11institutetext:  School of Computing Science, Sichuan Normal University,
Chengdu 610066, China 11email<EMAIL_ADDRESS><EMAIL_ADDRESS>
22institutetext: School of Computing Science, Simon Fraser University,
Burnaby, BC V5A 1S6, Canada 33institutetext: Visual Computing and Virtual
Reality Key Laboratory of Sichuan, Sichuan Normal University, Chengdu 610068,
China 44institutetext: Sichuan Key Laboratory of Translational Medicine of
Traditional Chinese Medicine, Chengdu 610041, China

# Efficient Monaural Speech Enhancement using Spectrum Attention Fusion

Jinyu Long( ✉ ) 11 Jetic Gū 22 Binhao Bai 11 Zhibo Yang 11 Ping Wei 44 Junli
Li( ✉ ) 1133

###### Abstract

Speech enhancement is a demanding task in automated speech processing
pipelines, focusing on separating clean speech from noisy channels.
Transformer based models have recently bested RNN and CNN models in speech
enhancement, however at the same time they are much more computationally
expensive and require much more high quality training data, which is always
hard to come by. In this paper, we present an improvement for speech
enhancement models that maintains the expressiveness of self-attention while
significantly reducing model complexity, which we have termed Spectrum
Attention Fusion. We carefully construct a convolutional module to replace
several self-attention layers in a speech Transformer, allowing the model to
more efficiently fuse spectral features. Our proposed model is able to achieve
comparable or better results against SOTA models but with significantly
smaller parameters (0.58M) on the Voice Bank + DEMAND dataset.

###### Keywords:

Speech Enhancement Noise Reduction Speech Transformer Spectrum Attention

##  1 Introduction

Real-world acoustic environments are often contaminated with different forms
of background disturbances, which can significantly hinder speech-based
applications such as phone calls and Automatic Speech Recognition (ASR).
Monaural speech enhancement techniques aim to recover clean speech from the
original noisy input, by often isolating speech related information from the
noisy background. While conventional statistical methods are primarily based
on specific speech and noise characteristics [31], in recent years neural
network based models have been shown to be much more versatile at handling
non-stationary noise as well as more challenging scenarios such as with low
Signal-to-Noise Ratio (SNR) and reverberation[38].

In a neural speech processing model, a common approach is to directly encode a
time-frequency representation (i.e. spectrogram, Mel spectrogram) as
input[20].After the transformation, the spectrogram, either in single vector
forms or as 2D image slices, is fed directly into the encoding layer of the
network. To extract features from diverse spectra, previous studies have used
Convolutional Neural Networks (CNNs) to capture local information in
neighbouring sub-bands in a duration [19, 12], Recurrent Neural Networks
(RNNs) for modelling sequential features in frequency or time dimensions [7,
33, 6], and hybrid architectures to collaboratively boost capacity [28, 25].
More recently, Transformer models built more on dynamic global context
interactions [35] have shown even more remarkable results in extracting
features from spectra [1, 40].

Despite being highly expressive, a major problem with transformers is their
quadratic computational complexity, which can be a constraint when dealing
with long inputs. Recently proposed Computer Vision model Conv2Former [8]
attempts to address this issue by computing the Hadamard product between a
large-kernel convolution’s and multi-head values to mimic self-attention,
dramatically reducing parameter size and computation cost. This shows that
vision transformers can indeed be optimized to work more efficiently, and we
hypothesise that this may be the same in the speech domain.

In this work, we introduce a new approach named Spectrum Attention Fusion,
which refines and enhances the Convolutional Modulation from Conv2Former to
more effectively integrate spectral features in speech processing. Our
Spectrum Attention Fusion module augments the speech Transformer’s self-
attention mechanism[40] with convolutional layers, so it can more efficiently
fuse these features while retaining access to a more global context. With
significantly fewer parameters (0.58M) than a full Transformer, our proposed
model achieves better performance than a vanilla Transformer, and remains
competitive or better (2.84 WB-PESQ, 94.3% STOI) on the Voice Bank + DEMAND
dataset compared to other previous models including ones with much more
parameters.

##  2 Related Work

Supervised speech enhancement models can generally be divided into two
categories based on the kind of input the model receives after preprocessing:
time-domain and frequency-domain. Time-domain methods [21, 18] directly encode
the normalised noisy speech waveform into the model, while frequency-domain
methods [19, 33, 6] first perform short-time Fourier transformation (STFT),
then feed the spectrogram into the model. For the latter approach, the
learning objective is often to generate a noise mask [37, 26, 39] that is
applied on the noisy spectrogram prior to reconstruction.

The predicted noise mask is typically calculated for and applied on the
magnitude spectrum, and during reconstruction needs to be combined with the
phase spectrum, which is usually completely unaltered. However, the importance
of recovering phase information in perception should not be overlooked,
especially when the SNR is low [5]. One way is to utilise complex neural
networks to directly process complex values, allowing them to learn features
directly from both spectra [9, 5], however complex neural network research is
itself a challenging research direction. Another powerful phase-aware
technique is to have sub-networks for refining spectrogram [15, 13], where
magnitude and phase information are processed with interconnections. In this
work, our modelling is based on the latter approach.

Many different types of noise typically exhibit different levels of non-
stationary characteristics. To improve a model’s ability to discern noise
signals and effectively capture the local spectral context, prior work has
suggested the incorporation of sub-band units for extracting features from
adjacent frequencies[7], but its performance lag behind that of transformers
which allows for more dynamic interactions for more complicated types of non-
stationary patterns spreading across multiple frequency bands [40].
Transformers as mentioned above, suffers however from high computational cost
and rely on more high quality training data to achieve peak performance. To
solve this problem, we augment the self-attention mechanism to focus on
specific frequency ranges while replacing more complicated heads with Temporal
Convolution Network (TCN) blocks [2]. As shown in our experimental results,
our proposed model remain highly competitive if not better than compared
previous SOTA models, and the reduced computational and memory overhead also
make our model suitable for a wider range of applications, including those
with limited resources or stringent efficiency constraints.

##  3 METHOD

Fig. 1 presents an overview of our proposed model, which consists of three
main components: spectrum encoders $\mathbf{E}_{MP}$ and $\mathbf{E}_{RI}$ for
magnitude-phase and real-imaginary spectra, Spectrum Attention Fusion
$\mathbf{AF}$, and decoders $\mathbf{D}_{IRM}$ and $\mathbf{D}_{Bias}$ for
decoding the ideal ratio mask and bias of the estimated spectrogram,
respectively.

Figure 1:  The overview of the proposed Spectrum Attention Fusion

Specifically, the proposed model accepts the magnitude, phase, real, and
imaginary spectra ${M}$, ${\theta}$, ${S}_{r}$, ${S}_{i}\in\mathbb{R}^{T\times
F}$ as input, where $T$ and $F$ denote the total number of frequency frames
and bins, respectively. Within the proposed $\mathbf{AF}$, the encoded
representations are first fed into the Convolutional Module, then through
Local Attention to perform sub-band contextual feature extraction, and finally
TCN blocks for temporal feature extraction, ultimately resulting in
$\widetilde{{X}}^{spec}\in\mathbb{R}^{64\times T\times F}$.

In the decoding stage, the mask decoder $\mathbf{D}_{IRM}$ uses
$\widetilde{{X}}^{spec}$ to generate the ideal ratio mask ${M}^{irm}$, which
is paired with the noisy phase $\theta$ to produce the initial estimated
spectrogram
$\left\\{\widetilde{{S}}^{init}_{r},\widetilde{{S}}^{init}_{i}\right\\}$,
while Bias Decoder $\mathbf{D}_{Bias}$ leverages $\widetilde{{X}}^{spec}$ to
compute the bias values
$\left\\{\widetilde{{S}}^{bias}_{r},\widetilde{{S}}^{bias}_{i}\right\\}$ of
the initial estimated spectrogram. We then obtain the enhanced spectrogram
$\left\\{\widetilde{\mathbf{S}}_{r},\widetilde{\mathbf{S}}_{i}\right\\}$ by
refining the initial estimated spectrum with the bias values. The entire
decoding process can be formulated as:

$\displaystyle\widetilde{M}={M}\otimes{M}^{irm}$ (1)
$\displaystyle\widetilde{S}^{init}_{r}=\widetilde{M}\otimes\cos\left(\theta\right)$
(2)
$\displaystyle\widetilde{S}^{init}_{i}=\widetilde{M}\otimes\sin\left(\theta\right)$
(3)
$\displaystyle\widetilde{S}_{r}=\widetilde{S}^{init}_{r}+\widetilde{S}^{bias}_{r}$
(4)
$\displaystyle\widetilde{S}_{i}=\widetilde{S}^{init}_{i}+\widetilde{S}^{bias}_{i}$
(5)

where the $\otimes$ denotes element-wise product.

###  3.1 Spectrum Encoder

Figure 2:  (a) The illustration of Spectrum Encoder. (b) The illustration of
the Mask/Bias Decoder. (c) The illustration of Convolutional Module. The “DW-
Conv” indicates a depth-wise convolution. The “PW-Conv” indicates a point-wise
convolution.The “DS-Conv” indicates a depth separable convolution.

The encoder directly takes input representations ${M}$, ${\theta}$, ${S}_{r}$,
${S}_{i}\in\mathbb{R}^{T\times F}$ and encodes them into hidden
representations for each frame. Specifically, ${M}$, ${\theta}$ are
concatenated into ${{X}}^{mp}\in\mathbb{R}^{2\times T\times F}$; similarly,
${S}_{r}$, ${S}_{i}$ produce ${{X}}^{ri}\in\mathbb{R}^{2\times T\times F}$,
the concatenation creates a new channel dimension. By up-sampling channels of
${X}^{ri},{X}^{mp}$ in $\mathbf{E}_{MP}$ and $\mathbf{E}_{RI}$, we can obtain
$\widetilde{X}^{mp},\widetilde{X}^{ri}\in\mathbb{R}^{64\times T\times F}$,
respectively. Next, we concatenate ${\widetilde{X}}^{ri}$,
${\widetilde{X}}^{mp}$ to form ${\widetilde{X}}^{mpri}\in\mathbb{R}^{128\times
T\times F}$, which serves as input to Attention Fusion $\mathbf{AF}$.

As shown in Fig. 2 (a), the Spectrum Encoder comprises two layers with 2-D
point-wise convolution, layer normalisation (LN), and parametric ReLU (PReLU)
activation. Subsequently, a four-layer depth-wise 2-D convolution with kernel
size (1,3) is applied, along with LN and PReLU. A final 2-D point-wise
convolutional layer is used, followed by LN and PReLU. This depth-separable
convolution-like architecture aims to up-sample channel of spectrum features
while reducing the number of parameters and computation needed.

###  3.2 Attention Fusion

As shown in Fig. 2 (c), the Convolutional Module employs a convolutional
attention mechanism to selectively concentrate on features across spectral
channels. For an input $X\in\mathbb{R}^{C\times T\times F}$, we compute the
value $V$ using a point-wise convolution (i.e. linear projection in channel
dimension). In parallel, $X$ is transformed through the same channel
projection and the GELU activation function to generate the similarity score
$A$ from a depth-wise convolution with an 11x11 kernel size. The final value
$Z$ is produced by taking the element-wise product of the value $V$ and the
similarity score $A$, followed by another channel projection. Convolutional
attention can be formulated as follows:

$\displaystyle V$ $\displaystyle=\mathrm{Conv2D}^{PW}_{1}(X),$ (6)
$\displaystyle A$
$\displaystyle=\mathrm{Conv2D}^{DW}(\mathrm{GELU}(\mathrm{Conv2D}^{PW}_{2}(X)))$
(7) $\displaystyle Z$ $\displaystyle=\mathrm{Conv2D}^{PW}_{3}(A\otimes V)$ (8)

where $\otimes$ stands for the element-wise product, while
$\mathrm{Conv2D}^{PW}_{[1:3]}$ represent three different point-wise
convolution layers. Additionally, $\mathrm{Conv2D}^{DW}$ signifies a depth-
wise convolution with an $11\times 11$ kernel size.

Convolutional attention allows each spectral feature to establish correlations
with all others in the $11\times 11$ window and the interaction among spectral
channels is enabled by channel projection. Following the Transformer, the
remainder of the Convolutional Modulation uses skip connections, channel
normalization as the norm, and point-wise convolution as the feedforward. In
comparison to self-attention, Convolutional Modulation takes advantage of
convolutions, leading to improved memory overhead for long audio processing.

Then local attention with a window size of three is implemented to focus on
extracting sub-band information. This is achieved by conducting dot-product
attention on three neighboring frequency bands for each given band, allowing
the model to capture local-level context. Following this step, TCN is utilized
to reconstruct the time-domain information, ensuring a comprehensive
representation of the original signal’s temporal dynamics.

###  3.3 Mask/Bias Decoder

As shown in Fig. 2 (b), the Mask/Bias Decoder begins with a depth-separable
convolution, followed by a dual-path convolutional structure. Each path
comprises a unique point-wise convolution; one is activated by a sigmoid
function, and the other is activated by a $tanh$ function. Subsequently, the
outputs of both paths are combined through an element-wise product, followed
by another point-wise convolution, LN, and sigmoid activation.

###  3.4 Loss Function

The loss function takes into account both the spectrogram and the magnitude
spectrum of the speech signal, which can be represented as follows:

$\displaystyle\mathcal{L}^{Mag}=\left\|M^{*}-M\right\|^{2},$ (9)
$\displaystyle\mathcal{L}^{RI}=\left\|S^{*}_{r}-\widetilde{S}_{r}\right\|^{2}+\left\|S^{*}_{i}-\widetilde{S}_{i}\right\|^{2},$
(10) $\displaystyle\mathcal{L}=0.5\mathcal{L}^{Mag}+0.5\mathcal{L}^{RI}$ (11)

Above, loss function $\mathcal{L}$ can be represented as a combination of two
individual loss terms, one for the magnitude and the other for spectrogram.
The first term $\mathcal{L}^{Mag}$ measures the squared error between the
target magnitude spectrum $M^{*}$ and the estimated magnitude spectrum $M$.
The second term $\mathcal{L}^{RI}$ measures the squared error between the
target and estimated spectrogram, namely the real components $S^{*}_{r}$ and
$\widetilde{S}_{r}$, and the imaginary components $S^{*}_{i}$ and
$\widetilde{S}_{i}$. The overall loss function $\mathcal{L}$ combines the
individual loss terms $\mathcal{L}^{Mag}$ and $\mathcal{L}^{RI}$ with equal
weights, ensuring that both the magnitude and spectrogram contribute to the
total error.

##  4 EXPERIMENTS

###  4.1 Data

To evaluate our model, we use the publicly available Voice Bank + DEMAND
dataset, which is a standard dataset for speech enhancement evaluation. The
Voice Bank + DEMAND dataset, as proposed by [32], combines clean speech
recordings from the Voice Bank corpus [36] with various acoustic noise
recordings from the Diverse Environments Multichannel Acoustic Noise Database
(DEMAND) [29] dataset, each recorded with 48kHz sampling rate. The training
set consists of 11,572 noisy-clean pairs, created by mixing clean speech from
28 Voice Bank speakers with 10 types of noise (2 synthetic and 8 from DEMAND)
at four different SNR settings $\left\\{0\ \rm{dB},5\ \rm{dB},10\ \rm{dB},15\
\rm{dB}\right\\}$, these settings result in 40 different patterns. For
evaluation, the test set contains 824 noisy-clean pairs which were created
with 2 unseen speakers and 5 unseen noise types at different SNR settings
$\left\\{2.5\ \rm{dB},7.5\ \rm{dB},12.5\ \rm{dB},17.5\ \rm{dB}\right\\}$. This
arrangement helps to create a more challenging evaluation setup, as the model
are exposed to diverse acoustic conditions and speaker variations.

###  4.2 Implementation setup

All audio samples are resampled from 48kHz to 16kHz and trimmed to a maximum
length of 3 seconds. By applying a 320-point STFT with a 25 ms Hanning window
length and a 10 ms hop length, we obtain spectrograms containing
161-dimensional spectral characteristics. Following the power compression
technique presented in [14], we apply it to the magnitude using a coefficient
of $0.5$, while leave the phase unchanged. We adopt the Adam optimiser [11]
with a learning rate of 5e-4, $\beta_{1}$ of 0.95, and $\beta_{2}$ of 0.999,
train the model for 50 epochs with a batch size of 4.

###  4.3 Evaluation Metrics

To evaluate our model, we use several evaluation metrics as follows:

  * •

WB-PESQ [30] (Wideband Perceptual Evaluation of Speech Quality): WB-PESQ is an
extension of the PESQ [22] metric for wideband speech signals. It is a widely
used objective measure that predicts the subjective quality of speech signals.
It ranges from -0.5 to 4.5, with higher scores indicating better perceived
quality.

  * •

STOI [27] (Short-Time Objective Intelligibility): STOI is an objective metric
that gauges speech intelligibility. The score ranges from 0 to 100%, with
higher percentages denoting better intelligibility.

  * •

CSIG [10] (Composite Signal-to-Noise Ratio): CSIG evaluates the overall
quality of the enhanced signal, taking into account both distortion and noise
suppression. Higher scores indicate better signal quality.

  * •

CBAK [10] (Composite Background-to-Noise Ratio): CBAK measures the
effectiveness of background noise suppression. Higher scores signify more
effective noise suppression.

  * •

COVL [10] (Composite Overall Quality): COVL is a comprehensive metric that
considers both signal quality and noise suppression. Higher scores indicate
better overall performance.

  * •

SSNR (Segmental Signal-to-Noise Ratio): SSNR is an objective metric that
assesses the quality of the enhanced speech signal by comparing it to the
original clean signal. Higher SSNR scores indicate better speech enhancement.

###  4.4 Comparison With Previous Methods

Table 1: This table compares the performance of our proposed model with
previous methods on the Voicebank+Demand test set. Missing values are denoted
by -.

| Model | Params(M) | WB-PESQ | STOI(%) | CSIG | CBAK | COVL | SSNR
---|---|---|---|---|---|---|---|---
Unprocessed
| Noisy | - | $1.97$ | $92.1$ | $3.35$ | $2.44$ | $2.63$ | $1.68$
Conventional Statistical-based Baseline
|  Wiener [23] | - | $2.22$ | - | $3.23$ | $2.68$ | $2.67$ | $5.07$
Previous Deep Learning Models
|  RNNoise [33] | 0.06 | $2.34$ | $92.2$ | $3.40$ | $2.51$ | $2.84$ | -
|  Wavenet [21] | - | - | - | $3.62$ | $3.23$ | $2.98$ | -
|  SETransformer [40] | - | $2.62$ | $93.0$ | - | - | - | -
|  NSNet2 [3] | $6.17$ | $2.47$ | $90.3$ | $3.23$ | $2.99$ | $2.90$ | -
|  PercepNet [34] | $8.00$ | $2.73$ | - | - | - | - | -
|  DCCRN [9] | $3.70$ | $2.54$ | $93.8$ | $3.74$ | $3.13$ | $2.75$ | -
|  DCCRN+ [17] | $3.30$ | $2.84$ | - | - | - | - | -
|  S-DCCRN [16] | $2.34$ | $2.84$ | $94.0$ | $4.03$ | 3.43 | $2.97$ | -
|  FullSubNet+ [4] | $8.67$ | 2.88 | $94.0$ | $3.86$ | $3.42$ | 3.57 | -
|  DeepFilterNet [24] | $1.78$ | $2.81$ | 94.2 | 4.14 | $3.31$ | $3.46$ | -
Ours
| Spectrum Attention Fusion | 0.58 | $2.84$ | $94.3$ | $4.04$ | $3.38$ | $3.43$ | $9.05$
|  \- Phase Spectrum Input | 0.58 | $2.71$ | $94.1$ | $3.87$ | $3.29$ | $3.28$ | $8.80$
|  \+ 1 Attention Fusion Layer | $1.16$ | $2.78$ | $94.1$ | $3.91$ | $3.31$ | $3.35$ | $8.92$
|  \+ skip connection | $1.16$ | 2.89 | 94.4 | 4.18 | 3.43 | 3.59 | 9.10

As reported in Table 1, we provide a comprehensive comparison of our proposed
model with previous methods on the Voicebank+Demand test set. Our model
exhibits competitive performance when compared to several previous deep
learning models. Specifically, the Spectrum Attention Fusion achieves a WB-
PESQ score of 2.84, outperforming the SETransformer model that employs a
vanilla Transformer. This score is also on par with leading models such as
FullSubNet+ and S-DCCRN. Moreover, our model attains the highest STOI score of
94.3% among all evaluated methods. It is important to note that our model
achieves these high performance scores with only 0.58 million parameters,
which is significantly lower than many previous models. Furthermore, the
ablation studies in Table 1 demonstrate the impact of different components in
our model. For instance, integrating the phase spectrum input contributes to
an improvement in the model’s performance. Interestingly, merely increasing
the number of attention fusion layers does not necessarily lead to improved
performance. On the other hand, combining the attention fusion layer with skip
connections results in a boost in performance.

###  4.5 The Influence of Phase Spectrum Input

Figure 3:  Comparison of STOI performance during training over 50 epochs for
models using Magnitude-only and Magnitude+Phase spectrum inputs.

It is important to note some previous work such as [13] only used the
magnitude spectrum as input. To see how much improvement do phase spectrum
provide, we also train our models with different combinations inputs: only
magnitude and magnitude combined with phase. We then compared their STOI
performance during 50 epochs of training. Fig. 3 demonstrate that the model
with both magnitude and phase inputs outperforms the magnitude-only model in
terms of STOI.

###  4.6 Scalability

As shown in Table 1, we also explore the scalability of our proposed model by
examining the effects of increasing the number of Attention Fusion layers and
incorporating skip connections in the architecture. We found that merely
increasing the number of layers in the Attention Fusion mechanism led to
decreased performance. This outcome suggests that simply deepening the model
does not ensure improved results, and can even make the learning process more
challenging. The drop in performance might be due to the vanishing gradient
problem, which causes gradients to become too small for effective weight
updates during back-propagation. On the other hand, when we employed skip
connections while increasing the number of layers, the model’s performance
improved. This finding highlights the benefits of using skip connections in
deeper architectures to mitigate the vanishing gradient problem and facilitate
gradient flow through the network. These results emphasise the need for
careful architectural design and optimisation to balance complexity and
performance. Furthermore, the results demonstrate the scalability of our
Spectrum Attention Fusion model, which can be improved further by leveraging
skip connections and other architectural innovations.

##  5 Conclusion

In this work, we proposed an efficient model named Spectrum Attention Fusion
for monaural speech enhancement. With a compact design, our model achieves
competitive performance with fewer parameters than previous approaches. The
incorporation of phase spectrum input and key components such as Attention
Fusion proved to be instrumental in enhancing the effectiveness of our model.
By providing high-quality speech enhancement with an efficient architecture,
our work contributes to the development of more streamlined models for speech
enhancement applications.

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# A convex dual variational formulation for non-convex optimization applied to
a non-linear model of plates

Fabio Silva Botelho
Department of Mathematics
Federal University of Santa Catarina - UFSC
Florianópolis, SC - Brazil

###### Abstract

This article develops duality principles applicable to the non-linear
Kirchhoff-Love model of plates. The results are obtained through standard
tools of convex analysis, functional analysis, calculus of variations and
duality theory. The main duality principle concerns a convex (in fact concave)
dual variational formulation and related new optimality conditions for the
model in question. Finally, in the last section we develop some global
existence results for a similar model in elasticity.

##  1 Introduction

In this article, in a first step, we develop a new existence proof and a dual
variational formulation for the Kirchhoff-Love thin plate model. Previous
results on existence in mathematical elasticity and related models may be
found in [7, 8, 9].

At this point we refer to the exceptionally important article ”A contribution
to contact problems for a class of solids and structures” by W.R. Bielski and
J.J. Telega, [2], published in 1985, as the first one to successfully apply
and generalize the convex analysis approach to a model in non-convex and non-
linear mechanics.

The present work is, in some sense, a kind of extension of this previous work
[2] and others such as [3], which greatly influenced and inspired my work and
recent books [5, 6].

Here we highlight that such earlier results establish the complementary energy
under the hypothesis of positive definiteness of the membrane force tensor at
a critical point (please see [2, 3] for details).

We have obtained a dual variational formulation which allows the global
optimal point in question not to be positive definite (for related results see
F.S. Botelho [5]), but also not necessarily negative definite. The approach
developed also includes sufficient conditions of optimality for the primal
problem. It is worth mentioning that the standard tools of convex analysis
used in this text may be found in [10, 5], for example.

At this point we start to describe the primal formulation.

Let $\Omega\subset\mathbb{R}^{2}$ be an open, bounded, connected set which
represents the middle surface of a plate of thickness $h$. The boundary of
$\Omega$, which is assumed to be regular (Lipschitzian), is denoted by
$\partial\Omega$. The vectorial basis related to the cartesian system
$\\{x_{1},x_{2},x_{3}\\}$ is denoted by
$(\textbf{a}_{\alpha},\textbf{a}_{3})$, where $\alpha=1,2$ (in general Greek
indices stand for 1 or 2), and where $\textbf{a}_{3}$ is the vector normal to
$\Omega$, whereas $\textbf{a}_{1}$ and $\textbf{a}_{2}$ are orthogonal vectors
parallel to $\Omega.$ Also, n is the outward normal to the plate surface.

The displacements will be denoted by

$\hat{\textbf{u}}=\\{\hat{u}_{\alpha},\hat{u}_{3}\\}=\hat{u}_{\alpha}\textbf{a}_{\alpha}+\hat{u}_{3}\textbf{a}_{3}.$

The Kirchhoff-Love relations are

$\displaystyle\hat{u}_{\alpha}(x_{1},x_{2},x_{3})=u_{\alpha}(x_{1},x_{2})-x_{3}w(x_{1},x_{2})_{,\alpha}\;$
$\displaystyle\text{ and }\hat{u}_{3}(x_{1},x_{2},x_{3})=w(x_{1},x_{2}).$ (1)

Here $-h/2\leq x_{3}\leq h/2$ so that we have $u=(u_{\alpha},w)\in U$ where

$\displaystyle U$ $\displaystyle=$ $\displaystyle\left\\{(u_{\alpha},w)\in
W^{1,2}(\Omega;\mathbb{R}^{2})\times W^{2,2}(\Omega),\right.$
$\displaystyle\;u_{\alpha}=w=\frac{\partial
w}{\partial\textbf{n}}=0\left.\text{ on }\partial\Omega\right\\}$
$\displaystyle=$ $\displaystyle W_{0}^{1,2}(\Omega;\mathbb{R}^{2})\times
W_{0}^{2,2}(\Omega).$

It is worth emphasizing that the boundary conditions here specified refer to a
clamped plate.

We define the operator $\Lambda:U\rightarrow Y\times Y$, where
$Y=Y^{*}=L^{2}(\Omega;\mathbb{R}^{2\times 2})$, by

$\Lambda(u)=\\{\gamma(u),\kappa(u)\\},$
$\gamma_{\alpha\beta}(u)=\frac{u_{\alpha,\beta}+u_{\beta,\alpha}}{2}+\frac{w_{,\alpha}w_{,\beta}}{2},$
$\kappa_{\alpha\beta}(u)=-w_{,\alpha\beta}.$

The constitutive relations are given by

$N_{\alpha\beta}(u)=H_{\alpha\beta\lambda\mu}\gamma_{\lambda\mu}(u),$ (2)
$M_{\alpha\beta}(u)=h_{\alpha\beta\lambda\mu}\kappa_{\lambda\mu}(u),$ (3)

where:

$\\{H_{\alpha\beta\lambda\mu}\\}=\left\\{h\left(\frac{4\lambda^{h}\mu^{h}}{\lambda^{h}+2\mu^{h}}\delta_{\alpha\beta}\delta_{\lambda\mu}+2\mu^{h}(\delta_{\alpha\lambda}\delta_{\beta\mu}+\delta_{\alpha\mu}\delta_{\beta\lambda})\right)\right\\}$

and

$\\{h_{\alpha\beta\lambda\mu}\\}=\frac{h^{2}}{3}\\{H_{\alpha\beta\lambda\mu}\\}$

are symmetric positive definite fourth order tensors. Here $\lambda^{h}>0$ and
$\mu^{h}>0$ are the Lamé constants which depend on $h$ and
$\\{\delta_{\alpha\beta}\\}$ is the Kronecker delta.

From now on, in an appropriate sense, we denote
$\\{\overline{H}_{\alpha\beta\lambda\mu}\\}=\\{H_{\alpha\beta\lambda\mu}\\}^{-1}$
and
$\\{\overline{h}_{\alpha\beta\lambda\mu}\\}=\\{h_{\alpha\beta\lambda\mu}\\}^{-1}$.

Furthermore $\\{N_{\alpha\beta}\\}$ denote the membrane force tensor and
$\\{M_{\alpha\beta}\\}$ the moment one. The plate stored energy, represented
by $(G\circ\Lambda):U\rightarrow\mathbb{R}$ is expressed by

$(G\circ\Lambda)(u)=\frac{1}{2}\int_{\Omega}N_{\alpha\beta}(u)\gamma_{\alpha\beta}(u)\;dx+\frac{1}{2}\int_{\Omega}M_{\alpha\beta}(u)\kappa_{\alpha\beta}(u)\;dx$
(4)

and the external work, represented by $F:U\rightarrow\mathbb{R}$, is given by

$F(u)=\langle w,P\rangle_{L^{2}(\Omega)}+\langle
u_{\alpha},P_{\alpha}\rangle_{L^{2}(\Omega)},$ (5)

where $P,P_{1},P_{2}\in L^{2}(\Omega)$ are external loads in the directions
$\textbf{a}_{3}$, $\textbf{a}_{1}$ and $\textbf{a}_{2}$ respectively. The
potential energy, denoted by $J:U\rightarrow\mathbb{R}$ is expressed by:

$J(u)=(G\circ\Lambda)(u)-F(u)$

Finally, we also emphasize from now on, as their meaning are clear, we may
denote $L^{2}(\Omega)$ and $L^{2}(\Omega;\mathbb{R}^{2\times 2})$ simply by
$L^{2}$, and the respective norms by $\|\cdot\|_{2}.$ Moreover derivatives are
always understood in the distributional sense, $\mathbf{0}$ may denote the
zero vector in appropriate Banach spaces and, the following and relating
notations are used:

$w_{,\alpha\beta}=\frac{\partial^{2}w}{\partial x_{\alpha}\partial
x_{\beta}},$ $u_{\alpha,\beta}=\frac{\partial u_{\alpha}}{\partial
x_{\beta}},$ $N_{\alpha\beta,1}=\frac{\partial N_{\alpha\beta}}{\partial
x_{1}},$

and

$N_{\alpha\beta,2}=\frac{\partial N_{\alpha\beta}}{\partial x_{2}}.$

##  2 On the existence of a global minimizer

At this point we present an existence result concerning the Kirchhoff-Love
plate model.

We start with the following two remarks.

######  Remark 2.1.

Let $\\{P_{\alpha}\\}\in L^{\infty}(\Omega;\mathbb{R}^{2})$. We may easily
obtain by appropriate Lebesgue integration $\\{\tilde{T}_{\alpha\beta}\\}$
symmetric and such that

$\tilde{T}_{\alpha\beta,\beta}=-P_{\alpha},\text{ in }\Omega.$

Indeed, extending $\\{P_{\alpha}\\}$ to zero outside $\Omega$ if necessary, we
may set

$\tilde{T}_{11}(x,y)=-\int_{0}^{x}P_{1}(\xi,y)\;d\xi,$
$\tilde{T}_{22}(x,y)=-\int_{0}^{y}P_{2}(x,\xi)\;d\xi,$

and

$\tilde{T}_{12}(x,y)=\tilde{T}_{21}(x,y)=0,\text{ in }\Omega.$

Thus, we may choose a $C>0$ sufficiently big, such that

$\\{T_{\alpha\beta}\\}=\\{\tilde{T}_{\alpha\beta}+C\delta_{\alpha\beta}\\}$

is positive definite $\text{ in }\Omega$, so that

$T_{\alpha\beta,\beta}=\tilde{T}_{\alpha\beta,\beta}=-P_{\alpha},$

where

$\\{\delta_{\alpha\beta}\\}$

is the Kronecker delta.

So, for the kind of boundary conditions of the next theorem, we do NOT have
any restriction for the $\\{P_{\alpha}\\}$ norm.

Summarizing, the next result is new and it is really a step forward concerning
the previous one in Ciarlet [8] and the previous results in [6]. We emphasize
this result and its proof through such a tensor $\\{T_{\alpha\beta}\\}$ are
new, even though the final part of the proof is established through a standard
procedure in the calculus of variations.

It is also worth mentioning in the concerning primal formulation we have
included a term denoted by

$J_{5}(u)=\langle\varepsilon_{\alpha},u_{\alpha}^{2}\rangle_{L^{2}(\Gamma_{t})}.$

This term, even in the case where the positive $\varepsilon_{\alpha}\in
L^{2}(\Gamma)$ is of small magnitude, has an amazing structural stabling
effect on the plate model in question and makes the primal energy functional
bounded below. This feature makes viable the proof of this existence result
for a more general set of boundary conditions. We highlight such a functional
$J_{5}$ corresponds to adding a distribution of springs on the portion
boundary $\Gamma_{t}$.

Finally, we also highlight the duality principles and concerning optimality
conditions are established through new functionals. Similar results may be
found in [5, 6]. Indeed the results here developed represent some advances
concerning those presented in [5] in 2014 and in [6] of 2020.

######  Theorem 2.2.

Let $\Omega\subset\mathbb{R}^{2}$ be an open, bounded, connected set with a
Lipschitzian boundary denoted by $\partial\Omega=\Gamma.$ Suppose
$(G\circ\Lambda):U\rightarrow\mathbb{R}$ is defined by

$G(\Lambda u)=G_{1}(\gamma(u))+G_{2}(\kappa(u)),\;\forall u\in U,$

where

$G_{1}(\gamma
u)=\frac{1}{2}\int_{\Omega}H_{\alpha\beta\lambda\mu}\gamma_{\alpha\beta}(u)\gamma_{\lambda\mu}(u)\;dx,$

and

$G_{2}(\kappa
u)=\frac{1}{2}\int_{\Omega}h_{\alpha\beta\lambda\mu}\kappa_{\alpha\beta}(u)\kappa_{\lambda\mu}(u)\;dx,$

where

$\Lambda(u)=(\gamma(u),\kappa(u))=(\\{\gamma_{\alpha\beta}(u)\\},\\{\kappa_{\alpha\beta}(u)\\}),$
$\gamma_{\alpha\beta}(u)=\frac{u_{\alpha,\beta}+u_{\beta,\alpha}}{2}+\frac{w_{,\alpha}w_{,\beta}}{2},$
$\kappa_{\alpha\beta}(u)=-w_{,\alpha\beta},$

where,

$\displaystyle U$ $\displaystyle=$
$\displaystyle\left\\{u=(u_{\alpha},w)=(u_{1},u_{2},w)\in
W^{1,2}(\Omega;\mathbb{R}^{2})\times W^{2,2}(\Omega)\;:\right.$ (6)
$\displaystyle\left.u_{\alpha}=w=\frac{\partial
w}{\partial\mathbf{n}}=0,\text{ on }\Gamma_{0}\right\\}.$

Here $\partial\Omega=\Gamma_{0}\cup\Gamma_{t}$ and the Lebesgue measures

$m_{\Gamma}(\Gamma_{0}\cap\Gamma_{t})=0,$

and

$m_{\Gamma}(\Gamma_{0})>0.$

We also define,

$\displaystyle\langle u,\mathbf{f}\rangle_{L^{2}}$ $\displaystyle=$
$\displaystyle\langle w,P\rangle_{L^{2}(\Omega)}+\langle
u_{\alpha},P_{\alpha}\rangle_{L^{2}(\Omega)}$ (7) $\displaystyle+\langle
P^{t}_{\alpha},u_{\alpha}\rangle_{L^{2}(\Gamma_{t})}+\langle
P^{t},w\rangle_{L^{2}(\Gamma_{t})},$ $\displaystyle F_{1}(u)$ $\displaystyle=$
$\displaystyle-\langle w,P\rangle_{L^{2}(\Omega)}-\langle
u_{\alpha},P_{\alpha}\rangle_{L^{2}(\Omega)}-\langle
P^{t}_{\alpha},u_{\alpha}\rangle_{L^{2}(\Gamma_{t})}$ (8)
$\displaystyle-\langle
P^{t},w\rangle_{L^{2}(\Gamma_{t})}+\langle\varepsilon_{\alpha},u_{\alpha}^{2}\rangle_{L^{2}(\Gamma_{t})}$
$\displaystyle=$ $\displaystyle-\langle
u,\mathbf{f}\rangle_{L^{2}}+\langle\varepsilon_{\alpha},u_{\alpha}^{2}\rangle_{L^{2}(\Gamma_{t})}$
$\displaystyle\equiv$ $\displaystyle-\langle
u,\mathbf{f}_{1}\rangle_{L^{2}}-\langle
u_{\alpha},P_{\alpha}\rangle_{L^{2}(\Omega)}+\langle\varepsilon_{\alpha},u_{\alpha}^{2}\rangle_{L^{2}(\Gamma_{t})},$

where

$\langle u,\mathbf{f}_{1}\rangle_{L^{2}}=\langle
u,\mathbf{f}\rangle_{L^{2}}-\langle
u_{\alpha},P_{\alpha}\rangle_{L^{2}(\Omega)},$

$\varepsilon_{\alpha}\in L^{2}(\Gamma)$ is such that
$\varepsilon_{\alpha}>0,\text{ in }\Gamma,\;\forall\alpha\in\\{1,2\\}$ and

$\mathbf{f}=(P_{\alpha},P)\in L^{\infty}(\Omega;\mathbb{R}^{3}).$

Let $J:U\rightarrow\mathbb{R}$ be defined by

$J(u)=G(\Lambda u)+F_{1}(u),\;\forall u\in U.$

Assume there exists $\\{c_{\alpha\beta}\\}\in\mathbb{R}^{2\times 2}$ such that
$c_{\alpha\beta}>0,\;\forall\alpha,\beta\in\\{1,2\\}$ and

$G_{2}(\kappa(u))\geq c_{\alpha\beta}\|w_{,\alpha\beta}\|_{2}^{2},\;\forall
u\in U.$

Under such hypotheses, there exists $u_{0}\in U$ such that

$J(u_{0})=\min_{u\in U}J(u).$

###### Proof.

Observe that we may find $\mathbf{T}_{\alpha}=\\{(T_{\alpha})_{\beta}\\}$ such
that

$div\mathbf{T}_{\alpha}=T_{\alpha\beta,\beta}=-P_{\alpha}$

an also such that $\\{T_{\alpha\beta}\\}$ is positive definite and symmetric
(please, see Remark 2.1).

Thus defining

$v_{\alpha\beta}(u)=\frac{u_{\alpha,\beta}+u_{\beta,\alpha}}{2}+\frac{1}{2}w_{,\alpha}w_{,\beta},$
(9)

we obtain

$\displaystyle J(u)$ $\displaystyle=$ $\displaystyle
G_{1}(\\{v_{\alpha\beta}(u)\\})+G_{2}(\kappa(u))-\langle
u,\mathbf{f}\rangle_{L^{2}}+\langle\varepsilon_{\alpha},u_{\alpha}^{2}\rangle_{L^{2}(\Gamma_{t})}$
(10) $\displaystyle=$ $\displaystyle
G_{1}(\\{v_{\alpha\beta}(u)\\})+G_{2}(\kappa(u))+\langle
T_{\alpha\beta,\beta},u_{\alpha}\rangle_{L^{2}(\Omega)}-\langle
u,\mathbf{f}_{1}\rangle_{L^{2}}+\langle\varepsilon_{\alpha},u_{\alpha}^{2}\rangle_{L^{2}(\Gamma_{t})}$
$\displaystyle=$ $\displaystyle
G_{1}(\\{v_{\alpha\beta}(u)\\})+G_{2}(\kappa(u))-\left\langle
T_{\alpha\beta},\frac{u_{\alpha,\beta}+u_{\beta,\alpha}}{2}\right\rangle_{L^{2}(\Omega)}$
$\displaystyle+\langle
T_{\alpha\beta}n_{\beta},u_{\alpha}\rangle_{L^{2}(\Gamma_{t})}-\langle
u,\mathbf{f}_{1}\rangle_{L^{2}}+\langle\varepsilon_{\alpha},u_{\alpha}^{2}\rangle_{L^{2}(\Gamma_{t})}$
$\displaystyle=$ $\displaystyle
G_{1}(\\{v_{\alpha\beta}(u)\\})+G_{2}(\kappa(u))-\left\langle
T_{\alpha\beta},v_{\alpha\beta}(u)-\frac{1}{2}w_{,\alpha}w_{,\beta}\right\rangle_{L^{2}(\Omega)}-\langle
u,\mathbf{f}_{1}\rangle_{L^{2}}+\langle\varepsilon_{\alpha},u_{\alpha}^{2}\rangle_{L^{2}(\Gamma_{t})}$
$\displaystyle+\langle
T_{\alpha\beta}n_{\beta},u_{\alpha}\rangle_{L^{2}(\Gamma_{t})}$
$\displaystyle\geq$ $\displaystyle
c_{\alpha\beta}\|w_{,\alpha\beta}\|_{2}^{2}+\frac{1}{2}\left\langle
T_{\alpha\beta},w_{,\alpha}w_{,\beta}\right\rangle_{L^{2}(\Omega)}-\langle
u,\mathbf{f}_{1}\rangle_{L^{2}}+\langle\varepsilon_{\alpha},u_{\alpha}^{2}\rangle_{L^{2}(\Gamma_{t})}+G_{1}(\\{v_{\alpha\beta}(u)\\})$
$\displaystyle-\langle
T_{\alpha\beta},v_{\alpha\beta}(u)\rangle_{L^{2}(\Omega)}+\langle
T_{\alpha\beta}n_{\beta},u_{\alpha}\rangle_{L^{2}(\Gamma_{t})}.$

From this, since $\\{T_{\alpha\beta}\\}$ is positive definite, clearly $J$ is
bounded below.

Let $\\{u_{n}\\}\in U$ be a minimizing sequence for $J$. Thus there exists
$\alpha_{1}\in\mathbb{R}$ such that

$\lim_{n\rightarrow\infty}J(u_{n})=\inf_{u\in U}J(u)=\alpha_{1}.$

From (10), there exists $K_{1}>0$ such that

$\|(w_{n})_{,\alpha\beta}\|_{2}<K_{1},\forall\alpha,\beta\in\\{1,2\\},\;n\in\mathbb{N}.$

Therefore, there exists $w_{0}\in W^{2,2}(\Omega)$ such that, up to a
subsequence not relabeled,

$(w_{n})_{,\alpha\beta}\rightharpoonup(w_{0})_{,\alpha\beta},\;\text{ weakly
in }L^{2},$

$\forall\alpha,\beta\in\\{1,2\\},\text{ as }n\rightarrow\infty.$

Moreover, also up to a subsequence not relabeled,

$(w_{n})_{,\alpha}\rightarrow(w_{0})_{,\alpha},\;\text{ strongly in
}L^{2}\text{ and }L^{4},$ (11)

$\forall\alpha,\in\\{1,2\\},\text{ as }n\rightarrow\infty.$

Also from (10), there exists $K_{2}>0$ such that,

$\|(v_{n})_{\alpha\beta}(u)\|_{2}<K_{2},\forall\alpha,\beta\in\\{1,2\\},\;n\in\mathbb{N},$

and thus, from this, (9) and (11), we may infer that there exists $K_{3}>0$
such that

$\|(u_{n})_{\alpha,\beta}+(u_{n})_{\beta,\alpha}\|_{2}<K_{3},\forall\alpha,\beta\in\\{1,2\\},\;n\in\mathbb{N}.$

From this and Korn’s inequality, there exists $K_{4}>0$ such that

$\|u_{n}\|_{W^{1,2}(\Omega;\mathbb{R}^{2})}\leq K_{4},\;\forall
n\in\mathbb{N}.$

So, up to a subsequence not relabeled, there exists $\\{(u_{0})_{\alpha}\\}\in
W^{1,2}(\Omega,\mathbb{R}^{2}),$ such that

$(u_{n})_{\alpha,\beta}+(u_{n})_{\beta,\alpha}\rightharpoonup(u_{0})_{\alpha,\beta}+(u_{0})_{\beta,\alpha},\;\text{
weakly in }L^{2},$

$\forall\alpha,\beta\in\\{1,2\\},\text{ as }n\rightarrow\infty,$ and,

$(u_{n})_{\alpha}\rightarrow(u_{0})_{\alpha},\;\text{ strongly in }L^{2},$

$\forall\alpha\in\\{1,2\\},\text{ as }n\rightarrow\infty.$

Moreover, the boundary conditions satisfied by the subsequences are also
satisfied for $w_{0}$ and $u_{0}$ in a trace sense, so that

$u_{0}=((u_{0})_{\alpha},w_{0})\in U.$

From this, up to a subsequence not relabeled, we get

$\gamma_{\alpha\beta}(u_{n})\rightharpoonup\gamma_{\alpha\beta}(u_{0}),\text{
weakly in }L^{2},$

$\forall\alpha,\beta\in\\{1,2\\},$ and

$\kappa_{\alpha\beta}(u_{n})\rightharpoonup\kappa_{\alpha\beta}(u_{0}),\text{
weakly in }L^{2},$

$\forall\alpha,\beta\in\\{1,2\\}.$

Therefore, from the convexity of $G_{1}$ in $\gamma$ and $G_{2}$ in $\kappa$
we obtain

$\displaystyle\inf_{u\in U}J(u)$ $\displaystyle=$ $\displaystyle\alpha_{1}$
(12) $\displaystyle=$ $\displaystyle\liminf_{n\rightarrow\infty}J(u_{n})$
$\displaystyle\geq$ $\displaystyle J(u_{0}).$

Thus,

$J(u_{0})=\min_{u\in U}J(u).$

The proof is complete. ∎

##  3 The duality principles

Considering the statements and definitions of the previous sections, define
again the functional $J:U\rightarrow\mathbb{R}$ by

$\displaystyle J(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}h_{\alpha\beta\mu\lambda}\kappa_{\alpha\beta}(u)\kappa_{\lambda\mu}(u)\;dx$
(13) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}H_{\alpha\beta\mu\lambda}\gamma_{\alpha\beta}(u)\gamma_{\lambda\mu}(u)\;dx$
$\displaystyle-\langle w,P\rangle_{L^{2}}-\langle
u_{\alpha},P_{\alpha}\rangle_{L^{2}}$

where

$(u_{\alpha},w)=(u_{1},u_{2},w)\in U=W_{0}^{1,2}(\Omega,\mathbb{R}^{2})\times
W_{0}^{2,2}(\Omega),$

$P\in L^{2}(\Omega),\;P_{\alpha}\in L^{2}(\Omega)$ for $\alpha\in\\{1,2\\}.$

Also

$\kappa_{\alpha\beta}(u)=-w_{,\alpha\beta},$
$\gamma_{\alpha\beta}(u)=\frac{1}{2}(u_{\alpha,\beta}+u_{\beta,\alpha})+\frac{1}{2}w_{,\alpha}w_{,\beta}.$

Here we define $\tilde{G}_{1}:U\rightarrow\mathbb{R}$ by

$\tilde{G}_{1}(u)=G_{1}(\\{w_{,\alpha\beta}\\})=\frac{1}{2}\int_{\Omega}h_{\alpha\beta\mu\lambda}\kappa_{\alpha\beta}(u)\kappa_{\lambda\mu}(u)\;dx,$

$\tilde{G}_{2}:U\rightarrow\mathbb{R}$ by

$\tilde{G}_{2}(u)=G_{2}(\\{u_{\alpha,\beta},w_{,\alpha}\\})=\frac{1}{2}\int_{\Omega}H_{\alpha\beta\mu\lambda}\gamma_{\alpha\beta}(u)\gamma_{\lambda\mu}(u)\;dx+\frac{K}{2}\langle
w_{,\alpha},w_{,\alpha}\rangle_{L^{2}},$

and $F:U\rightarrow\mathbb{R}$ by

$F(u)=\frac{K}{2}\langle w_{,\alpha},w_{,\alpha}\rangle_{L^{2}}.$

Moreover, denoting $Y=Y^{*}=W^{2,2}_{0}(\Omega),$
$Y_{1}=Y_{1}^{*}=L^{2}(\Omega;\mathbb{R}^{2\times 2}),$
$Y_{2}=Y_{2}^{*}=L^{2}(\Omega;\mathbb{R}^{2})$ and $Y_{3}=Y_{3}^{*}=Y_{1},$
define also $G_{1}^{*}:Y_{3}^{*}\times Y^{*}\rightarrow\mathbb{R}$ by

$\displaystyle G_{1}^{*}(\tilde{M},z^{*})$ $\displaystyle=$
$\displaystyle\sup_{v_{3}\in
Y_{3}}\\{\langle(v_{3})_{\alpha\beta},\tilde{M}_{\alpha\beta}+z^{*}\delta_{\alpha\beta}\rangle_{L^{2}}-G_{1}(\\{(v_{3})_{\alpha\beta}\\})$
(14) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}\overline{h}_{\alpha\beta\lambda\mu}(\tilde{M}_{\alpha\beta}+z^{*}\delta_{\alpha\beta})(\tilde{M}_{\lambda\mu}+z^{*}\delta_{\lambda\mu})\;dx,$

$G_{2}^{*}:Y_{1}^{*}\times Y_{2}^{*}\rightarrow\mathbb{R}$ by

$\displaystyle G_{2}^{*}(N,Q)$ $\displaystyle=$
$\displaystyle\sup_{(v_{1},v_{2})\in Y_{1}\times
Y_{2}}\\{\langle(v_{1})_{\alpha\beta},N_{\alpha\beta}\rangle_{L^{2}}+\langle(v_{2})_{\alpha},Q_{\alpha}\rangle_{L^{2}}-G_{2}(\\{(v_{1})_{\alpha\beta},(v_{2})_{\alpha}\\})$
(15) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}\overline{N^{K}_{\alpha\beta}}Q_{\alpha}Q_{\beta}\;dx+\frac{1}{2}\int_{\Omega}\overline{H}_{\alpha\beta\lambda\mu}N_{\alpha\beta}N_{\lambda\mu}\;dx,$

in $B^{*}$ where

$B^{*}=\\{v^{*}=(N,Q,M)\in Y_{1}^{*}\times Y_{2}^{*}\times
Y_{3}^{*}\;:\;\\{N_{\alpha\beta}+K\delta_{\alpha\beta}\\}\text{ is positive
definite in }\Omega\\}.$

Here we have denoted

$\\{\overline{N^{K}_{\alpha\beta}}\\}=\\{N_{\alpha\beta}+K\delta_{\alpha\beta}\\}^{-1}.$

Finally, define

$F^{*}:Y^{*}\rightarrow\mathbb{R}$ by

$\displaystyle F^{*}(z^{*})$ $\displaystyle=$ $\displaystyle\sup_{u\in
U}\\{-\langle\nabla w,\nabla z^{*}\rangle_{L^{2}}-F(u)\\}$ (16)
$\displaystyle=$ $\displaystyle\frac{1}{2K}\int_{\Omega}|\nabla
z^{*}|^{2}\;dx.$

Furthermore, define

$J^{*}:B^{*}\times Y^{*}\rightarrow\mathbb{R},$

by

$J^{*}(N,Q,\tilde{M},z^{*})=-G_{1}^{*}(\tilde{M},z^{*})-G_{2}^{*}(N,Q)+F^{*}(z^{*}),$
$J_{1}^{*}:B^{*}\times Y^{*}\rightarrow\mathbb{R}$

by

$\displaystyle J_{1}^{*}(N,Q,\tilde{M},z^{*})$ $\displaystyle=$ $\displaystyle
J^{*}(N,Q,\tilde{M},z^{*})$ (17)
$\displaystyle+\frac{1}{2}\int_{\Omega}[C_{0}(\overline{h}_{11\lambda\mu}(\tilde{M}_{\lambda\mu}+z^{*}\delta_{\lambda\mu})_{,22}-\overline{h}_{22\lambda\mu}(\tilde{M}_{\lambda\mu}+z^{*}\delta_{\lambda\mu})_{,11})]$
$\displaystyle\times(\overline{h}_{11\lambda\mu}(\tilde{M}_{\lambda\mu}+z^{*}\delta_{\lambda\mu})_{,22}-\overline{h}_{22\lambda\mu}(\tilde{M}_{\lambda\mu}+z^{*}\delta_{\lambda\mu})_{,11})\;dx,$

where

$C_{0}=(1-\varepsilon_{3})(\overline{h}_{2222}D_{1111}-2\overline{h}_{1122}D_{11}D_{22}+\overline{h}_{1111}D_{2222})^{-1}$

and $0<\varepsilon_{3}<1$.

We suppose $\varepsilon_{3}$ is sufficiently close to $1$ so that $J_{1}^{*}$
is concave in $v^{*}=(N,Q,\tilde{M}),$ on $B^{*}$ (in fact, for this kind of
tensor $\\{h_{\alpha\beta\lambda\mu}\\},$ through an analysis of the Hessian
in question with the help of the softwares MATHEMATICA or MAPLE, we may infer
that $J_{1}^{*}$ is concave in $v^{*}$ on $B^{*}$ for any value
$0<\varepsilon_{3}<1).$

Here we remark to generically denote for $y\in L^{2}(\Omega),$

$(\overline{h}_{2222}D_{1111}-2\overline{h}_{1122}D_{11}D_{22}+\overline{h}_{1111}D_{2222})^{-1}(y)=w,$

if and only if

$w\in W_{0}^{2,2}(\Omega)$

and

$(\overline{h}_{2222}D_{1111}-2\overline{h}_{1122}D_{11}D_{22}+\overline{h}_{1111}D_{2222})[w]=y,$

where also generically we have denoted

$D_{\alpha\beta\gamma\mu}[w]=w_{,\alpha\beta\lambda\mu}=\frac{\partial^{4}}{\partial
x_{\alpha}\;\partial x_{\beta}\;\partial x_{\lambda}\;\partial x_{\mu}}[w].$

Moreover, define

$J_{2}^{*}:Y^{*}\rightarrow\mathbb{R}$

by

$\displaystyle J_{2}^{*}(z^{*})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{\Omega}\overline{h}_{\alpha\beta\lambda\mu}(z^{*}\delta_{\alpha\beta})(z^{*}\delta_{\lambda\mu})\;dx$
(18)
$\displaystyle+\frac{1}{2}\int_{\Omega}[C_{0}(\overline{h}_{11\lambda\mu}(z^{*}\delta_{\lambda\mu})_{,22}-(\overline{h}_{22\lambda\mu}z^{*}\delta_{\lambda\mu})_{,11})]$
$\displaystyle\times(\overline{h}_{11\lambda\mu}(z^{*}\delta_{\lambda\mu})_{,22}-(\overline{h}_{22\lambda\mu}z^{*}\delta_{\lambda\mu})_{,11})\;dx$
$\displaystyle+F^{*}(z^{*})$

and $J_{3}^{*}:B^{*}\times Y^{*}\times U\rightarrow\mathbb{R}$ by

$\displaystyle J_{3}^{*}(N,Q,\tilde{M},z^{*},u)$ $\displaystyle=$
$\displaystyle J^{*}(N,Q,\tilde{M},z^{*})$ (19) $\displaystyle+\langle
w,\tilde{M}_{\alpha\beta,\alpha\beta}-Q_{\alpha,\alpha}-P\rangle_{L^{2}}$
$\displaystyle-\langle
u_{\alpha},N_{\alpha\beta,\beta}+P_{\alpha}\rangle_{L^{2}}$

Also, define $A^{*}=A_{1}\cap A_{2}$ where

$A_{1}=\\{v^{*}=(N,Q,\tilde{M})\in
B^{*}\;:\;N_{\alpha\beta,\beta}+P_{\alpha}=0,\text{ in }\Omega\\},$
$A_{2}=\\{v^{*}=(N,Q,\tilde{M})\in
B^{*}\;:\;M_{\alpha\beta,\alpha\beta}-Q_{\alpha,\alpha}-P=0,\text{ in
}\Omega\\},$

With such statements and definitions in mind we establish our main result,
which is summarized by the following theorem.

######  Theorem 3.1.

Denoting $v_{0}^{*}=(N_{0},Q_{0},\tilde{M}_{0}),$ suppose
$(v_{0}^{*},z_{0}^{*},u_{0})\in B^{*}\times Y^{*}\times U$ is such that

$\delta J_{3}^{*}(v_{0}^{*},z_{0}^{*},u_{0})=\mathbf{0}.$

Under such hypotheses we have,

$\delta J(u_{0})=\mathbf{0},$ $v_{0}^{*}=(N_{0},Q_{0},\tilde{M}_{0})\in A^{*}$

and

$\displaystyle J(u_{0})$ $\displaystyle=$ $\displaystyle\inf_{u\in
U}\left\\{J(u)+\frac{K}{2}\int_{\Omega}|\nabla w-\nabla
w_{0}|^{2}\;dx\right\\}$ (20) $\displaystyle=$ $\displaystyle
J^{*}(v_{0}^{*},z_{0}^{*})$ $\displaystyle=$ $\displaystyle\sup_{v^{*}\in
A^{*}}J^{*}(v^{*},z_{0}^{*})=J_{1}^{*}(v_{0}^{*},z_{0}^{*}).$

###### Proof.

From the hypotheses,

$\delta J_{3}^{*}(v_{0}^{*},z_{0}^{*},u_{0})=\mathbf{0}.$

From the variation in $\tilde{M}$ we have

$-\overline{h}_{\alpha\beta\lambda\mu}((\tilde{M}_{0})_{\lambda\mu}+z_{0}^{*}\delta_{\lambda\mu})+(w_{0})_{,\alpha\beta}=0,\text{
in }\Omega.$

Thus,

$(\tilde{M}_{0})_{\alpha\beta}=h_{\alpha\beta\lambda\mu}(w_{0})_{\lambda\mu}-z_{0}^{*}\delta_{\alpha\beta}.$

From the variation in $Q$ we obtain

$-\overline{N^{K}_{\alpha\beta}}(Q_{0})_{\beta}+(w_{0})_{,\alpha}=0,\text{ in
}\Omega$

so that

$(Q_{0})_{\alpha}=(N_{0})_{\alpha\beta}(w_{0})_{\beta}+K(w_{0})_{\alpha}.$

From the variation in $N$ we get

$(\overline{N_{0}^{K}})_{\alpha\rho}(Q_{0})_{\rho}(\overline{N_{0}^{K}})_{\alpha\mu}(Q_{0})_{\mu}+\frac{1}{2}((u_{0})_{\alpha,\beta}+(u_{0})_{\beta,\alpha})-\overline{H}_{\alpha\beta\lambda\mu}(N_{0})_{\lambda\mu}=0,\text{
in }\Omega$

and hence

$(N_{0})_{\alpha\beta}=H_{\alpha\beta\lambda\mu}\left(\frac{1}{2}((u_{0})_{\lambda,\mu}+(u_{0})_{\lambda,\mu})+\frac{1}{2}(w_{0})_{\lambda}(w_{0})_{\mu}\right).$

Considering the variation in $z^{*}$ we obtain

$-\frac{\nabla^{2}z^{*}_{0}}{K}-(H_{\alpha\beta\lambda\mu}((\tilde{M}_{0})_{\lambda\mu}+z_{0}^{*}\delta_{\lambda\mu}))\delta_{\alpha\beta}=0,\text{
in }\Omega$

so that

$-\frac{\nabla^{2}z^{*}_{0}}{K}-((w_{0})_{11}+(w_{0})_{22})=0,$

that is,

$-\frac{\nabla^{2}z^{*}_{0}}{K}-\nabla^{2}w_{0}=0,$

and thus

$z_{0}^{*}=-Kw_{0},\text{ in }\Omega.$

Finally, from the variation in $u$ we obtain

$(\tilde{M}_{0})_{\alpha\beta,\alpha\beta}-(Q_{0})_{\alpha,\alpha}-P=0,$

and

$(N_{0})_{\alpha\beta,\beta}+P_{\alpha}=0,\text{ in }\Omega,$

so that

$v_{0}^{*}\in A^{*}.$

From these last results and the Legendre transform properties we get

$G_{1}^{*}(\tilde{M}_{0},z_{0}^{*})=\langle(w_{0})_{\alpha\beta},(\tilde{M}_{0})_{\alpha\beta}+z^{*}_{0}\delta_{\alpha\beta}\rangle_{L^{2}}-G_{1}(\\{(w_{0})_{\alpha\beta}\\}),$
$G_{2}^{*}(N_{0},Q_{0})=\langle(u_{0})_{\alpha,\beta},(N_{0})_{\alpha\beta}\rangle_{L^{2}}+\langle(w_{0})_{,\alpha},(Q_{0})_{\alpha}\rangle_{L^{2}}-G_{2}(\\{(u_{0})_{\alpha,\beta},(w_{0})_{,\alpha}\\})$

and

$F^{*}(z^{*}_{0})=-\langle\nabla w_{0},\nabla
z^{*}_{0}\rangle_{L^{2}}-F(u_{0}).$

From such results we may infer that

$\displaystyle J^{*}(v_{0}^{*},z_{0}^{*})$ $\displaystyle=$ $\displaystyle-
G_{1}^{*}(\tilde{M}_{0},z_{0}^{*})-G_{2}^{*}(N_{0},Q_{0})+F^{*}(z^{*}_{0})$
(21) $\displaystyle=$
$\displaystyle-\langle(w_{0})_{,\alpha\beta},(\tilde{M}_{0})_{\alpha\beta}+z^{*}_{0}\delta_{\alpha\beta}\rangle_{L^{2}}+G_{1}(\\{(w_{0})_{,\alpha\beta}\\})$
$\displaystyle-\langle(u_{0})_{\alpha,\beta},(N_{0})_{\alpha\beta}\rangle_{L^{2}}-\langle(w_{0})_{,\alpha},(Q_{0})_{\alpha}\rangle_{L^{2}}+G_{2}(\\{(u_{0})_{\alpha,\beta},(w_{0})_{,\alpha}\\})$
$\displaystyle-\langle\nabla w_{0},\nabla z^{*}_{0}\rangle_{L^{2}}-F(u_{0})$
$\displaystyle=$ $\displaystyle-\langle
w_{0},P\rangle_{L^{2}}-\langle(u_{0})_{\alpha},P_{\alpha}\rangle_{L^{2}}$
$\displaystyle+\frac{1}{2}\int_{\Omega}h_{\alpha\beta\lambda\mu}\kappa_{\alpha\beta}(u_{0})\kappa_{\lambda\mu}(u_{0})\;dx$
$\displaystyle+\frac{1}{2}\int_{\Omega}H_{\alpha\beta\lambda\mu}\gamma_{\alpha\beta}(u_{0})\gamma_{\lambda\mu}(u_{0})\;dx$
$\displaystyle=$ $\displaystyle J(u_{0}).$

Also

$(\overline{h}_{11\lambda\mu}((\tilde{M}_{0})_{\lambda\mu}+z^{*}_{0}\delta_{\lambda\mu}))_{,22}-(\overline{h}_{22\lambda\mu}((\tilde{M}_{0})_{\lambda\mu}+z^{*}_{0}\delta_{\lambda\mu})_{,11}=(w_{0})_{1122}-(w_{0})_{2211}=0,$

in distributional sense, so that

$J^{*}(v_{0}^{*},z_{0}^{*})=J_{1}^{*}(v_{0}^{*},z_{0}^{*})=J(u_{0}).$

Now observe that

$\displaystyle J^{*}(v_{0}^{*},z_{0}^{*})$ $\displaystyle=$ $\displaystyle-
G_{1}^{*}(\tilde{M}_{0},z_{0}^{*})-G_{2}^{*}(N_{0},Q_{0})+F^{*}(z^{*}_{0})$
(22) $\displaystyle\leq$ $\displaystyle-\langle
w_{,\alpha\beta},(\tilde{M}_{0})_{\alpha\beta}+z^{*}_{0}\delta_{\alpha\beta}\rangle_{L^{2}}+G_{1}(\\{w_{,\alpha\beta}\\})$
$\displaystyle-\langle
u_{\alpha,\beta},(N_{0})_{\alpha\beta}\rangle_{L^{2}}+\langle
w_{,\alpha},(Q_{0})_{\alpha}\rangle_{L^{2}}+G_{2}(\\{u_{\alpha,\beta},w_{,\alpha}\\})$
$\displaystyle+F^{*}(z_{0}^{*})$ $\displaystyle=$ $\displaystyle-\langle
w,P\rangle_{L^{2}}-\langle u_{\alpha},P_{\alpha}\rangle_{L^{2}}$
$\displaystyle+G_{1}(\\{w_{,\alpha\beta}\\})+G_{2}(\\{u_{\alpha,\beta},w_{,\alpha}\\})-F(u)+F(u)+\langle\nabla
w,\nabla z_{0}^{*}\rangle_{L^{2}}+F^{*}(z_{0}^{*})$ $\displaystyle=$
$\displaystyle-\langle w,P\rangle_{L^{2}}-\langle
u_{\alpha},P_{\alpha}\rangle_{L^{2}}$
$\displaystyle+G_{1}(\\{w_{,\alpha\beta}\\})+G_{2}(\\{u_{\alpha,\beta},w_{,\alpha}\\})-F(u)$
$\displaystyle+\frac{K}{2}\int_{\Omega}|\nabla w|^{2}\;dx-K\langle\nabla
w,\nabla w_{0}\rangle_{L^{2}}+\frac{K}{2}\int_{\Omega}|\nabla w_{0}|^{2}\;dx$
$\displaystyle=$ $\displaystyle J(u)+\frac{K}{2}\int_{\Omega}|\nabla w-\nabla
w_{0}|^{2}\;dx,\;\forall u\in U.$

Therefore we have got

$J(u_{0})=J^{*}(v_{0}^{*},z_{0}^{*})\leq\inf_{u\in
U}\left\\{J(u)+\frac{K}{2}\int_{\Omega}|\nabla w-\nabla
w_{0}|^{2}\;dx\right\\},$ (23)

so that, from this we may infer that

$\displaystyle J(u_{0})$ $\displaystyle=$ $\displaystyle\inf_{u\in
U}\left\\{J(u)+\frac{K}{2}\int_{\Omega}|\nabla w-\nabla
w_{0}|^{2}\;dx\right\\}$ (24) $\displaystyle=$ $\displaystyle
J^{*}(v_{0}^{*},z_{0}^{*})$ $\displaystyle=$ $\displaystyle\sup_{v^{*}\in
A^{*}}J^{*}(v^{*},z_{0}^{*})$ $\displaystyle=$ $\displaystyle
J_{1}^{*}(v_{0}^{*},z_{0}^{*}).$

The proof is complete. ∎

Our final result in this section refers to a concave dual variational
formulation for the plate model in question.

######  Theorem 3.2.

Denoting $v_{0}^{*}=(N_{0},Q_{0},\tilde{M}_{0}),$ suppose
$(v_{0}^{*},z_{0}^{*},u_{0})\in B^{*}\times Y^{*}\times U$ is such that

$\delta J_{3}^{*}(v_{0}^{*},z_{0}^{*},u_{0})=\mathbf{0}.$

Assume $K>0$ is such that

$J_{2}^{*}(z^{*})>0,\;\forall z^{*}\in Y^{*}\text{ such that
}z^{*}\neq\mathbf{0}.$

Under such hypotheses we have,

$\delta J(u_{0})=\mathbf{0},$ $v_{0}^{*}=(N_{0},Q_{0},\tilde{M}_{0})\in A^{*}$

and

$\displaystyle J(u_{0})$ $\displaystyle=$ $\displaystyle\inf_{u\in U}J(u)$
(25) $\displaystyle=$ $\displaystyle\sup_{v^{*}\in
A^{*}}\left\\{\inf_{z^{*}\in Y^{*}}J_{1}^{*}(v^{*},z^{*})\right\\}$
$\displaystyle=$ $\displaystyle J_{1}^{*}(v_{0}^{*},z_{0}^{*}).$

###### Proof.

The proof that $\delta J(u_{0})=\mathbf{0}$ and $v_{0}^{*}\in A^{*}$ may be
done exactly as in the proof of the last theorem. First recall that we have
assumed $J_{2}^{*}(z^{*})>0$ for all $z^{*}\in Y^{*}$ such that
$z^{*}\neq\mathbf{0}.$

Therefore, denoting $J_{4}^{*}:B^{*}\times Y^{*}\times U\rightarrow\mathbb{R}$
by

$\displaystyle J_{4}^{*}(v^{*},z^{*},u)$ $\displaystyle=$ $\displaystyle
J_{1}^{*}(v^{*},z^{*})$ (26) $\displaystyle+\langle
w,\tilde{M}_{\alpha\beta,\alpha\beta}-Q_{\alpha,\alpha}-P\rangle_{L^{2}}$
$\displaystyle-\langle
u_{\alpha},N_{\alpha\beta,\beta}+P_{\alpha}\rangle_{L^{2}},$

since $J_{1}^{*}$ is concave in $v^{*}$ and convex in $z^{*}$, we have got

$\displaystyle J(u_{0})$ $\displaystyle=$ $\displaystyle
J^{*}_{1}(v_{0}^{*},z_{0}^{*})$ (27) $\displaystyle=$
$\displaystyle\sup_{v^{*}\in A^{*}}\left\\{\inf_{z^{*}\in
Y^{*}}J_{1}^{*}(v^{*},z^{*})\right\\}$ $\displaystyle\leq$
$\displaystyle\sup_{v^{*}\in A^{*}}J_{1}^{*}(v^{*},z^{*})$ $\displaystyle\leq$
$\displaystyle\sup_{v^{*}\in B^{*}}J_{4}^{*}(v^{*},z^{*},u),\;\forall u\in
U,\;z^{*}\in Y^{*}.$

From this we get

$\displaystyle J(u_{0})$ $\displaystyle=$ $\displaystyle
J^{*}_{1}(v_{0}^{*},z_{0}^{*})$ (28) $\displaystyle\leq$
$\displaystyle\inf_{z^{*}\in Y^{*}}\left\\{\sup_{v^{*}\in
B^{*}}J_{4}^{*}(v^{*},z^{*},u)\right\\},\;\forall u\in U.$

Now observe that denoting

$L(v^{*},z^{*})=(\overline{h}_{11\lambda\mu}(\tilde{M}_{\lambda\mu}+z^{*}\delta_{\lambda\mu}))_{,22}+-(\overline{h}_{22\lambda\mu}(\tilde{M}_{\lambda\mu}+z^{*}\delta_{\lambda\mu}))_{,11},$

from the variations in $\tilde{M}_{11}$ and $\tilde{M}_{22}$, since the
operator $C_{0}$ is self-adjoint, we may obtain,

$\displaystyle-\overline{h}_{11\lambda\mu}(\tilde{M}_{\lambda\mu}+z^{*}\delta_{\lambda\mu})+w_{,11}$
$\displaystyle+\\{\overline{h}_{1111}[C_{0}[L(v^{*},z^{*})]]\\}_{,22}-\\{\overline{h}_{2211}[C_{0}[L(v^{*},z^{*})]]\\}_{,11}=0,$
(29)

and

$\displaystyle-\overline{h}_{22\lambda\mu}(\tilde{M}_{\lambda\mu}+z^{*}\delta_{\lambda\mu})+w_{,22}$
$\displaystyle+\\{\overline{h}_{1122}[C_{0}[L(v^{*},z^{*})]]\\}_{,22}-\\{\overline{h}_{2222}[C_{0}[L(v^{*},z^{*})]]\\}_{,11}=0.$
(30)

From this, recalling that

$L(v^{*},z^{*})=(\overline{h}_{11\lambda\mu}(\tilde{M}_{\lambda\mu}+z^{*}\delta_{\lambda\mu}))_{,22}-(\overline{h}_{22\lambda\mu}(\tilde{M}_{\lambda\mu}+z^{*}\delta_{\lambda\mu}))_{,11}$

we get

$\displaystyle-(\overline{h}_{11\lambda\mu}(\tilde{M}_{\lambda\mu}+z^{*}\delta_{\lambda\mu}))_{,22}+(\overline{h}_{22\lambda\mu}(\tilde{M}_{\lambda\mu}+z^{*}\delta_{\lambda\mu}))_{,11}$
(31)
$\displaystyle+\left\\{\overline{h}_{1111}[C_{0}[L(v^{*},z^{*})]]_{,22}-\overline{h}_{2211}[C_{0}[L(v^{*},z^{*})]]_{,11}\right\\}_{,22}$
$\displaystyle-\left\\{\overline{h}_{1122}[C_{0}[L(v^{*},z^{*})]]_{,22}-\overline{h}_{2222}[C_{0}[L(v^{*},z^{*})]]_{,11}\right\\}_{,11}$
$\displaystyle=$ $\displaystyle-w_{1122}+w_{2211}=0,$

so that through the equation related to the variation in $\tilde{M}$ satisfied
we have obtained

$L(v^{*},z^{*})=0,\;\forall v^{*}\in B^{*},z^{*}\in Y^{*}.$

From such results we may infer that if

$\\{N_{\alpha\beta}(u)+K\delta_{\alpha\beta}\\}$

is positive definite $\text{in }\Omega$, then

$\inf_{z^{*}\in Y^{*}}\left\\{\sup_{v^{*}\in
B^{*}}J_{4}^{*}(v^{*},z^{*},u)\right\\}=J(u),$

where

$N_{\alpha\beta}(u)=H_{\alpha\beta\lambda\mu}\gamma_{\lambda\mu}(u).$

Therefore, since $J_{1}^{*}$ is concave in $v^{*}$ and convex in $z^{*}$, from
these last results and from the min-max theorem we have

$\displaystyle J(u_{0})$ $\displaystyle=$ $\displaystyle
J^{*}_{1}(v_{0}^{*},z_{0}^{*})$ (32) $\displaystyle=$
$\displaystyle\sup_{v^{*}\in A^{*}}\left\\{\inf_{z^{*}\in
Y^{*}}J_{1}^{*}(v^{*},z^{*})\right\\}$ $\displaystyle\leq$
$\displaystyle\inf_{z^{*}\in Y^{*}}\left\\{\sup_{v^{*}\in
B^{*}}J_{4}^{*}(v^{*},z^{*},u)\right\\}$ $\displaystyle\leq$ $\displaystyle
J(u),\;\forall u\in U.$

Thus,

$\displaystyle J(u_{0})$ $\displaystyle=$ $\displaystyle\inf_{u\in U}J(u)$
(33) $\displaystyle=$ $\displaystyle\sup_{v^{*}\in
A^{*}}\left\\{\inf_{z^{*}\in Y^{*}}J_{1}^{*}(v^{*},z^{*})\right\\}$
$\displaystyle=$ $\displaystyle J_{1}^{*}(v_{0}^{*},z_{0}^{*}).$

The proof is complete.

∎

##  4 An auxiliary theoretical result in analysis

In this section we state and prove some theoretical results in analysis which
will be used in the subsequent sections.

######  Theorem 4.1.

Let $\Omega\subset\mathbb{R}^{3}$ be an open, bounded and connected set with a
regular (Lipschitzian) boundary denoted by $\partial\Omega.$

Assume $\\{u_{n}\\}\subset W^{1,4}(\Omega)$ be such that

$\|u_{n}\|_{1,4}\leq K,\;\forall n\in\mathbb{N},$

for some $K>0.$

Under such hypotheses there exists $u_{0}\in W^{1,4}(\Omega)\cap
C(\overline{\Omega})$ such that, up to a not relabeled subsequence,

$u_{n}\rightharpoonup u_{0},\text{ weakly in }W^{1,4}(\Omega),$
$u_{n}\rightarrow u_{0}\text{ uniformly in }\overline{\Omega}$

and

$u_{n}\rightarrow u_{0},\text{ strongly in }W^{1,3}(\Omega).$

###### Proof.

Since $W^{1,4}(\Omega)$ is reflexive, from the Kakutani and Sobolev Imbedding
theorems, up to a not relabeled there exists $u_{0}\in W^{1,4}(\Omega)$ such
that

$u_{n}\rightharpoonup u_{0},\text{ weakly in }W^{1,4}(\Omega),$

and

$u_{n}\rightarrow u_{0},\text{ strongly in }L^{4}(\Omega).$

From the Rellich-Kondrachov Theorem, since for $m=1$, $p=4$ and $n=3$, we have
$mp>n$, the following imbedding is compact,

$W^{1,4}(\Omega)\hookrightarrow C(\overline{\Omega}).$

Thus,

$\\{u_{n}\\}\subset C(\overline{\Omega}),$

and again up to a not relabeled subsequence,

$u_{n}\rightarrow u_{0}\text{ uniformly in }\overline{\Omega},$

and also

$u_{0}\in C(\overline{\Omega}),$

so that

$u_{0}\in W^{1,4}(\Omega)\cap C(\overline{\Omega}).$

Let $\varepsilon>0$. Hence, there exists $n_{0}\in\mathbb{N}$ such that if
$n>n_{0}$, then

$|u_{n}(x)-u_{0}(x)|<\varepsilon,\text{ for almost all }x\in\Omega.$

Let

$\varphi\in C^{1}_{c}(\Omega).$

Choose $j\in\\{1,2,3\\}.$

Therefore, we may obtain

$\displaystyle\left|\left\langle\frac{\partial u_{n}}{\partial
x_{j}}-\frac{\partial u_{0}}{\partial
x_{j}},\varphi\right\rangle_{L^{2}}\right|$ $\displaystyle=$
$\displaystyle\left|\left\langle u_{n}-u_{0},\frac{\partial\varphi}{\partial
x_{j}}\right\rangle_{L^{2}}\right|$ (34) $\displaystyle\leq$
$\displaystyle\left\langle|u_{n}-u_{0}|,\left|\frac{\partial\varphi}{\partial
x_{j}}\right|\right\rangle_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\varepsilon\left\|\frac{\partial\varphi}{\partial
x_{j}}\right\|_{1},\;\forall n>n_{0}.$

From this we may infer that

$\lim_{n\rightarrow\infty}\left\langle\frac{\partial u_{n}}{\partial
x_{j}}-\frac{\partial u_{0}}{\partial
x_{j}},\varphi\right\rangle_{L^{2}}=0,\forall\varphi\in C^{1}_{c}(\Omega).$

At this point we claim that

$\lim_{n\rightarrow\infty}\left\langle\frac{\partial u_{n}}{\partial
x_{j}}-\frac{\partial u_{0}}{\partial
x_{j}},\varphi\right\rangle_{L^{2}}=0,\forall\varphi\in C_{c}(\Omega).$

To prove such a claim, let $\varphi\in C_{c}(\Omega).$

Let a new $\varepsilon>0$ be given.

Hence, for each $r>0$ there exists $n_{r}\in\mathbb{N}$ such that if
$n>n_{r}$, then

$\|u_{n}-u_{0}\|_{\infty}<\varepsilon r.$

Observe that by density, we may obtain $\varphi_{1}\in C^{1}_{c}(\Omega)$ such
that

$\|\varphi-\varphi_{1}\|_{\infty}<\varepsilon.$

Hence,

$\displaystyle\left|\left\langle\frac{\partial u_{n}}{\partial
x_{j}}-\frac{\partial u_{0}}{\partial
x_{j}},\varphi\right\rangle_{L^{2}}\right|$ (35) $\displaystyle\leq$
$\displaystyle\left\langle\left|\frac{\partial u_{n}}{\partial
x_{j}}-\frac{\partial u_{0}}{\partial
x_{j}}\right|,|\varphi-\varphi_{1}|\right\rangle_{L^{2}}$
$\displaystyle+\left|\left\langle\frac{\partial u_{n}}{\partial
x_{j}}-\frac{\partial u_{0}}{\partial
x_{j}},\varphi_{1}\right\rangle_{L^{2}}\right|$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}\left|\frac{\partial u_{n}}{\partial
x_{j}}-\frac{\partial u_{0}}{\partial
x_{j}}\right|\;dx\|\varphi-\varphi_{1}\|_{\infty}$
$\displaystyle+\left|\left\langle
u_{n}-u_{0},\frac{\partial\varphi_{1}}{\partial
x_{j}}\right\rangle_{L^{2}}\right|$ $\displaystyle\leq$ $\displaystyle
2K_{1}\varepsilon+\varepsilon\left\|\frac{\partial\varphi_{1}}{\partial
x_{j}}\right\|_{1}$ $\displaystyle=$
$\displaystyle\left(2K_{1}+\left\|\frac{\partial\varphi_{1}}{\partial
x_{j}}\right\|_{1}\right)\varepsilon,\forall n>n_{1}.$

where $K_{1}>0$ is such that

$\|u_{0}\|_{1}<K_{1},\|u_{0}\|_{2}<K_{1}$

and

$\|u_{n}\|_{1}<K_{1},\;\|u_{n}\|_{2}<K_{1}\;\forall n\in\mathbb{N}.$

From this we may infer that

$\lim_{n\rightarrow\infty}\left\langle\frac{\partial u_{n}}{\partial
x_{j}}-\frac{\partial u_{0}}{\partial
x_{j}},\varphi\right\rangle_{L^{2}}=0,\;\forall\varphi\in C_{c}(\Omega)$ (36)

so that the claim holds.

Since $\Omega$ is bounded, we have $W^{1,4}(\Omega)\subset W^{1,2}(\Omega)$.

From the Gauss-Green Formula for such a latter space, we obtain

$\displaystyle\lim_{n\rightarrow\infty}\left|\frac{\partial u_{n}(x)}{\partial
x_{j}}-\frac{\partial u_{0}(x)}{\partial x_{j}}\right|$ (37) $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\left(\lim_{r\rightarrow
0^{+}}\frac{\left|\int_{B_{r}(x)}\left(\frac{\partial u_{n}(y)}{\partial
x_{j}}-\frac{\partial u_{0}(y)}{\partial
x_{j}}\right)\;dy\right|}{m(B_{r}(x))}\right)$ $\displaystyle\leq$
$\displaystyle\limsup_{n\rightarrow\infty}\left(\limsup_{r\rightarrow
0^{+}}\frac{\left|\int_{B_{r}(x)}\left(\frac{\partial u_{n}(y)}{\partial
x_{j}}-\frac{\partial u_{0}(y)}{\partial
x_{j}}\right)\;dy\right|}{m(B_{r}(x))}\right)$ $\displaystyle=$
$\displaystyle\limsup_{r\rightarrow
0^{+}}\left(\limsup_{n\rightarrow\infty}\frac{\left|\int_{B_{r}(x)}\left(\frac{\partial
u_{n}(y)}{\partial x_{j}}-\frac{\partial u_{0}(y)}{\partial
x_{j}}\right)\;dy\right|}{m(B_{r}(x))}\right)$ $\displaystyle=$
$\displaystyle\limsup_{r\rightarrow
0^{+}}\left(\limsup_{n\rightarrow\infty}\frac{\left|\int_{\partial
B_{r}(x)}(u_{n}(y)-u_{0}(y))\nu_{i}\;dS(y)\right|}{m(B_{r}(x))}\right)$
$\displaystyle=$ $\displaystyle\limsup_{r\rightarrow
0^{+}}\left(\limsup_{n\rightarrow\infty}\frac{\left|(u_{n}(\tilde{y})-u_{0}(\tilde{y}))\int_{\partial
B_{r}(x)}\nu_{i}\;dS(y)\right|}{m(B_{r}(x))}\right)$ $\displaystyle\leq$
$\displaystyle\varepsilon\limsup_{r\rightarrow 0^{+}}\frac{\int_{\partial
B_{r}(x)}r|\nu_{i}|\;dS(y)}{m(B_{r}(x))}$ $\displaystyle\leq$ $\displaystyle
K_{1}\varepsilon,\text{ for almost all }x\in\Omega,$

where $\tilde{y}\in\overline{B_{r}(x)}$ depends on $r$ and $n$.

Therefore, we may infer that

$\lim_{n\rightarrow\infty}\frac{\partial u_{n}(x)}{\partial
x_{j}}=\frac{\partial u_{0}(x)}{\partial x_{j}},\text{ a. e. in }\Omega.$

Here we define

$A_{n,\varepsilon}=\left\\{x\in\Omega\;:\;\left|\frac{\partial
u_{n}(x)}{\partial x_{j}}-\frac{\partial u_{0}(x)}{\partial
x_{j}}\right|<\varepsilon\right\\}.$

Define also

$B_{n}=\cap_{k=n}^{\infty}A_{k,\varepsilon}.$

Observe that for almost all $x\in\Omega$, there exists $n_{x}\in\mathbb{N}$
such that if $n>n_{x}$, then

$\left|\frac{\partial u_{n}(x)}{\partial x_{j}}-\frac{\partial
u_{0}(x)}{\partial x_{j}}\right|<\varepsilon,$

so that almost all $x\in B_{n},\;\forall n>n_{x}.$

From this

$\Omega=\left(\cup_{n=1}^{\infty}B_{n}\right)\cup B_{0},$

where $m(B_{0})=0.$

Also

$\cup_{k=1}^{n}B_{k}=B_{n},$

so that

$\lim_{n\rightarrow\infty}m(B_{n})=m(\Omega).$

Observe that there exists $n_{0}\in\mathbb{N}$ such that if $n>n_{0}$, then

$\sqrt[4]{m(\Omega\setminus B_{n})}<\varepsilon/K^{3}.$

Consequently fixing $n>n_{0}$, from the generalized Hölder inequality, if
$m>n$, we have

$\displaystyle\int_{\Omega}\left|\frac{\partial u_{m}}{\partial
x_{j}}-\frac{\partial u_{0}}{\partial x_{j}}\right|^{3}\;dx$ (38)
$\displaystyle=$ $\displaystyle\int_{\Omega\setminus
B_{n}}\left|\frac{\partial u_{m}}{\partial x_{j}}-\frac{\partial
u_{0}}{\partial x_{j}}\right|^{3}\;dx$
$\displaystyle+\int_{B_{n}}\left|\frac{\partial u_{m}}{\partial
x_{j}}-\frac{\partial u_{0}}{\partial x_{j}}\right|^{3}\;dx$
$\displaystyle\leq$ $\displaystyle\left\|\frac{\partial u_{m}}{\partial
x_{j}}-\frac{\partial u_{0}}{\partial
x_{j}}\right\|_{4}^{3}\|\chi_{\Omega\setminus
B_{n}}\|_{4}+\varepsilon^{3}m(\Omega)$ $\displaystyle\leq$
$\displaystyle\varepsilon+\varepsilon^{3}m(\Omega).$

Summarizing, we may infer that

$\int_{\Omega}\left|\frac{\partial u_{m}}{\partial x_{j}}-\frac{\partial
u_{0}}{\partial x_{j}}\right|^{3}\;dx\rightarrow 0,\text{ as
}m\rightarrow\infty,\;\forall j\in\\{1,2,3\\}.$

so that

$u_{n}\rightarrow u_{0},\text{ strongly in }W^{1,3}(\Omega).$

The proof is complete.

∎

##  5 An existence result for a model in elasticity

In this section we present an existence result for a non-linear elasticity
model.

######  Theorem 5.1.

Let $\Omega\subset\mathbb{R}^{3}$ be an open, bounded and connected set with a
regular (Lipschitzian) boundary denoted by $\partial\Omega.$

Consider the functional $J:U\rightarrow\mathbb{R}$ defined by

$\displaystyle J(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}H_{ijkl}\left(\frac{u_{i,j}+u_{j,i}}{2}+\frac{u_{m,i}u_{m,j}}{2}\right)\left(\frac{u_{k,l}+u_{l,k}}{2}+\frac{u_{p,k}u_{p,l}}{2}\right)\;dx$
(39) $\displaystyle-\langle P_{i},u_{i}\rangle_{L^{2}},$

where $U=W_{0}^{1,4}(\Omega;\mathbb{R}^{3})$, $P_{i}\in
L^{\infty}(\Omega),\;\forall i\in\\{1,2,3\\}.$

Moreover, $\\{H_{ijkl}\\}$ is a fourth order constant tensor such that

$H_{ijkl}t_{ij}t_{kl}\geq c_{0}t_{ij}t_{ij},\;\forall\text{ symmetric
}t\in\mathbb{R}^{2\times 2}$

and

$H_{ijkl}t_{mi}t_{mj}t_{kp}t_{lp}\geq
c_{1}\sum_{i,j=1}^{3}t_{ij}^{4},\;\forall\text{ symmetric
}t\in\mathbb{R}^{2\times 2},$

for some real constants $c_{0}>0,c_{1}>0$.

Under such hypotheses, there exists $u_{0}\in U$ such that

$J(u_{0})=\min_{u\in U}J(u).$

###### Proof.

First observe that we may find a positive definite tensor $\\{T_{ij}\\}\subset
L^{\infty}(\Omega;\mathbb{R}^{2\times 2})$ such that

$T_{ij,j}+P_{i}=0,\text{ in }\Omega.$

Hence, denoting

$v_{ij}(u)=\frac{u_{i,j}+u_{j,i}}{2}+\frac{u_{m,i}u_{m,j}}{2},$

we have

$\frac{u_{i,j}+u_{j,i}}{2}=v_{ij}(u)-\frac{u_{m,i}u_{m,j}}{2},$

so that

$\displaystyle J(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}H_{ijkl}v_{ij}(u)v_{kl}(u)\;dx+\langle
T_{ij,j},u_{i}\rangle_{L^{2}}$ (40) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}H_{ijkl}v_{ij}(u)v_{kl}(u)\;dx-\left\langle
T_{ij},\frac{u_{i,j}+u_{j,i}}{2}\right\rangle_{L^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}H_{ijkl}v_{ij}(u)v_{kl}(u)\;dx-\left\langle
T_{ij},v_{ij}(u)-\frac{u_{m,i}\;u_{m,j}}{2}\right\rangle_{L^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}H_{ijkl}v_{ij}(u)v_{kl}(u)\;dx-\left\langle
T_{ij},v_{ij}(u)\right\rangle_{L^{2}}+\left\langle
T_{ij},\frac{u_{m,i}\;u_{m,j}}{2}\right\rangle_{L^{2}},\;\forall u\in U.$

From this and the hypotheses on $\\{H_{ijkl}\\}$ it is clear that $J$ is
bounded below so that there exists $\alpha\in\mathbb{R}$ such that

$\alpha=\inf_{u\in U}J(u).$

Let $\\{u_{n}\\}\subset U$ be a minimizing sequence for $J$, that is, let such
a sequence be such that

$J(u_{n})\rightarrow\alpha,\;\text{ as }n\rightarrow\infty.$

Also from the hypotheses on $\\{H_{ijkl}\\}$ and the Poincaré inequality, we
have that there exists $K>0$ such that

$\|u_{n}\|_{1,4}\leq K,\;\forall n\in\mathbb{N}.$

From the auxiliary result in the last section, there exists $u_{0}\in
C^{0}(\overline{\Omega};\mathbb{R}^{3})\cap W^{1,4}(\Omega;\mathbb{R}^{3})$
such that, up to a not relabeled subsequence,

$u_{n}\rightarrow u_{0},\text{ strongly in }W^{1,3}(\Omega:\mathbb{R}^{3}).$

From such a latter result, up to a not relabeled subsequence, we may obtain

$\frac{(u_{n})_{i,j}+(u_{n})_{j,i}}{2}+\frac{(u_{n})_{m,i}(u_{n})_{m,j}}{2}\rightharpoonup\frac{(u_{0})_{i,j}+(u_{0})_{j,i}}{2}+\frac{(u_{0})_{m,i}(u_{0})_{m,j}}{2},\text{
weakly in }L^{3/2}(\Omega).$

Since $L^{3/2}(\Omega)$ is reflexive, from the convexity of $J$ in $v_{ij}(u)$
and since $\\{T_{ij}\\}$ is positive definite, we have that

$\alpha=\liminf_{n\rightarrow\infty}J(u_{n})\geq J(u_{0}),$

so that

$J(u_{0})=\min_{u\in U}J(u).$

The proof is complete.

∎

##  6 Another existence result for a model in elasticity

In this section we present another existence result for a similar (to the
previous one) non-linear elasticity model.

######  Theorem 6.1.

Let $\Omega\subset\mathbb{R}^{3}$ be an open, bounded and connected set with a
regular (Lipschitzian) boundary denoted by $\partial\Omega.$

Consider the functional $J:U\rightarrow\mathbb{R}$ defined by

$\displaystyle J(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}H_{ijkl}\left(\frac{u_{i,j}+u_{j,i}}{2}+\frac{u_{m,i}u_{m,j}}{2}\right)\left(\frac{u_{k,l}+u_{l,k}}{2}+\frac{u_{p,k}u_{p,l}}{2}\right)\;dx$
(41) $\displaystyle-\langle P_{i},u_{i}\rangle_{L^{2}}-\langle
P^{t}_{i},u_{i}\rangle_{L^{2}(\Gamma_{t})},$

where

$\partial\Omega=\Gamma=\Gamma_{0}\cup\Gamma_{t},$
$\Gamma_{0}\cap\Gamma_{t}=\emptyset,$

$m_{\Gamma}(\Gamma_{0})>0,$ $m_{\Gamma}(\Gamma_{t})>0,$ $P_{i}\in
L^{\infty}(\Omega)\cap W^{1,2}(\Omega),\;P^{t}_{i}\in
L^{\infty}(\Gamma_{t}),\;\forall i\in\\{1,2,3\\}.$

Moreover

$U=\\{u\in W^{1,4}(\Omega;\mathbb{R}^{3})\;:\;u=\hat{u}_{0}\text{ on
}\Gamma_{0}\\},$

where we assume $\hat{u}_{0}\in W^{1,4}(\Omega).$

Furthermore, $\\{H_{ijkl}\\}$ is a fourth order symmetric constant tensor such
that

$H_{ijkl}t_{ij}t_{kl}\geq c_{0}t_{ij}t_{ij},\;\forall\text{ symmetric
}t\in\mathbb{R}^{2\times 2}$

and

$H_{ijkl}t_{mi}t_{mj}t_{kp}t_{lp}\geq
c_{1}\sum_{i,j=1}^{3}t_{ij}^{4},\;\forall\text{ symmetric
}t\in\mathbb{R}^{2\times 2},$

for some real constants $c_{0}>0,c_{1}>0$.

Under such hypotheses, there exists $u_{0}\in U$ such that

$J(u_{0})=\min_{u\in U}J(u).$

###### Proof.

First observe that we may find a positive definite tensor $\\{T_{ij}\\}\subset
L^{\infty}(\Omega;\mathbb{R}^{2\times 2})\cap
W^{1,2}(\Omega;\mathbb{R}^{2\times 2})$ such that

$T_{ij,j}+P_{i}=0,\text{ in }\Omega.$

Hence, denoting

$v_{ij}(u)=\frac{u_{i,j}+u_{j,i}}{2}+\frac{u_{m,i}u_{m,j}}{2},$

we have

$\frac{u_{i,j}+u_{j,i}}{2}=v_{ij}(u)-\frac{u_{m,i}u_{m,j}}{2},$

so that, from this, the Gauss-Green formula and the Trace Theorem,

$\displaystyle J(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}H_{ijkl}v_{ij}(u)v_{kl}(u)\;dx+\langle
T_{ij,j},u_{i}\rangle_{L^{2}}-\langle P_{i},u_{i}\rangle_{L^{2}}-\langle
P^{t}_{i},u_{i}\rangle_{L^{2}(\Gamma_{t})}$ (42) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}H_{ijkl}v_{ij}(u)v_{kl}(u)\;dx-\left\langle
T_{ij},\frac{u_{i,j}+u_{j,i}}{2}\right\rangle_{L^{2}}$ $\displaystyle+\langle
u_{i},T_{ij}\nu_{j}\rangle_{L^{2}(\Gamma_{t})}+\langle(u_{0})_{i},T_{ij}\nu_{j}\rangle_{L^{2}(\Gamma_{0})}-\langle
P_{i},u_{i}\rangle_{L^{2}}-\langle P^{t}_{i},u_{i}\rangle_{L^{2}(\Gamma_{t})}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}H_{ijkl}v_{ij}(u)v_{kl}(u)\;dx-\left\langle
T_{ij},v_{ij}(u)-\frac{u_{m,i}\;u_{m,j}}{2}\right\rangle_{L^{2}}$
$\displaystyle+\langle
u_{i},T_{ij}\nu_{j}\rangle_{L^{2}(\Gamma_{t})}+\langle(\hat{u}_{0})_{i},T_{ij}\nu_{j}\rangle_{L^{2}(\Gamma_{0})}-\langle
P_{i},u_{i}\rangle_{L^{2}}-\langle P^{t}_{i},u_{i}\rangle_{L^{2}(\Gamma_{t})}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\int_{\Omega}H_{ijkl}v_{ij}(u)v_{kl}(u)\;dx-\langle
T_{ij},v_{ij}(u)\rangle_{L^{2}}+\left\langle
T_{ij},\frac{u_{m,i}\;u_{m,j}}{2}\right\rangle_{L^{2}}$ $\displaystyle-
K_{3}\sum_{i=1}^{3}\|u_{i}\|_{1,4}-K_{3}\|\hat{u}_{0}\|_{1,4},\;\forall u\in
U,$

for some appropriate $K_{3}>0$.

From this, the hypotheses on $\\{H_{ijkl}\\}$ and a Poincaré type inequality,
since $\\{T_{ij}\\}$ is positive definite, it is clear that $J$ is bounded
below so that there exists $\alpha\in\mathbb{R}$ such that

$\alpha=\inf_{u\in U}J(u).$

Let $\\{u_{n}\\}\subset U$ be a minimizing sequence for $J$, that is, let such
a sequence be such that

$J(u_{n})\rightarrow\alpha,\;\text{ as }n\rightarrow\infty.$

Also from the hypotheses on $\\{H_{ijkl}\\}$ and a Poincaré type inequality,
we have that there exists $K>0$ such that

$\|u_{n}\|_{1,4}\leq K,\;\forall n\in\mathbb{N}.$

From the auxiliary result in the last section, there exists $u_{0}\in
C^{0}(\overline{\Omega};\mathbb{R}^{3})\cap W^{1,4}(\Omega;\mathbb{R}^{3})$
such that, up to a not relabeled subsequence,

$u_{n}\rightarrow u_{0},\text{ strongly in }W^{1,3}(\Omega;\mathbb{R}^{3}).$

From such a latter result, up to a not relabeled subsequence, we may obtain

$\frac{(u_{n})_{i,j}+(u_{n})_{j,i}}{2}+\frac{(u_{n})_{m,i}(u_{n})_{m,j}}{2}\rightharpoonup\frac{(u_{0})_{i,j}+(u_{0})_{j,i}}{2}+\frac{(u_{0})_{m,i}(u_{0})_{m,j}}{2},\text{
weakly in }L^{3/2}(\Omega).$

Also from the continuity of the Trace operator we get

$u_{0}=\hat{u}_{0},\text{ on }\Gamma_{0},$

so that $u_{0}\in U.$

Since $L^{3/2}(\Omega)$ is reflexive, from the convexity of $J$ in
$\\{v_{ij}(u)\\}$ and since $\\{T_{ij}\\}$ is positive definite, we have that

$\alpha=\liminf_{n\rightarrow\infty}J(u_{n})\geq J(u_{0}),$

so that,

$J(u_{0})=\min_{u\in U}J(u).$

The proof is complete.

∎

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