
|∇(u` − u∞ )|2 ≤ Ae−B`
Ω`
2
3
where u∞ is the solution to the problem
− divX2 (A22 ∇X2 u∞ ) = f
u∞ = 0 on ∂ω2 .
in ω2 ,
Their proof relies on a suitable choice of test function in the weak formulation of (1) and an iteration technique.
In [10], the authors studied similar issues for the case of F (ξ) = |ξ|q where
q ≥ 2. In their case the minimizer satisfies the following Euler-Lagrange
equation:
(
− div |∇u` |q−2 ∇u` = f in Ω` ,
u` = 0 on ∂Ω` .
In this case also they obtained the convergence of u` toward the solution of
an associated problem set on the cross section ω2 .
Theorem 1.2. [Chipot-Xie] For q ≥ 2, there exists some constants A, r > 0,
such that
ˆ
|∇(u` − u∞ )|q ≤ A`−r
Ω`
2
where u∞ is the solution to the problem
(
− divX2 |∇X2 u∞ |q−2 ∇u∞ = f
u = 0 on ∂ω2 .
in ω2 ,
We emphasize that since we do not assume any regularity on F (except
that the condition (1.2) forces F to be differentiable at 0), our problem (1.3)
is purely variational in nature (u` does not satisfy any Euler type equation).
Hence this work is a generalization of the work [7] and [10].
The following relation between the Problem (1.3) and (1.5) (for large `)
is the main result of this paper. One has :
Theorem 1.3. Under the assumption p = 1, q > 1,(1.1) and (1.2), if u`
and u∞ satisfy (1.3) and (1.5) respectively then
ˆ
A
|∇(u` − u∞ )|q ≤ 1 ,
Ω`
`q
2
for some constant A > 0(independent of `).
For the case when q = 2, note that the rate of convergence in Theorem
1
1.1 is exponential, where as in our case it is like O(`− 2 ). But it is possible to
recover an exponential rate of convergence under some additional assumption on F . This assumption is also satisfied by F in the work of [7]. More
precisely in this direction our result is the following.
Theorem 1.4. Under the assumption (1.1) and
ξ+η
(1.6)
2F (
) + β|ξ − η|q ≥ F (ξ) + F (η) ∀ξ, η ∈ Rn , β > 0,
2
4
M. CHIPOT, A. MOJSIC, AND PROSENJIT ROY
if u` and u∞ satisfy (1.3) and (1.5) respectively then one has
ˆ
|∇(u` − u∞ )|q ≤ Ae−B` ,
Ω`
2
for some constant A, B > 0 (independent of `).
From the point of view of applications, many problems of mathematical physics are set on large cylindrical domains. For instance, these are
porous media flows in channels, plate theory, elasticity theory, etc. Many
problems of the type “` → ∞” were studied in the past. Apart from second order elliptic equations [7] already mentioned, this includes eigenvalue
problems, parabolic problems, variational inequalities, Stokes problem, hyperbolic problems and many others. We refer to [2, 3, 4, 5, 6, 8, 10, 13, 14]
and the references there for the literature available in this direction.
The work of this paper is organized as follows. In the next section we
study two problems similar to (1.3) in dimension 1 to develop a better understanding of this kind of issues in a simpler situation. In section 3 we will
present the proof of Theorem 1.3. In section 4, we will give the proof of
Theorem 1.4. A partial result is obtained when the cylinder goes to infinity
in more than 1 directions [see, Theorem 4.1]. We will also
o some
n present
results regarding the asymptotic behavior of the sequence
tends to infinity.
EΩ` (u` )
2`
`
as `
2. A one dimensional problem
In this section we assume that Ω` = (−`, `), F is a function satisfying
(1.2). The functional, analogous to (1.4) in 1 dimensional situation, is defined as
ˆ
`
EΩ` (u) =
−`
F (u0 ) − γu, u ∈ W01,q (Ω` )
where γ ∈ R. To motivate the possible limit of the minizers u` , we take
2
F (x) = x2 . Then one can write down explicitly the solution, after solving
the associated Euler-Lagrange equation, as u` (x) = γ2 (`2 − x2 ), x ∈ (−`, `).
This implies that u` → ±∞ pointwise depending on the sign of γ.
Now we consider the case of a general F satisfying (1.2) and γ > 0 (we
are restricting ourselves to a constant γ but the proof goes through for a
positive function γ bounded away from zero uniformly). We do not assume
any convexity on F but the existence of a minimizer u` that we consider
next. First note then that
u` ≥ 0,
in Ω` .
This is because, if u` < 0 on A ⊂ Ω` , where |A| > 0. Then defining w` =
max{0, u` }, one gets EΩ` (w` ) < EΩ` (u` ), which contradicts the definition of
u` .
5
Define
u` (x` ) = max u` (x).
x∈[−`,`]
Note that since u` is a continuous function, such a x` ∈ [−`, `] always exists.
Lemma 2.1. u` is non decreasing on (−`, x` ) and non increasing on (x` , `).
Proof. If u` is not non decreasing on (−`, x` ) there exist y0 ∈ (−`, x` ) and
y1 such that
y0 < y1 < x` , u` (y0 ) > u` (y1 ).
Let z be a point in (y0 , x` ) such that u` (z) = min(y0 ,x` ) u` and Iz the connected component of {x | u` (x) < u` (y0 )} containing z. Define then φ`
by
u`
on Ω` \ Iz ,
φ` =
u` (y0 )
on Iz .
Clearly this function satisfies EΩ` (φ` ) < EΩ` (u` ) which contradicts the definition of u` . The proof that u` is non increasing on (x` , `) follows the same
way.
Theorem 2.1. u` (x) → ∞ pointwise ∀x, and for fixed a < b, one has
´b s
a u` → ∞, for all s > 0.
Proof. Let us argue by contradiction. Suppose that there exist K > 0,
z ∈ R and a subsequence {lk } of {`} (which is again labeled by {`}), such
that 0 ≤ u` (z) ≤ K. Let us assume that x` ≤ z for all `. This implies from
the previous lemma that u` (x) ≤ K, x ≥ z. Let δ > 0, define the function

u
on I1 := (−`, z],


 `
u` (z) + (x − z)(u` (z + 1) − u` (z) + δ) on I2 := (z, z + 1],
φ` (x) =
u` (z + 1) + δ
on I3 := [z + 1, ` − 1),



(` − x)(u` (z + 1) + δ)
on I4 := (` − 1, `].
It is easy to see that
(2.1)
EI1 (u` ) − EI1 (φ` ) = 0, EI3 (φ` ) − EI3 (u` ) ≤ −γδ(` − z − 2).
Now using the fact that 0 ≤ φ` ≤ K + δ on I3 ∪ I4 , it is posible to find a
constant C > 0 (independent of `) such that
EI2 ∪I4 (φ` ) − EI2 ∪I4 (u` ) ≤ C.
(2.2)
Adding (2.1) and (2.2), we obtain , when ` is sufficiently large
EΩ` (φ` ) − EΩ` (u` ) ≤ C − γδ(` − z − 2) < 0,
which is a contradiction to the definition of u` . If x` ≥ z for some ` large
defining φ` analogously on the other side of x` we arrive to the same contradiction and the proof follows.
Then, using the monotonicity property of u` , one has
ˆ b
us` ≥ min{u` (a), u` (b)}s (b − a) → ∞.
a
6
M. CHIPOT, A. MOJSIC, AND PROSENJIT ROY
This completes the proof of the theorem.
The situation can be different if γ = 0 and if some coerciveness is added
(compare to [9]). Suppose α, β > 0, define the energy functional, EΩ` :
1,q
Wα,β
(Ω` ) → R as
ˆ `
F (u0 ) + |u|q
EΩ` (u) =
−`
1,q
where Wα,β
(Ω` ) := u ∈ W 1,q (Ω` ) u(−`) = α, u(`) = β . Consider then
the following minimization problem:
(2.3)
EΩ` (v` ) =
inf
1,q
u∈Wα,β
(Ω` )
EΩ` (u).
For existence and uniqueness of such function v` , we refer to [11] but we just
assume here existence of some minimizer v` that we consider next.
Lemma 2.2. It holds that
ˆ
`
0 ≤ v` ≤ max{α, β} and
−`
|v`0 |q + v`q ≤ C
where C is a positive constant, independent of `.
Proof. Let v` < 0 on A ⊂ Ω` , where |A| > 0. Define w` = max{0, v` }.
Then EΩ` (w` ) < EΩ` (v` ), which contradicts the definition of v` . The fact
that v` ≤ max{α, β} follows with a similar argument, with the function
W` = min{v` , max{α, β}} playing the role of w` in the previous part.
Clearly the function

on (−`, −` + 1],
 −αx1 + α − α`
0
on (−` + 1, ` − 1),
φ` (x1 ) =

βx1 + β − `β
on [` − 1, `),
1,q
(Ω` ). Then it holds also that for all ` ≥ 1,
is in the space Wα,β
EΩ` (φ` ) = EΩ1 (φ1 ).
Using φ` as test function in (2.3), one gets
EΩ` (v` ) ≤ EΩ` (φ` ) = EΩ1 (φ1 ).
This proves the lemma.
Now we state the main theorem of this section which shows exponential
rate of convergence of the solutions v` towards 0. The method of proof
depends on an iteration scheme.
Theorem 2.2. It holds that
ˆ `
2
− 2`
0
|v` |q + v`q ≤ Ce−α` ,
where C, α are some positive constants, independent of `.
7
Proof. Suppose `0 ≤ `.
Ω` = (−`, `) by




φ`0 (x1 ) =



Consider the following test function φ`0 defined on
1
−x1 − `0 + 1
x 1 − `0 + 1
0
on
on
on
on
[−`, −`0 ] ∪ [`0 , `],
(−`0 , −`0 + 1),
(`0 − 1, −`0 ),
[−`0 + 1, `0 − 1].
The graph of φ`0 is shown below.
6
r
−`
@
@
r
@r
r
−`0 −`0 + 1
r
r - x1
`0 − 1 `0
0
`
1,q
Clearly the function φ`0 v` ∈ Wα,β
(Ω` ). Hence from (2.3), we get EΩ` (v` ) ≤
EΩ` (φ`0 v` ). Since φ`0 = 1 on the set Ω` \ Ω`0 , we have
ˆ
ˆ
0
q
F (v` ) + v` ≤
F ((φ`0 v` )0 ) + (φ`0 v` )q .
Ω`0
Ω`0
Set D`0 := Ω`0 \ Ω`0 −1 and observing that φ`0 = 0 on Ω`0 −1 , we have
ˆ
ˆ
0
F ((φ`0 v` )0 ) + (φ`0 v` )q .
F (v` ) + v`q ≤
D`0
Ω`0
Using (1.2), convexity of the function xq and properties of φ`0 we have for
some constant C > 0 (independent of ` )
!
ˆ
ˆ
ˆ
q
0 q
min{λ, 1}
|v` | + v`
|(φ`0 v` )0 |q +
(φ`0 v` )q
≤ Λ
Ω`0
D`0
D`0
ˆ
|φ`0 v`0
≤ Λ
D`0
0
+ v` φ` 0 | +
ˆ
≤ C
D`0
The above inequality implies for Λ̃ =
ˆ
|v`0 |q
Ω`0 −1
+
v`q
|v`0 |q + v`q .
C
min{λ,1}
Λ̃
≤
Λ̃ + 1
ˆ
!
|v`0 |q
Ω`0
+
v`q
ˆ
q
.
D`0
v`q
8
M. CHIPOT, A. MOJSIC, AND PROSENJIT ROY
Now choosing `0 = 2` + 1, 2` + 2, . . . , 2` + [ 2` ] and iterating the above formula,
we obtain

![ ` ] ˆ
ˆ
2
Λ̃

|v`0 |q + v`q ≤
|v`0 |q + v`q  .
1 + Λ̃
Ω`
Ω ` +[ ` ]
2
since
`
2
−1<
2
[ 2` ]
≤
ˆ
`
2
Ω`
2
there are positive constants C1 and C2 such that
ˆ
q
q
0 q
0 q
−C2 `
|v` | + v` .
|v` | + v` ≤ C1 e
Ω`
2
The theorem then follows from the last lemma.
3. Proof of Theorem 1.3
We define the spaces
1,q
(Ω` ) := u ∈ W 1,q (Ω` ) u = 0 on `ω1 × ∂ω2
Wlat
and
V
1,q
(Ω` ) =
u∈
1,q
(Ω` )
Wlat
u(Y, X2 ) = u(Z, X2 ),
a.e.(Y, X2 ), (Z, X2 ) ∈ ∂(`ω1 ) × ω2
.
We consider the following minimization problem on V 1,q (Ω` ):
(3.1)
EΩ` (w` ) =
min
u∈V 1,q (Ω` )
EΩ` (u).
Existence, uniqueness of such w` again follows as for the case of Problem
(1.3), assuming (1.2) and F strictly convex.
Theorem 3.1. One has ∀`, w` = u∞ , where u∞ is as in (1.5).
Proof. Let u ∈ V 1,q (Ω` ). Set
1
v(X2 ) =
u(., X2 ) :=
|`ω1 |
`ω1
ˆ
u(., X2 )
`ω1
where |`ω1 | denotes the measure of `ω1 . It is easy to see that v ∈ W01,q (ω2 ).
Thus
ˆ
Eω2 (u∞ ) ≤ Eω2 (v) =
F (0, ∇X2 v) − f v.
ω2
Now using the fact that u(., X2 ) is constant on the set ∂ω1 and from the
divergence theorem, one has
∇X1 u.
0=
`ω1
Also by differentiation under the integral, we get
∇X2 v =
∇X2 u.
`ω1
9
Therefore we have by Jensen’s inequality
ˆ
ˆ −
f
u
∇X2 u
F
∇X1 u,
Eω2 (u∞ ) ≤
ω2
`ω1
`ω1
ω2
`ω1
ˆ
EΩ` (u)
{F (∇u) − f u} =
≤
.
|`ω1 |
ω2 `ω1
Clearly equality holds in the above inequality if u = u∞ . Then the claim
follows from the uniqueness of the minimizer.
We now consider the case when p = 1, that is to say we assume that the
cylinder Ω` becomes unbounded in one direction only. Points in ω1 are now
simply denoted by x1 . ∇, ∇X2 denote the gradients in X and X2 variables
respectively. By
ˆ
1
q
q
,
|u|q,Ω =
|u|
Ω
we will denote the Lq norm of u on any domain Ω.
We need a few preliminaries before proceeding to the proof of Theorem
1.3. We know that a convex function is locally Lipschitz continuous on Rn ,
the next lemma provides an estimate on the growth of the Lipschitz constant
for convex functions satisfying (1.2).
Proposition 3.1. For convex function F : Rn → R satisfying (1.2), for
every P, Q ∈ Rn , we have
n
o
(3.2) |F (Q) − F (P )| ≤ 2q Λ max |P |q−1 , |Q|q−1 |Q − P |
≤ 2q Λ(|P |q−1 + |Q|q−1 )|P − Q|.
Proof. First note that the second inequality holds trivially. For a convex
function f : R → R and a < b < c, we known that
f (c) − f (b)
f (b) − f (a)
≤
.
b−a
c−b
The function f : R → R defined as
(3.3)
f (t) = F ((1 − t) P + tQ) ,
is clearly convex. Using the inequality (3.3) for a = 0, b = 1 and c = t > 1
we have
F (Q) − F (P ) ≤
Now for t = 1 +
F ((1 − t) P + tQ) − F (Q)
Λ |(1 − t) P + tQ|q
≤
.
t−1
t−1
|Q|
|Q−P | ,
we have
F (Q) − F (P ) ≤
Λ Q +
|Q|
|Q−P |
|Q|
q
(Q − P )
|Q − P |
10
M. CHIPOT, A. MOJSIC, AND PROSENJIT ROY
Λ |Q| +
≤
|Q|
|Q−P |
q
|Q − P |
|Q − P |
|Q|
≤ 2q Λ |Q|q−1 |Q − P | .
Now changing the roles of P and Q, we have
F (P ) − F (Q) ≤ 2q Λ |P |q−1 |P − Q| ,
and (3.2) follows.
The following result is well known [2], we state it without proof.
Lemma 3.1. [Poincaré’s Inequality] Let 0 < s < t and q ≥ 1, then there
exists λ1 = λ1 (q, ω2 ) > 0 (independent of s and t) such that
|u|q,(s,t)×ω2 ≤ λ1 ||∇u||q,(s,t)×ω2 ,
for all u ∈ W 1,q ((s, t) × ω2 ) with u = 0 on (s, t) × ∂ω2 .
Now our main aim is to prove the Corollary 3.1 below, which we will use
in the proof of Theorem 1.3.
For s < t, ρs,t = ρs,t (x1 ) denotes a real, Lipschitz continuous function
such that
(3.4) 0 ≤ ρs,t ≤ 1, |ρ0s,t | ≤ C, ρs,t = 1 on (s, t), ρs,t = 0 outside (s−1, t+1)
and Ds,t denotes the set defined as
Ds,t = {(s − 1, t + 1) × ω2 \(s, t) × ω2 } ∩ Ω` .
Proposition 3.2. Let p = 1, for −` < s < t ≤ `,
q
E(s,t)×ω2 (u` ) ≤ C|f |qq−1
0 ,ω .
2
In particular it implies that for `0 < `,
q
ED`0 (u` ) ≤ 2C|f |qq−1
0 ,ω
2
(3.5)
where D`0 := Ω` ∩ (Ω`0 +1 \ Ω`0 ) and u` is as in (1.3).
Proof. First we claim that if Ds,t is not empty and if EDs,t (u` ) ≤ 0 then
q
E(s,t)×ω2 (u` ) ≤ C(λ, λ1 , Λ)|f |qq−1
0 ,ω .
2
(3.6)
Indeed if EDs,t (u` ) ≤ 0 then one has
ˆ
ˆ
ˆ
q
λ
|∇u` | ≤
F (∇u` ) ≤
Ds,t
Ds,t
f u` ≤ |f |q0 ,Ds,t |u` |q,Ds,t
Ds,t
≤ λ1 |f |q0 ,Ds,t ||∇u` ||q,Ds,t
by the Poincaré inequality. It follows that
q
q
λ1 q−1
q
(3.7)
||∇u` ||q,Ds,t ≤
|f |qq−1
.
0 ,D
s,t
λ
11
Since (1 − ρs,t )u` ∈ W01,q (Ω` ) we have
EΩ` (u` ) ≤ EΩ` ((1 − ρs,t )u` ),
that is to say
E(s,t)×ω2 (u` ) + EDs,t (u` ) + EΩ` \(s−1,t+1)×ω2 (u` ) ≤ E(s,t)×ω2 ((1 − ρs,t )u` )
+ EDs,t ((1 − ρs,t )u` ) + EΩ` \(s−1,t+1)×ω2 ((1 − ρs,t )u` ).
Since ρs,t = 0 outside (s − 1, t + 1) ρs,t = 1 on (s, t) we get
E(s,t)×ω2 (u` ) ≤ EDs,t ((1 − ρs,t )u` ) − EDs,t (u` )
ˆ
=
F (∇(1 − ρs,t )u` ) − F (∇u` ) + f ρs,t u`
Ds,t
n
o
≤ Λ ||∇(1 − ρs,t )u` ||qq,Ds,t + ||∇u` ||qq,Ds,t + |f |q0 ,Ds,t |u` |q,Ds,t
o
n
≤ Λ ||(1 − ρs,t )∇u` − u` ∇ρs,t ||qq,Ds,t + ||∇u` ||qq,Ds,t
+ λ1 |f |q0 ,Ds,t ||∇u` ||q,Ds,t .
Now using the triangle inequality we obtain for some constants K(q) and
C(λ, λ1 , Λ, q),
o
n
E(s,t)×ω2 (u` ) ≤ ΛK |u` |qq,Ds,t + ||∇u` ||qq,Ds,t + λ1 |f |q0 ,Ds,t |∇u` |q,Ds,t
q
≤ C(λ, λ1 , Λ, q)|f |qq−1
0 ,ω
2
by (3.7). This completes the claim i.e. shows (3.6).
Next we can complete the proof of the theorem. Indeed, let m be the first
nonnegative integer such that Ds−m,t+m is non empty and
EDs−m,t+m (u` ) ≤ 0.
If there is no such integer, then
E(s,t)×ω2 (u` ) ≤ E(s−1,t+1)×ω2 (u` ) ≤ ... ≤ EΩ` (u` ) ≤ 0.
In the last inequality we used the fact that
EΩ` (u` ) ≤ EΩ` (0) = 0.
If such a m exists then, by the first part of this proof,
q
E(s,t)×ω2 (u` ) ≤ E(s−m,t+m)×ω2 (u` ) ≤ C|f |qq−1
0 ,ω .
2
This proves the proposition.
Corollary 3.1. [Gradient Estimate] For every `0 ≤ ` we have,
ˆ
|∇u` |q ≤ C̃,
D`0
for some constant C̃ = C̃(λ, λ1 , Λ, f )
12
M. CHIPOT, A. MOJSIC, AND PROSENJIT ROY
Proof. Applying Hölder’s, Young’s inequality and (1.2), to (3.5), we obtain
ˆ
ˆ
q
F (∇u` )
|∇u` | ≤
λ
D`0
D`0
q
q−1
0
q ,ω
≤ 2C|f |
ˆ
2
q
f u`
+
D`0
q
1
q0
+
|u` |qq,D`
|f
|
0
0
q
q0 q 0 q ,D`0
q q
0
1
λ + q0 0 |f |qq0 ,D` + 1 ||∇u` ||qq,D` .
0
0
q
q
≤ 2C|f |qq−1
0 ,ω +
2
q
≤ 2C|f |qq−1
0 ,ω
2
This implies that
(λ −
λq1 q
)
q
ˆ
1
0
D`0
|∇u` |q ≤ 2C|f |qq0 ,ω2 +
q0 q 0
0
|f |qq0 ,D` .
0
The corollary then follows after choosing small enough such that λ−
1
2.
Proof of Theorem 1.3
functions
λq1 q
q
=
For some 0 ≤ `0 ≤ ` − 1, we define the
ψ̂`0 ,` = u` +
(3.8)
ψ̌`0 ,` = u∞ +
ρ`0
2
(u∞ − u` ) ,
ρ`0
2
(u` − u∞ ) ,
where ρ`0 is defined before in (3.4). Because ψ̂`0 ,` ∈ W01,q (Ω` ) , we have from
(1.3)
(3.9)
EΩ` (u` ) ≤ EΩ` ψ̂`0 ,` .
Also since ψ̌`0 ,` ∈ V 1,q (Ω` ), from Theorem 3.1
EΩ` (u∞ ) ≤ EΩ` ψ̌`0 ,` .
(3.10)
From (3.8) it is easy to see that
(3.11)
ψ̂`0 ,` + ψ̌`0 ,` = u` + u∞ .
Adding (3.9) and (3.10), we have
EΩ` (u` ) + EΩ` (u∞ ) ≤ EΩ` (ψ̂`0 ,` ) + EΩ` ψ̌`0 ,` ,
which implies
ˆ
ˆ
F (∇u` ) + F (∇u∞ ) ≤
Ω`
F (ψ̂`0 ,` ) + F (ψ̌`0 ,` ).
Ω`
Using the fact that u` = ψ̂`0 ,` and u∞ = ψ̌`0 ,` on Ω` \ Ω`0 +1 , we get
ˆ
ˆ
F (∇u` ) + F (∇u∞ ) ≤
F (∇ψ̂`0 ,` ) + F ∇ψ̌`0 ,` .
Ω`0 +1
Ω`0 +1
13
∞
Note that ψ̂`0 ,` = ψ̌`0 ,` = u` +u
on Ω`0 , which gives
2
ˆ
∇u` + ∇u∞
(3.12)
F (∇u` ) + F (∇u∞ ) − 2F
2
Ω`0
ˆ
F (∇ψ̂`0 ,` ) + F ∇ψ̌`0 ,` − F (∇u` ) − F (∇u∞ ) := I.
≤
D`0
Using the convexity of F and (3.11), we have
ˆ
F (∇ψ̂`0 ,` ) + F ∇ψ̌`0 ,` − F (∇u` ) − F (∇u∞ )
I=
D`0
ˆ
≤
F (∇ψ̂`0 ,` ) + F ∇ψ̌`0 ,` − 2F
D`0
∇u` + ∇u∞
2
ˆ
F (∇ψ̂`0 ,` ) + F ∇ψ̌`0 ,` − 2F
=
D`0
∇ψ̂`0 ,` + ∇ψ̌`0 ,`
2
!
.
Now using Proposition 3.1, we obtain for some constant C = C(q, Λ) > 0,
(3.13)
ˆ
n
o
∇ψ̂` ,` q−1 + ∇(ψ̂` ,` + ψ̌` ,` )q−1 + ∇ψ̌` ,` q−1 ∇(ψ̂` ,` −ψ̌` ,` )
0
0
0
0
0
0
|I| ≤ C
D`0
Since q > 1, using the monotonicity of the function |X|q−1 , we have for
a, b > 0
(a + b)q−1 ≤ (2max(a, b))q−1 = 2q−1 max(a, b)q−1 ≤ 2q−1 (aq−1 + bq−1 ).
This implies from (3.13) that,
ˆ
∇ψ̂` ,` q−1 + ∇ψ̌` ,` q−1 ∇(ψ̂` ,` − ψ̌` ,` ).
I≤C
0
0
0
0
D`0
Then using Hölder’s inequality, we obtain
q
q
q0
q0
(3.14)
I ≤ C ||∇ψ̂`0 ,` ||q,D` + ||∇ψ̌`0 ,` ||q,D` ||∇(ψ̂`0 ,` − ψ̌`0 ,` )||q,D`0
0
0
Since F is uniformly convex of power q-type, we get
(3.15)
ˆ
ˆ
∇u` + ∇u∞
q
.
α
|∇(u` − u∞ )| ≤
F (∇u` ) + F (∇u∞ ) − 2F
2
Ω`
Ω`
0
0
First combining (3.12),(3.14) and (3.15) and then using (3.13), one gets
ˆ
(3.16) α
|∇(u` − u∞ )|q
Ω`0
q
q
q0
q0
≤ C ||∇ψ̂`0 ,` ||q,D
+
||∇
ψ̌
||
||∇(ψ̂`0 ,` − ψ̌`0 ,` )||q,D`0 .
`0 ,` q,D`
`
0
0
14
M. CHIPOT, A. MOJSIC, AND PROSENJIT ROY
Now we estimate each integral on the right hand side of the above inequality.
One has for some constant C > 0,
ˆ
ˆ
q
|∇(ψ̂`0 ,` − ψ̌`0 ,` )| =
|∇{(1 − ρ`0 )(u` − u∞ )}|q
D`0
D`0
ˆ
≤2
ˆ
q
q−1
q
(1 − ρ`0 ) |∇(u` − u∞ )| + 2
(u` − u∞ )q |∇ρ`0 |q .
q−1
D`0
D`0
Now using Poincaré’s inequality and the properties of ρ`0 , one has for
some K = K(λ1 )
ˆ
ˆ
q
|∇(u` − u∞ )|q .
|∇(ψ̂`0 ,` − ψ̌`0 ,` )| ≤ K
D`0
D`0
Together with the Corollary 3.1, one can estimate the terms ||∇ψ̂`0 ,` ||q,D`0 ,
||∇ψ̂`0 ,` ||q,D`0 similarly, to obtain for some constant K1 = K1 (f, λ1 )
ˆ
ˆ
q
|∇ψ̂`0 ,` | ,
|∇ψ̌`0 ,` |q ≤ K1 .
D`0
D`0
For some other constant M (which depends only on K, K1 ), (3.16) then
becomes
!1
ˆ
ˆ
q
(3.17)
|∇(u` − u∞ )|q ≤ M
|∇(u` − u∞ )|q
.
Ω`0
D`0
Applying Corollary 3.1, we get for some other constant M1
ˆ
(3.18)
|∇(u` − u∞ )|q ≤ M1 .
Ω`0
Choosing `0 = 2` and for m = 1, 2, ., ., ., [ 2` ] − 2, we have from (3.17)
q 
q

ˆ
ˆ

|∇(u` − u∞ )|q  ≤ 
|∇(u` − u∞ )|q 
Ω ` +m
Ω`
2
ˆ
2
≤M
q
|∇(u` − u∞ )|q .
D ` +m
2
Summing up over m, one gets

q
ˆ
`
[ ]−2 
|∇(u` − u∞ )|q 
2
Ω`
2
[ 2` ]−2 ˆ
≤M
q
ˆ
X
m=1
q
|∇(u` − u∞ )| = M
D ` +m
2
q
|∇(u` − u∞ )|q
Ω`−1 \Ω `
2
15
ˆ
|∇(u` − u∞ )|q ≤ M q M1 ,
≤ Mq
Ω`−1
where in the last inequality we have used (3.18). Hence we can find a
constant A > 0, which depend only on the previous constants, such that
ˆ
−1
|∇(u` − u∞ )|q ≤ A` q ,
Ω`
2
This completes the proof of the theorem.
4. Proof of Theorem 1.4 and Some additional Results
Now we proceed to the proof of Theorem 1.4. We do not restrict ourself
to the assumption that p = 1. We assume that ω1 is open, bounded subset
of Rp , which is star shaped around the origin.
Proof of Theorem 1.4 We have EΩ` (u` ) ≤ EΩ` (0) = 0, which implies
that
ˆ
ˆ
ˆ
λ
|∇u` |q ≤
F (∇u` ) ≤
f u` .
Ω`
Ω`
Ω`
Using Hölder’s inequality and then Poincaré’s inequality, we have for some
constant C > 0,
ˆ
(4.1)
|∇u` |q ≤ C`p .
Ω`
For `0 < ` − 1, choose ρ`0 = ρ`0 (X1 ) satisfying ρ`0 = 1 on `0 ω1 and 0
outside Ω`0 +1 . Also assume |∇X1 ρ`0 | ≤ C, for some C > 0.
Then one can proceed exactly as in the proof of Theorem 1.3, until to get
ˆ
ˆ
F (∇u` ) + F (∇u∞ ) ≤
F (∇ψ̂`0 ,` ) + F ∇ψ̌`0 ,` .
Ω`0 +1
Ω`0 +1
Then using the convexity of q-type, the assumption on F , and arguing as
below (3.12) we arrive to
ˆ
ˆ
α
|∇(u` − u∞ )|q ≤ β
|∇(ψ̂`0 ,` − ψ̌`0 ,` )|q .
Ω`0 +1
Ω`0 +1
Now since ψ̂`0 ,` − ψ̌`0 ,` = (1 − ρ`0 )(u` − u∞ ) and ρ`0 = 1 on Ω`0 , this implies
that
ˆ
ˆ
q
α
|∇(u` − u∞ )| ≤ β
|∇(ψ̂`0 ,` − ψ̌`0 ,` )|q .
Ω`0 +1
D`0
Using Poincaré’s and Young’s Inequality as done in step (3), one obtains for
some constant C = C(λ, λ1 , Λ, α, β)
ˆ
ˆ
q
|∇(u` − u∞ )| ≤ C
|∇(u` − u∞ )|q ,
Ω`0
D`0
16
M. CHIPOT, A. MOJSIC, AND PROSENJIT ROY
which is same as
ˆ
|∇(u` − u∞ )|q ≤
(4.2)
Ω`0
`
2
Choosing `0 =
ˆ
+ m for m =
C
C +1
2
Finally setting r =
C
C+1
ˆ
ˆ
|∇(u` − u∞ )|q .
Ω`0 +1
0, 1, ..., [ 2` ]
|∇(u` − u∞ )|q ≤
Ω`
C
C +1
− 1 and iterating (4.2), we get
[ ` ]−1 ˆ
2
|∇(u` − u∞ )|q .
Ω`
< 1 and from (4.1), we have
`
|∇(u` − u∞ )|q ≤ C`p e([ 2 ]−1) log r .
Ω`
2
Since log r < 0, the theorem follows.
The next proposition gives a sufficient criterion for (1.6) to hold true.
More precisely we have :
Proposition 4.1. If F ∈ C 1 (Rn ) is such that for some α, β > 0
α|ξ − η|q ≤ (∇F (ξ) − ∇F (η)) · ξ − η
∀ξ, η ∈ Rn
or
(∇F (ξ) − ∇F (η)) · ξ − η ≤ β|ξ − η|q , ∀ξ, η ∈ Rn
then the condition (1.1) or respectively (1.6) holds.
Proof. One has
ˆ 1
d
ξ + η
η−ξ
F (ξ) − F
=−
F (ξ + t(
))dt
2
2
0 dt
ˆ
1 1
η−ξ
=−
∇F (ξ + t(
)) · (η − ξ)dt.
2 0
2
Exchanging the roles of ξ and η we get
ˆ 1
ξ + η
d
ξ−η
F (η + t(
))dt
F (η) − F
=−
2
2
0 dt
ˆ
1 1
ξ−η
=−
∇F (η + t(
)) · (ξ − η)dt.
2 0
2
Then adding the two equalities above we obtain
ξ + η
F (ξ) + F (η) − 2F
2
ˆ 1
1
ξ−η
η−ξ
=
(∇F (η+t(
))−∇F (ξ+t(
)))·(η−ξ)dt
2 0
2
2
Noting that
ξ−η
η−ξ
[η + t(
)] − [ξ + t(
)] = (1 − t)(η − ξ)
2
2
17
we obtain from our assumptions
ˆ
α 1
ξ + η
(1 − t)q−1 |η − ξ|q dt ≤ F (ξ) + F (η) − 2F
2 0
2
or
ˆ
ξ + η β 1
≤
(1 − t)q−1 |η − ξ|q dt
F (ξ) + F (η) − 2F
2
2 0
i.e.
α
ξ + η
|η − ξ|q ≤ F (ξ) + F (η) − 2F
2q
2
or
ξ + η
β
≤ |η − ξ|q .
2
2q
This completes the proof of the proposition.
F (ξ) + F (η) − 2F
The next theorem provides us with a pointwise estimate from above and
below for the functions u` when f is non negative and when the problems
are assumed to have a unique solution.
Lemma 4.1. [A pointwise estimate] If f ≥ 0, then 0 ≤ u` ≤ u∞ a.e. for
all `.
Proof. Fix ` > 0. First we claim that u` , u∞ ≥ 0. We will prove the
claim only for u` , since the proof for u∞ is identical. Define the function
w` = max{0, u` }. Clearly w` ∈ W01,q (Ω` ) is non negative and since f ≥ 0,
we have
EΩ` (w` ) ≤ EΩ` (u` ).
The claim then follows from the uniqueness of u` . Set
A` := {X ∈ Ω` | u` (X) > u∞ (X) a.e.}.
We claim that
EA` (u` ) ≤ EA` (u∞ ).
Indeed, if not, setting v` = u` − (u` − u∞ )+ one has v` ∈ W01,q (Ω` ) and
EΩ` (v` ) = EA` (v` ) + EΩ` \A` (v` ) = EA` (u∞ ) + EΩ` \A` (u` )
< EA` (u` ) + EΩ` \A` (u` ) = EΩ` (u` )
and a contradiction with the definition of u` . Setting then w` = u∞ + (u` −
u∞ )+ one has w` ∈ V 1,q (Ω` ) and
EΩ` (w` ) = EA` (w` ) + EΩ` \A` (w` ) = EA` (u` ) + EΩ` \A` (u∞ ) ≤ EΩ` (u∞ ).
Thus w` = u∞ and (u` − u∞ )+ = 0 which completes the proof.
Using similar argument as in the last theorem one can prove the following
monotonicity property of the solutions u` . We emphasize that such a monotonicity property holds true for any general domains, but we will present
the result only for the family Ω` .
18
M. CHIPOT, A. MOJSIC, AND PROSENJIT ROY
Lemma 4.2. [Monotonicity] If ` < `0 then u` ≤ u`0 , a.e. in Ω`0 , where u`
is extended by 0 outside Ω`0 .
Our main result Theorem 1.3 works only in the case when p = 1. The
next result provides some partial result in the case when 0 < p < n and
Ω` = (−`, `)p × ω2 are hypercubes.
Theorem 4.1. Let Ω` = (−`, `)p × ω, then for some function ũ∞ = ũ∞ (X2 )
it holds
u` → ũ∞ (X2 )
pointwise.
Proof. In the statement of the theorem it is understood that u` are extended
by 0 outside Ω` . For h ∈ R, we set
τhi v := v(X − hei ) i = 1, ..., p
where ei denotes the unit vector in i-th direction. First we claim that
u`+h ≥ τhi u` .
In order to prove (4) one has to to show that (τhi u` − u`+h )+ = 0. Set
A := X ∈ Ω`+h τhi u` (X) > u`+h (X) .
We have then EA (u`+h ) ≤ EA (τhi u` ). Indeed if this is not true, then setting
v := u`+h + (τhi u` − u`+h )+
one has v ∈ W01,q (Ω`+h ) and
EΩ`+h (v) = EA (τhi u` ) + EΩ`+h \A (u`+h )
≤ EA (u`+h ) + EΩ`+h \A (u`+h ) = EΩ`+h (u`+h )
and a contradiction with the definition of u`+h .
i
(u`+h )(X) < u` (X) . We claim that
Define A0 := X ∈ Ω` τ−h
(4.3)
A = A0 + hei .
Indeed for X such that xi < −` + h one has τhi u` = 0 that is X does not
belongs to A and for xi − h ≥ −`,
τhi (u` (X)) > u`+h (X) ⇐⇒ u` (X − hei ) > u`+h (X),
i.e. X − hei ∈ A0 which proves (4.3).
i u
+ ∈ W 1,q (Ω ). Clearly the
We consider then w = u` − (u` − τ−h
`+h )
`
0
function vanishes when u` = 0 since u`+h ≥ 0. Then one has
i
EΩ` (w) = EA0 (τ−h
u`+h ) + EΩ\A0 (u` )
ˆ
F (∇u`+h (X + hei )) − f u`+h (X + hei )
= EΩ\A0 (u` ) +
A0
ˆ
= EΩ\A0 (u` ) +
F (∇u`+h (X)) − f u`+h (X)
A
19
= EA (u`+h ) + EΩ\A0 (u` ) ≤ EA (τhi u` ) + EΩ\A0 (u` )
= EA0 (u` ) + EΩ\A0 (u` ) = EΩ` (u` ).
By the definition and uniqueness of u` this implies that
i
w = u` ⇐⇒ (u` − τ−h
u`+h )+ = 0
and A0 , A are of measure 0. This proves (4). With similar argument one
can show that
i
u`+h ≥ τ−h
u` .
(4.4)
Since u` is a monotone increasing sequence of functions which are bounded
above by u∞ (X2 ) (from Lemma 4.2), one has for some ũ∞ ,
u` → ũ∞
pointwise. From (4.4) it follows that
ũ∞ (X) ≥ ũ∞ (X − hei ), ũ∞ (X) ≥ ũ∞ (X + hei ) ∀h, i ∈ {1, ..., p}
and thus ũ∞ is independent of the variable X1 .
nE
Ω
(u
o
)
`
`
Now we are interested in asymptotic behavior of the sequence
|`ω1 |
as ` → ∞. (| | denotes the measure of a set). In particular we will prove the
following theorem.
Theorem 4.2. [Convergence of energy] One has for some constant C > 0
and sufficiently large `,
EΩ` (u` )
C
Eω2 (u∞ ) ≤
≤ Eω2 (u∞ ) +
|`ω1 |
`
where u` , u∞ is as in (1.3) and (1.5) respectively.
Proof. Set
1
v` (X2 ) =
u` (., X2 ) =
|`ω1 |
`ω1
ˆ
u` (., X2 ).
`ω1
It is easy to see that v` ∈ W01,q (ω2 ). Thus
ˆ
Eω2 (u∞ ) ≤ Eω2 (v` ) =
F (0, ∇X2 v` ) − f v` .
ω2
By the divergence theorem, one has
∇X1 u`
0=
`ω1
and by differentiation under the integral
∇X2 v` =
∇X2 u` .
`ω1
Therefore we have by Jensen’s inequality
ˆ
ˆ Eω2 (u∞ ) ≤
F
∇X1 u` ,
∇X2 u`
−
ω2
`ω1
`ω1
ω2
f
u`
`ω1
20
M. CHIPOT, A. MOJSIC, AND PROSENJIT ROY
ˆ
{F (∇u` ) − f u` } =
≤
ω2
`ω1
EΩ` (u` )
.
|`ω1 |
This proves the first inequality.
For the second one, first we consider a Lipschitz continuous function ρ` =
ρ` (X1 ), such that ρ` = 1 on (` − 1)ω1 and ρ` = 0 on ∂(`ω1 ). We also assume
that there exists a constant C > 0 (independent of `) such that
|∇X1 ρ` | ≤ C and 0 ≤ ρ` ≤ 1.
Thus ρ` u∞ ∈ W01,q (Ω` ). Then from (1.3), we have
EΩ` (u` ) ≤ EΩ` (ρ` u∞ ).
Now for some constant K = K(Λ, f, u∞ ), one has
ˆ
EΩ` (ρ` u∞ ) = EΩ`−1 (u∞ ) +
F ∇ρ` u∞ − f u∞ ρ`
Ω` \Ω`−1
ˆ
≤ EΩ` (u∞ ) +
F ∇ρ` u∞ − F ∇u∞ − f u∞ (ρ` − 1)
Ω` \Ω`−1
ˆ
≤ |`ω1 |Eω2 (u∞ ) +
Λ|∇ρ` u∞ |q + Λ|∇u∞ |q + |f ||u∞ |
Ω` \Ω`−1
ˆ
0
≤ |`ω1 |Eω2 (u∞ ) + C
|∇u∞ |q + |u∞ |q + |f |q
Ω` \Ω`−1
ˆ
0
≤ |`ω1 |Eω2 (u∞ ) + C{|`ω1 | − |(` − 1)ω1 |}
|∇u∞ |q + |u∞ |q + |f |q .
ω2
Dividing by |`ω1 | and the result follows, i.e. one has
C
EΩ` (u` )
≤ Eω2 (u∞ ) + .
|`ω1 |
`
This finishes the proof of the theorem.
Acknowledgment The research leading to these results has received
funding from Lithuanian-Swiss cooperation programme to reduce economic
and social disparities within the enlarged European Union under project
agreement No CH-3-SMM-01/0. The research of the first author was supported also by the Swiss National Science Foundation under the contracts #
200021-129807/1 and 200021-146620. The research of the second author was
supported by the European Initial Training Network FIRST under the grant
agreement # PITN-GA-2009-238702. The research of the third author is
funded by “Innovation in Science Pursuit for Inspired Research(INSPIRE)”
under the IVR Number: 20140000099.
21
References
[1] P. G. Ciarlet, Introduction to linear shell theory, Series in Applied Mathematics, 1.
North-Holland, Amsterdam, 1998.
[2] M. Chipot, ` goes to plus infinity, Birkhäuser, 2002.
[3] M. Chipot and A. Rougirel, On the asymptotic behaviour of the solution of elliptic
problems in cylindrical domains becoming unbounded, Commun. Contemp. Math. 4,
p. 15–44, 2002.
[4] M. Chipot and A. Rougirel, On the asymptotic behavior of the eigenmodes for elliptic
problems in domain becoming unbounded, Trans. AMS, p. 3579–3602, 2008.
[5] M. Chipot and A. Rougirel, Remarks on the asymptotic behaviour of the solution to
parabolic problems in domains becoming unbounded, Nonlinear Analysis 47, p. 3–12,
2001.
[6] M. Chipot, P. Roy and I. Shafrir Asymptotics of eigenstates of elliptic problems with
mixed boundary data on domains tending to infinity, Asymptotic Analysis, 85, no.
3-4, 199–227, 2013.
[7] M. Chipot and K.Yeressian, Exponential rates of convergence by an iteration technique. C. R. Math. Acad. Sci. Paris 346, no. 1-2, 21–26, 2008.
[8] M. Chipot and K. Yeressian, On the asymptotic behavior of variational inequalities
set in cylinders. Discrete Contin. Dyn. Syst. 33 , no. 11–12, 2013.
[9] M. Chipot and K. Yeressian, Asymptotic behaviour of the solution to variational
inequalities with joint constraints on its value and its gradient Contemporary Mathematics 594, p. 137-154, 2013.
[10] M. Chipot and Y. Xie, On the asymptotic behaviour of the p-Laplace equation in
cylinders becoming unbounded, Nonlinear partial differential equations and their applications, p. 16–27, GAKUTO Internat. Ser. Math. Sci. Appl., 20, Gakkotosho,
Tokyo, 2004.
[11] I. Ekeland and R. Temam, Convex analysis and variational problems, North-Holland
Publishing Co., Amsterdam, 1984.
[12] L. C. Evans, Partial differential equations, Graduate Studies in Mathematics 19,
AMS, 1998.
[13] S. Guesmia,Some convergence results for quasilinear parabolic boundary value problems in cylindrical domains of large size. Nonlinear Anal. 70, no. 9, 3320–3331, 2009.
[14] S. Guesmia, Some results on the asymptotic behaviour for hyperbolic problems in
cylindrical domains becoming unbounded. J. Math. Anal. Appl. 341 , no. 2, 1190–
1212, 2008.
[15] J. Borwein, A. J. Guirao, P. Hajek, J. Vanderwerff, Uniformly convex functions on
Banach spaces. Proc. Amer. Math. Soc. 137, no. 3, 1081–1091, 2009.
22
M. CHIPOT, A. MOJSIC, AND PROSENJIT ROY
(Michel Chipot)
Institute für Mathematik, Universität Zürich,
Winterthurerstr. 190, CH-8057 Zürich, Switzerland.
E-mail address: [email protected]
(Aleksandar Mojsic)
Helmut Schmidt University / University of the Federal Armed Forces Hamburg,
Department of Mechanical Engineering,
Holstenhofweg 85
22043 Hamburg, Germany.
E-mail address: [email protected]
(Prosenjit Roy)
Tata Institute of Fundamental Research
Sharadanagar, GKVK Campus, Postbox - 560065,
Bangalore, India .
E-mail address: [email protected]