Some Boundary Problems, in Sobolev Spaces with Constant Exponents,
Governed by the Nonlinear Operator of Plate Vibrations
BOUZEGHAYA FOUZIA, MEROUANI BOUBAKEUR
Department of Mathematics, Faculty of sciences, Applied Mathematics Laboratory,
University Ferhat Abbas Setif 1,
Campus, El Bez, Setif 19137,
ALGERIA
Abstract: - In this paper, we aim to investigate certain nonlinear boundary problems within Sobolev spaces,
where the exponents remain constant. We focus on the dynamically modified operator, incorporating a
viscosity term into the nonlinear vibrations of plates. Vibrating plates have a broad range of applications. To
address user requirements comprehensively, we've taken into account factors such as the geometric
configuration, material density, plate thickness, and Poisson's ratio. After formulating the problems, our method
involves converting them into hyperbolic-type nonlinear problems. In this study, we examine six boundary
value problems, establishing existence and uniqueness theorems for each. Lastly, we establish the existence of a
solution for the stationary problem by employing a variation of Brouwer's fixed point theorem.
Key-Words: - Airy function, Coupled Problem, Elliptic-Hyperbolic, Sobolev Spaces with Constant Exponents,
Existence and uniqueness, Faedo-Galerkin method, vibrating plate, nonlinear vibrations, Weak
Solutions.
1 Introduction
The applications of vibrating plates span a broad
spectrum, encompassing various sectors such as:
Personal use in households, beauty salons, for
relaxation, well-being, and massage.
Utilization in sports facilities, fitness centers,
athletic clubs, and healthcare providers for
health and rehabilitation purposes.
Machinery engineered for soil compaction,
back-filling trenches, and paving or flagging
surfaces.
This paper delves into six boundary problems
characterized by a nonlinear, dynamic, and
stationary modifying operator, which integrates a
viscosity term into the equations governing plate
vibrations. We establish a functional framework
within the Sobolev Spaces constructed on LP (Ω).
Notably, the case for p=2 has been extensively
discussed in [1] and [2]. In addressing user
requirements, we will consider factors such as the
geometric shape, material density, plate thickness,
and Poisson's ratio. After formulating the problems,
our method involves converting them into nonlinear
hyperbolic-type problems. Building upon the study
in [2], this work extends further by establishing
existence and uniqueness theorems for each problem
in the dynamic case.
Lastly, we establish the solution's existence for
the stationary problem employing a variant of
Brouwer's fixed-point theorem. The methodologies
employed draw from [1] and [2], specifically
utilizing techniques akin to the well-known Faedo-
Galerkin method showcased in [3], [4], [5], [6] and
[7], for analyzing nonlinear boundary value
problems of both elliptic and hyperbolic nature.
While newer techniques like homogenization or
compensated compactness have emerged, retaining
their advantageous properties, it's worth noting that
these classical techniques remain integral. Notably,
these methods are currently part of the curriculum in
most leading universities worldwide, with
references such as [8] and [9] being prime examples.
The bibliography provided here does not aim to be
comprehensive, and any lack of completeness
should be attributed to the author's lack of
knowledge rather than any deliberate omission.
Given the coupling between the Airy function and
the transverse displacement in various problems, our
approach involves simplifying the equations
governed by hyperbolic-type equations. To achieve
this, we eliminate the Airy function from the system
and establish, for each problem, the existence and
uniqueness theorem for these modified evolution
Received: August 19, 2023. Revised: March 27, 2024. Accepted: April 22, 2024. Published: May 20, 2024.
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equations using the techniques mentioned earlier.
The models showcased here are pivotal in artificial
intelligence design. Put simply, mathematics
underpins the establishment of foundational rules
for artificial intelligence, drawing from four core
pillars: linear algebra, probability, statistics, and
calculus.
Let's examine an isotropic, homogeneous
vibrating plate situated within an open domain on a
lipschitzian boundary denoted as Γ, composed of
two measurable and disjoint parts Γ0 and Γ1.
The normal and tangential derivatives are
expressed as follows:
Where Mi (u) and Ni (u), i=0, 1, let's label the
subsequent boundary differential operators
as:
and
where is the trace map on Гi, i=0, 1. We will
denote by ai, i=1, to 3, the following positive
constants
Here, represents the Young modulus,
stands for the Poisson ratio, ρ denotes the
density of the material forming the plate, and
signifies the plate's thickness. In the upcoming
sections, we will designate and as two
functions within the domain Ω:
We denote by the iterated Laplacian with
respect to the variable x, decomposed based on the
Poisson ratio , as illustrated in [10]:
2 Position of Problems (
We examine a set of six problems dictated by the
dynamic equations of nonlinear plate vibrations,
specifically for f
∈
LP (Q), 1≤ p <+∞. Our objective
is to seek a pair of functions , defined within
,,0 TQ
with boundary that
satisfy the problem:
where the various notations are defined as follows:
for , - is a viscosity term, [11], for
see (5), for see (7) and for
see (6).
The boundary operators and , to
are given by:
Boundary conditions on unknown u:
For further elaboration,
represents the boundary operator on Σ, where the
first component is the restriction to Σ₀, and the
second one is the restriction to Σ₁. Moreover, when
k=1, we encounter Dirichlet conditions on Σ. When
k=2, we observe mixed Dirichlet conditions on Σ₀ -
solely supported on Σ₁. For k=3, we encounter
mixed Dirichlet conditions on Σ₀ - coupled with
Neumann conditions on Σ₁. When k=4, the
conditions resemble those of a simply supported
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plate. In the case of k=5, we find mixed conditions,
solely supported on Σ₀, coupled with Neumann
conditions on Σ₁. Lastly, for k=6, we observe
Neumann conditions on Σ.
For numerical examples stemming from physics
or engineering sciences, pertinent to the boundary
problems under examination here, and where our
methodology proves advantageous, please refer to
[12].
Boundary conditions on the unknown :
Physical interpretation: represents the provided
volumetric force density, denotes the transverse
displacement, stands for the normal displacement
or Airy function, signifies the bending
moment, and embodies the transverse force
consisting of both shear force and twisting moment.
The boundary conditions (9) to (10) indicate that the
plate is:
indented at the boundary, for the first problem,
indented at the boundary and supported simply
at the boundary, for the second problem,
indented at the boundary and unrestricted at the
boundary, for the third problem,
supported simply at the boundary, for the fourth
problem,
supported simply at the boundary and
unrestricted at the boundary, for the fifth
problem,
unrestricted at the boundary, for the sixth
problem.
Remark 2.1. In , there exists no initial
condition for ; its determination relies on the
absence of derivative terms of with respect to time
t within the system of partial differential equations.
The interconnected nature of the system allows
for simplification through the elimination of .
Moreover, given that the domain is bounded and
possesses a Lipschitzian boundary, there are no
inherent obstacles in applying the variational
method to solve the elliptic equation with its
corresponding boundary conditions, [13].
Additionally, the renowned regularity theorems
outlined in [14] and [15], validate the regularity of
, indicating that F ∈ W4,P (Q). Concerning the
Neumann problem, for and
0
, we
presume the necessary existence condition:
is confirmed. Thus, if represents the Green
operator, namely the inverse operator, i.e. of in
within the context of the problem:
then
and the first equation of (7) becomes
therefore the problem to becomes
of the hyperbolic type that said, we have the
following:
Theorem 2.1. Under the assumptions:
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Then, there exists a unique solution of ,
to such that
where they are the variational spaces of the
problems to ([13], Chapter IV)
We have
The to are closed subspaces of
W2,P(Ω), hence Banach spaces containing W02,P(Ω).
Remark 2.2.
It can be deduced from equations (16) and (18)
along with definition (5)
that and so the first
equation of (13) implies that
and consequently, the initial conditions in
to are meaningful. Indeed, to
demonstrate this, it is sufficient to observe that
In fact, if we have
Remark 2.3.In this observation, we examine the
spaces Ws,p(Ω), where is non-integer, as developed
in [16]. The function in Theorem 2.1 satisfies
In fact, let arbitrarily small. So
Also, if we have:
seen that W1+ε,p (Ω)⊂L∞ (Ω) for and
[13]. Then
and as
we derive (21) employing the solution of the
problems to and the fact that send
Ws,P (Ω) in
WS+4,p (Ω)∩Vk , s ≥ 0.
3 Demonstration of Existence
Before giving the proof of Theorem 2.1, we need
the following Lemma:
Lemma 3.1.
1. The mapping
),(,
6 to1 ,
vuLvu
kVVV kkk
is bilinear and continuous.
2. The expression
exhibits trilinearity and continuity over
For the rest of the demonstration, we follow the
same techniques from, [1].
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i) The approximate solutions.
The space Vk, where 𝑘=1k=1, is defined by the
mapping
to a closed subspace of
Being separable and uniformly convex, Vk allows
projection of a dense countable set onto the
subspace. Let be a basis of
(for instance, V₁=W₀2,p(Ω)) and let be such
that
The being to be determined by the conditions:
where
we use the notations of (11) and (12), with
If we define by
or by
thus, equation (24) can be expressed as
Certainly, is not typically valued in
Following the general findings in
the theory of systems of differential equations, one
is ensured the existence of and hence
over an interval for some
ii) A priori estimates
By multiply (29) by
)(tg jm
and sum over . This
yields:
But according to Lemma 3.1, we have
,)()),(),((
4
1
)()),(),(()()),(),((
tFtutuL
dt
d
tFtutuLtutFtuL
mmm
mmmmmm
and according to (28), this equals
.)()1()(
2
))(),((
2
2
33 tFtF
dt
d
a
tFtFA
a
mmmm
So (30) is written again:
and so
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We use the inequality for (a,b) in ℝ⁺² :
to get :
But according to (25),
constant.)1( 2
002
2
1
2
11 mmmm uuauua
According the definition (27), we find
As remains within a bounded subset of
and as Vk′⊂W-2,p(Ω), where 𝑘=1 to 6, then
remains within a bounded subset of , and
thus in (32)
According to (14), and of f ∈ Lq (0,T,W-2,q(Ω)),from
this, we infer the existence of a constant C>0 such
that:
To conclude, we use Gronwall’s inequality for the
terms: um, Fm and u‘m.
So, we have
iii) Taking the limit
From (34) and (35), we can derive a sequence
such that
Let be functions of
such that and
By integrating the first term of (29) by parts, we
deduce for that
Now, from Lemma 3.1, we obtain:
L (ψ,Fμ)→L (ψ,F) in LP (Q) weakly
for example and so since strongly in LP (Q)
we see that
,)),,(()),,()),,((
000
dtFuLdtuFLdtFuL TTT
and
dtuadtwua T
j
T
),(),(
00
.
Thus, (38) implies upon taking the limit:
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and this is true of the form (37).
By taking the limit, we conclude that (40) still holds
for all ψ∈ LP (0,T;Vk), k=1 to such as
ψ′∈ LP (0,T;LP (Ω)) and
This shows that and are related by the first
equation of to and The
only task left is to demonstrate the second equation
of to . We can go directly to the limit
on (28) (for noting that
in for example; indeed we have
,)),,(()),,((
00
dtuuLdtuuL TT
)(QD
and we proceed with the limit as previously
described.
4 Proof of the Uniqueness
Let
Fu,
and
Fu ,
be two solutions, let us
say:
Next, we obtain the algebraic relationships
obviously with
a) Estimates for
Here, we apply remark 2.3. From (22) and (43), we
obtain:
However,
as
we find
It's worth noting that is a norm that is
equivalent to over to
such that (46) delineates a norm and is also
equivalent to:
For , we have
(This holds obviously for and, [17]
so
and according to (47)
In (42), let
let us show that
Indeed, let
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then, by the fractional Sobolev Theorem [17], we
obtain, for fixed :
But
This is true according to (49), (52) and
to Then
thus (51) follows.
b) Energy Equality
From (42), (50) and (51), we derive:
Certainly, employing the approach outlined in
Theorem 1.6 of [1] yields
.everywherealmost ,),(),(
))()1()()()((
11
0
2
112
2
1
2
11
2
1
dsvKdsvK
tvtvatvtva
t
But
,),(
))()1()()()((
2
1
1
0
2
112
2
1
2
11
dsvK
vvavva
for nearly all and
According to (44), we can extend by for
and being similarly extended In (56),
we repeat the same process for . Thus, we
arrive at (55).
c) Proof of Uniqueness
We readily conclude the proof from (55) and (51).
Indeed,
and then, we deduce from (55) that
With the Gronwall’s Lemma, we acquire
Hence, deducing from (47), we establish
that
5 Stationary Problems (Sk ), k=1 to 6
Next, we aim to demonstrate an existence theorem
for a solution by employing a variation of Brouwer's
fixed-point theorem [1], applicable to stationary
problems corresponding to (7). Consequently, we
seek a pair of functions k=1 to 6, such
that
Where the various notations align with those
introduced in the preceding paragraphs 1, 2: (4), (5),
(6), (8), and (9).
For our further analysis, we will rely on the
following lemma:
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Lemma5.1. ([1]) Consider as a
continuous mapping, such that, for a given ,
we have:
where if
Then there exists
,
||
, such that
)(
P
.0
Proof.
We reason through the absurd; if P (ξ) ≠0, in the ball
K= {ξ, |ξ|≤ρ}, we can consider the application
Where this mapping is consequently continuous;
Brouwer's fixed point theorem then guarantees the
existence of a ξ such that
hence, |ξ|=ρ and taking the scalar product of the two
sides by ξ:
which implies
which contradicts (60), because
We employ Lemma 5.1 to establish the following
theorem:
Theorem 5.1. Suppose belongs to then the
problem to is assured to possess a
solution.
Proof.
1) Approximate solutions.
Let , a « base » of to
formed, for instance, by functions of as in
the dynamic scenario. We seek to
i.e.
ii
m
i
mwu
1
,
such that
If we define by
Then (62) is equivalent to
We need to demonstrate that (62) possesses a
solution. We employ Lemma 5.1 for this purpose as
follows: For each , we associate
ii
m
i
mwu
1
then
and we put
So,
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But by Lemma 3.1 and by (64),
so that (49) gives
But
thus
So if |∆um| ≥ (where
see (4)). Condition satisfied if
,||
is
sufficiently large. Therefore, we can utilize Lemma
5.1; consequently, there exists a solution
of (62), or equivalently, of (64).
Furthermore, if is a solution, we have
and (68), (69) provide
2) Passing to the limit.
From (70), we conclude that
We can therefore extract two sequences such
that
and, the injection of
being compact,
Considered i be fixed,
i
; then we have:
however
in LP (
) weak,
which, with (73), gives
Consequently
We derive the first equation of (59) and apply the
same process to the limit in the second equation of
(64).
6 Conclusion and Perspectives
In the initial section of this study, focusing on solid
mechanics and particularly within the realm of plate
theory, we've established an existence theorem for
diverse dynamic problems governed by the operator
of nonlinear vibrating plates. Additionally, we've
demonstrated an existence and uniqueness theorem
for modified evolution equations utilizing the
compactness approach.
As a potential avenue for future exploration, it
would be intriguing to expand this research to
scenarios where:
Sobolev spaces W1,p(x)( , with variable
exponents, are considered, [18].
The plates exhibit polygonal boundaries.
Acknowledgments:
The authors would like to thank the referees for their
constructive comments, which have further enriched
our article.
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Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
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Conflict of Interest
The authors declare that they have no competing
interests.
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