# Narrative Review of Support for Emotional Expressions in Virtual Reality:
Psychophysiology of speech-to-text interfaces

Sunday D. Ubur, Denis Gracanin

###### Abstract

Keywords: Emotional expression, VR, Psychophysiology, Speech-to-text, Empathic
machine.

This narrative review on emotional expression in Speech-to-Text (STT)
interfaces with Virtual Reality (VR) aims to identify advancements,
limitations, and research gaps in incorporating emotional expression into
transcribed text generated by STT systems. Using a rigorous search strategy,
relevant articles published between 2020 and 2024 are extracted and
categorized into themes such as communication enhancement technologies,
innovations in captioning, emotion recognition in AR and VR, and empathic
machines. The findings reveal the evolution of tools and techniques to meet
the needs of individuals with hearing impairments, showcasing innovations in
live transcription, closed captioning, AR, VR, and emotion recognition
technologies. Despite improvements in accessibility, the absence of emotional
nuance in transcribed text remains a significant communication challenge. The
study underscores the urgency for innovations in STT technology to capture
emotional expressions. The research discusses integrating emotional expression
into text through strategies like animated text captions, emojilization tools,
and models associating emotions with animation properties. Extending these
efforts into AR and VR environments opens new possibilities for immersive and
emotionally resonant experiences, especially in educational contexts. The
study also explores empathic applications in healthcare, education, and human-
robot interactions, highlighting the potential for personalized and effective
interactions. The multidisciplinary nature of the literature underscores the
potential for collaborative and interdisciplinary research.

##  1 Introduction

In the ever-evolving landscape of communication technologies, Speech-to-Text
(STT) interfaces play a pivotal role in enhancing accessibility and
inclusivity, particularly for individuals with hearing impairments. The
ability to convert spoken language into text facilitates communication in
various contexts, ranging from online meetings to educational settings.
However, amidst the strides made in improving accessibility, a critical aspect
often overlooked is the incorporation of emotional expression into transcribed
text generated by STT systems.

This study embarks on a comprehensive exploration of the existing literature
surrounding emotional expression in STT interfaces, with a specific emphasis
on the psychophysiology aspect within Virtual Reality (VR). The objective is
to identify advancements, limitations, and research gaps in the incorporation
of emotional expression in transcribed text generated by STT systems. As
communication technologies continue to advance, the ability to convey not only
the semantic content but also the emotional nuances of speech becomes
paramount for fostering richer and more meaningful interactions.

The ubiquity of STT applications, exemplified by tools like Live Transcribe,
has significantly contributed to breaking communication barriers for the Deaf
and Hard of Hearing (DHH) community. These applications offer real-time
transcriptions that are invaluable in various scenarios, from professional
communication to educational contexts. However, the inherent challenge lies in
the loss of emotional nuance during the transcription process, posing a
communication hurdle that this study seeks to address.

To unravel the complexities surrounding emotional expression in STT
interfaces, our investigation spans various dimensions. From examining
innovations in live transcription and closed captioning to delving into
advancements in augmented reality (AR), emotive captioning, emotion
recognition, and empathic machines, this study aims to provide a holistic
understanding of the evolving tools and techniques. Additionally, the
exploration extends into the realms of VR and AR, where unique opportunities
arise to create immersive and emotionally resonant experiences, especially in
the context of education and training. In the subsequent sections, we detail
the methodology employed in conducting the literature review, present the
background encompassing communication enhancement technologies, and delve into
innovations in captioning, emotion recognition, and empathic systems. The
synthesis of these findings not only identifies current advancements but also
points towards avenues for future research, emphasizing the ongoing need for
inclusive and accessible communication technologies.

##  2 Methods

#### Search Strategy:

The search involved a comprehensive search of scholarly databases, including
ACM Digital Library, IEEE Explore, Scopus, and Google Scholar, to identify
relevant articles published between 2020 and 2024. The search terms used
included (”Emotional Expressions” AND ”Virtual Reality” AND ”Psychophysiology”
AND ”speech-to-text Interfaces” OR ”Captions”), and the inclusion criteria
encompassed studies related to emotional expressions in psychophysiology of
STT interfaces. We searched individual databases, however, much of the data
came from the Publish or Perish software [1] as it extracts results across
these databases mentioned.

#### Inclusion and Exclusion Criteria:

The rest of the methods process was conducted using Rayyan software [2]. Most
of the papers included are peer-reviewed articles, books, and conference
proceedings that provided insights into the background and direction of
implementing emotional expressions in STT. Studies with a focus on generating
emotional expressions in text were prioritized. Non-English language
publications and articles without full-text availability were excluded.

#### Study Selection:

The initial search yielded 1046 articles. After removing duplicates and
conducting title and abstract screening, 31 articles were considered for full-
text review. The final selection was based on the relevance of the content to
the research question and the quality of the study design.

#### Data Extraction:

Data extraction was performed to gather key information from the selected
literature, including future work recommendations. This process involved
summarizing findings, identifying common themes, and noting variations across
studies.

#### Quality Assessment:

Given the narrative nature of this review, a formal quality assessment was not
conducted. However, efforts were made to critically appraised each study’s
methodology, sample size, and key findings to gauge the overall reliability of
the evidence.

#### Synthesis:

The synthesis involved organizing the selected literature into thematic
categories and identifying overarching trends and patterns, and employed a
narrative approach to present the key findings and critically discuss the
implications of the literature on the research question.

##  3 Background

This section provides the narrative review of the literature, with study
limitations and comparisions among related studies.

###  3.1 Communication Enhancement Technologies for DHH Individuals

The exploration of communication enhancement technologies for DHH individuals
has seen significant strides, particularly with the non-traditional
applications of Live Transcribe, an automatic speech recognition (ASR)
application [3]. This tool proves invaluable in various scenarios, from
technical support and communication with colleagues to note-taking during
meetings, especially during the prevalence of online meetings amid COVID-19
lockdowns. The adaptability of Live Transcribe for immobile DHH individuals by
mounting the tablet on a boom mic stand is a notable advancement [4],
emphasizing its pivotal role in promoting inclusivity and accessibility for
this demographic.

Closed captioning is a technology that has seen minimal innovation since its
inception in the 1970s, however recent studies are shedding light on potential
enhancements. One such is the animated text in captions [5] which offers a
promising avenue, providing improved access to emotive information often
overlooked in traditional captions, such as music, sound effects, and
intonation. This innovation, preferred by both hard of hearing and hearing
participants, emphasizes the need to bridge the gap in conveying nonverbal
nuances. This study and other related works was inspired by [6] which delved
into the impact of captions on the affective reactions of hearing-impaired
children to television programming, revealing the potential of captions in
enhancing emotional perception.

###  3.2 Innovations in Captioning for Enhanced Emotional Communication

Shifting focus to innovations in captioning for enhanced emotional
communication, Rashid’s framework [7] introduces the concept of using
animation and standard properties to express basic emotions. This framework,
associating emotions with animation properties, establishes a consistent
method for applying animation to text captions. Similarly, to enhance emotions
in captions [8] incorporating graphics, color, and animation to illustrate
sound and emotive information in television and film, a user study compared
viewer reactions to video samples with emotive captions against conventional
captioning, showcasing positive responses, particularly among hard of hearing
viewers.

In the realm of Voice User Interfaces (VUI), Hu’s paper [9] highlights the
lack of emotional information in STT by proposing an emojilization tool. This
tool automatically attaches emojis to generated text, compensating for
emotional loss in the conversion process. The pilot study indicates that
emojilized text enhances perceived emotions compared to plain text.

We can also understand how to represent emotions in captions by analyzing how
emotions are expressed in text. A study along this direction is the
representation of emotions in text-based messaging applications [10] through
visualizations in facial expression recognition in WhatsApp Web, and findings
underscore users’ preference for maintaining control over their emotions in
private settings, emphasizing the need for exclusive presentation and sharing
of emotions in certain contexts. An important application of emotional
expressions in text and imagery on social media during the COVID-19 pandemic
[11] adds a temporal dimension to the narrative, shedding light on how
emotions are expressed in different modalities during challenging times.

###  3.3 Emotion Recognition in AR and VR

VR has gained attention in various fields due to its unique advantages in
manipulating perceived scenarios and providing controlled experiences [12]. It
has been found that VR can generate emotional reactions and experiences in
users, as demonstrated by studies on 360 degree videos and VR educational
tools [13]. In fact, emotional responses in VR games have been observed to be
more intense compared to desktop games, both psychologically and
physiologically [14]. This suggests that VR has the potential to enhance
positive emotions, and it is worth to study if similar outcome can be
experienced from STT in an AR or VR environment.

More studies have discussed advancements in AR technologies for DHH
individuals, particularly in the classroom setting [15]. The AR visual-
captions system, implemented using Unity, ARKit, and AR Foundation, aims to
provide real-time STT and visual elements around the teacher, creating an
immersive learning experience. The study not only introduced the prototype but
also outlined future research directions, especially on refining the design,
conducting user studies, and extending the system’s capabilities for group
conversations. Similarly, Real-time augmented reality visual-captions [16] for
DHH children in classrooms builds on the AR-based system, providing a real-
time solution for STT and keyword extraction, addressing the challenges faced
by DHH children in mainstream education.

In higher education application, Pirker’s exploration into the potential of VR
for computer science education [17] sheds light on the positive impact of VR
environments on learning outcomes. The user study reveals higher engagement,
immersion, and positive emotions in the VR application compared to a web-based
alternative, emphasizing VR’s potential to enhance computer science education.
Given the importance of AR and VR in education training, making the virtual
environment convey emotional expression is essential. Chen’s paper [18]
focuses on facial expression recognition within immersive environments. The
proposed solution involves collecting real facial expression data using an
infrared camera and light source, showcasing promising results for
understanding human emotions in VR contexts.

###  3.4 Speech Emotion Recognition: Techniques and Challenges

An important technique found in the literature is the use of color coding and
visualization of emotion in speech, which offer a unique perspective on
conveying emotional information. [19] utilizes haptic feedback vests and
immersive virtual reality to quantify human emotion, providing insights for
the development of emotionally relevant input devices. The use of bubble
coloring in chat platforms [20, 21] explores the visual representation of
speech emotion, considering color alterations and participant feedback for
conveying specific emotions effectively. Other techniques in machine learning
are being employed to include emotional recognition in text. One is Human-
robot interaction and perception of emotions [22], demonstrating the
effectiveness of animated text associated with emotional information in
enhancing user interactions. The system’s potential applications in healthcare
and education underscore its versatile impact.

Further, Schiano’s paper [23] delves into the perception of facial affect,
bridging the gap between human facial expressions and prototype robot faces.
The study informs the design of affective robot faces, contributing valuable
insights into how emotions can be effectively communicated through facial
expressions.In a related study,the analysis of humanoid avatar representations
and emotions [24] delves into the uncanniness factor, highlighting the
importance of selecting appropriate avatar types for accurate expression
communication. The study’s findings support the need for avatar
representations that effectively convey emotional cues in interactive systems.
Finally, gestures can also be generated directly from speech using GANs [25]
thereby opening up new possibilities in human-computer interaction. The user
study using Turing test-inspired evaluation demonstrates the potential of the
proposed technique in creating realistic gestures from speech. In conclusion,
the narrative literature review traverses a spectrum of communication
enhancement technologies, showcasing the evolution of tools and techniques to
address the unique needs of deaf and hard of hearing individuals. From
innovations in live transcription and closed captioning to advancements in
augmented reality, emotive captioning, and speech emotion recognition, the
literature reflects a dynamic landscape of research and development. The
exploration of humanoid avatars, voice user interfaces, virtual reality,
human-robot interaction, and the color coding of speech emotions further
emphasizes the interdisciplinary nature of this field. The review also
underscores the significance of understanding emotional expressions in text,
images, and gestures, offering a holistic perspective on communication
technologies that cater to diverse sensory and cognitive needs. As we navigate
the intricacies of enhancing communication for individuals with hearing
impairments, these technological advancements pave the way for a more
inclusive and accessible future.

###  3.5 Empathic Machine

Emotions serve as an implicit communication channel among humans, conveyed
through spoken words, facial expressions, behavior, and physiological
responses. This empathic form of communication enables individuals to
recognize cues and respond empathetically. Despite computing devices excelling
in understanding user context, there are ongoing challenges in the scientific
community to create emotion-sensing and empathic applications for human-
computer interaction [26]. In this section we look at the literature
contributions in tackling these challenges and support empathic ability in
machines.

#### Enhancing Empathic Abilities with Chatbots:

The evolution of chatbots began with the creation of ELIZA in 1966 [27], and
today they are commonly used in customer service but have limitations in
showing empathy. Empathy, defined as understanding and sharing another’s
feelings, is deemed challenging for conversational agents. Recent research,
however, indicates the feasibility of generating empathic responses in
chatbots, particularly in customer service contexts, including in the
healthcare, especially in providing physical health diagnosis through short
text conversations.

Several studies available in the literature to compensate for the shortcomings
of lack of empathic expressions in computer and AI employed the use of
chatbots to convey emptional expressions. One study used chatbots to mediate
social presence and trust in consumer emotions [28], creating a sense of
social presence using emoticons, appropriate language styles, and other cues
that contribute to a more authentic and human-like interaction. Another study
used empathic chatbot to understand users’ emotional states and generate
responses that convey understanding and addressing challenges in handling
multi-layered, context-sensitive, and implicitly expressed emotions in text
[29]. It utilized advanced models and tools involving a benchmark bot and an
empathic bot. To enhance empathic capabilities, they fine-tuned the language
model on empathic conversations, incorporating the user’s emotional state into
the input. The findings from the empathic bot include the effectiveness of
transformer-based language models, the positive impact of training on empathic
conversations, and the influence of emotional valence on perceived empathy.

Similarly, chatbot application is finding its way into gender voice
discrimination. [30] explores the tension between designing empathic agents
and the gender assignment of chatbots and how they can relate to the design of
the metaphor of the chatbots in conversational agents (Cas). Computers lack
genuine emotions, however CAs can simulate and trigger empathy by adhering to
human-social rules during interactions. This simulation involves techniques
like sentiment analysis, emotion detection, and mimicking emotions to enhance
user engagement. Notably, the paper explores whether CAs, specifically those
perceived as having feminine qualities, evoke more empathy in human-chatbot
interactions compared to other gender perceptions.

#### Digitizing Human Emotions:

As computing machines and AI need to be programmed to work, inculcating
empathy into machines is through digitizing human emotions. One study used
non-invasive techniques like electroencephalography (EEG), specifically the
Emotive Epoc headset to digitize human emotions [31] and the primary goal was
to conduct a proof-of-concept experiment enabling a humanoid robot’s control
through digitized emotions, emphasizing potential applications in healthcare .
Their contribution lies in adapting mature image recognition tools from
Artificial Neural Networks (ANNs) for emotion recognition, and the resulting
Brain-Computer Interface (BCI) system is designed to enable the robot to
emulate empathy and interact with subjects based on predefined behavioral
models. Similarly, [32] explores the significance of recognizing emotions in
face-to-face conversations and discussed the limitations of traditional cues
like facial expressions and gestures, emphasizing the potential of analyzing
brain activity and physiological signals for more reliable emotion
recognition, especially in situations where emotions may be concealed. The
study presents an experimental setup for inducing spontaneous emotions in
conversations and creating a dataset incorporating EEG, Photoplethysmography
(PPG), and Galvanic Skin Response (GSR) signals. The researchers developed an
intelligent user interface for video conferencing and systems with
conversational digital humans capable of recognizing and responding to
emotions, using PEGCONV) dataset to train these systems to enhance their
ability to understand and appropriately respond to human behavior.

Further, Robots can also respond to user emotional states. [33] enhanced
human-robot interactions (HRI) by proposing an Automatic Cognitive Empathy
Model (ACEM) for humanoid robots. The goal was to achieve more extended and
engaging interactions by having robots respond appropriately to users’
emotional states using ACEM model to continuously detects users’ affective
states based on facial expressions, utilizing a stacked autoencoder network
trained on the RAVDESS dataset. The model generates empathic behaviors adapted
to users’ personalities, either in parallel or as reactive responses.

#### Empathic Systems in AR/VR:

Empathic machine is also finding applications into wearable. In exploring the
use of smart glasses equipped with an emotion recognition system to enhance
human-to-human communication, with a focus on doctors and autistic adults
[34], this study identifies the potential benefits for doctors in improving
patient-doctor communication and for autistic adults facing challenges in
adhering to neurotypical communication norms. The proposed system combines
emotion recognition, AI, and smart glasses to provide real-time feedback on
the emotional state of conversation partners. User evaluations indicate
positive responses from both doctors and autistic adults, highlighting the
potential for improving empathy and communication. Design considerations
include customizable output preferences and addressing privacy and social
impact concerns. The authors emphasize the system’s potential for educational
purposes and ongoing development in the context of social impact and user
experience. In a related study, [35] explores the use of virtual reality (VR)
with wearable physiological sensors to examine how recalling autobiographical
memories (AM) with emotional content affects an individual’s physiological
state. By replicating the Autobiographical Memory Test (AMT) in VR,
participants were presented with positive, negative, and neutral words to
trigger memory recall. The study observed a positive influence of AM recall on
electrodermal activity (EDA) peak amplitude, EDA peak number, and pupil
diameter compared to situations without recall. However, emotional AM recall
did not produce a significant impact. The article concludes by discussing the
potential and limitations of utilizing autobiographical memory to enhance
personalized mobile VR experiences in conjunction with physiological sensors.
The findings reveal a significant effect of AM recall on EDA mean peak
amplitude, EDA peak number, and pupil diameter compared to a no-recall
condition. However, no significant differences were identified in emotional AM
recall (positive, negative, neutral).

##  4 Discussion

The purpose of this review is to explore existing literature on emotional
expression in STT interfaces, with a specific focus on psychophysiology in VR.
As mentioned earlier in the introduction, this review aims to identify
advancements, limitations, and research gaps in the incorporation of emotional
expression in transcribed text generated by STT systems. The overarching goal
is to inspire further research in the field.

The review was conducted to address a critical aspect overlooked in
traditional language transcription services—emotional expression. While STT
systems enhance accessibility for the DHH, the loss of emotional nuance in
transcribed text poses a communication challenge. The review seeks to analyze
and synthesize available literature to understand the current state of
research, advancements, and gaps in reintegrating emotional expression into
transcribed text.

Through an extensive literature review spanning databases such as ACM Digital
Library, IEEE Explore, Scopus, and Google Scholar, the study identified 31
relevant papers. The review explored communication enhancement technologies,
innovations in captioning, emotion recognition in AR and VR, and empathic
machines. The review successfully organized the literature into thematic
categories, identified trends, and presented key findings. It critically
discussed the implications of the literature on the research objective. The
limited relevant studies found highlights the need for more research in this
area. Also, of the relevant papers found, there is no study that addressed
incorporating emotional expression in STT in an AR or VR environment, thus
further supporting more research in this area.

The main findings of the narrative review highlight the evolution of tools and
techniques aimed at addressing the unique needs of individuals with hearing
impairments and other users of language transcription text. This includes
innovations in live transcription, closed captioning, AR, VR, and emotion
recognition technologies. The review underscores the interdisciplinary nature
of the field, emphasizing the significance of understanding emotional
expressions in diverse modalities such as text, images, and gestures.

However, the notable absence of emotional expression in these applications, as
highlighted by [9], adds a layer of complexity to communication. Despite their
commendable impact on improving accessibility, the lack of emotional nuance in
transcribed text [6] limits the richness of communication. The results
emphasize the urgency for innovations in STT technology to capture emotional
expressions from various sources, including music, sound effects, and speech
emphasis [5].

The exploration of strategies to integrate emotional expression into text,
such as Rashid’s framework [7, 5], signifies a shift beyond traditional
captioning methods. Animated text captions, enriched with graphics and color,
not only show positive responses among hard of hearing viewers [8] but also
present a potential avenue for more inclusive and emotionally expressive
communication.

The extension of these efforts into Augmented Reality (AR) and Virtual Reality
(VR) environments, where technologies offer unique advantages [12], opens new
possibilities for immersive and emotionally resonant experiences. While
studies dedicated to enhancing STT captions in AR and VR environments are
promising [16, 15], the results underscores the need for continued research to
infuse emotional expression into these advanced platforms, particularly in
educational training contexts [18, 17].

The implementation of the emojization tool [9], despite its limitations in
covering a sufficient vocabulary, demonstrates a tangible effort to restore
emotion to text in the STT transcription process. This example serves as a
valuable step toward addressing the emotional gap in communication
technologies, particularly within the AR/VR environment.

Furthermore, the application of emotional recognition through machine learning
and robotics in healthcare and education [21, 22, 23, 24] highlights the
broader impact of these results. The effective communication of emotions in
interactive systems has implications for personalized and empathetic
interactions, potentially improving patient care and educational experiences.

The digitization of human emotions, as demonstrated in studies such as Roshdy
et al. [31], has direct implications for healthcare applications. The
development of Brain-Computer Interface (BCI) systems capable of interpreting
and responding to human emotions opens new possibilities for enhancing patient
care and interaction with humanoid robots, suggesting a potential revolution
in healthcare practices.

The exploration of models like the Automatic Cognitive Empathy Model (ACEM)
for humanoid robots [33] contributes to more extended and engaging human-robot
interactions. This has implications for various fields, including customer
service, education, and entertainment, where robots can adapt their behavior
based on users’ emotional states, leading to more personalized and effective
interactions.

Wearable technologies, such as smart glasses with emotion recognition systems
[34], have the potential to significantly improve communication in specific
contexts, such as doctor-patient interactions and communication challenges
faced by autistic adults. The real-time feedback on emotional states can lead
to more empathetic and tailored communication, addressing specific needs in
these domains.

The discussed systems, particularly those involving wearable technologies and
intelligent interfaces, have implications for educational settings. The
potential for improving empathy and communication, as well as considerations
for customizable output preferences, privacy, and social impact, suggests that
these technologies could play a role in educational environments, fostering
better communication and understanding.

The exploration of virtual reality (VR) experiences with wearable
physiological sensors [35] provides insights into the potential and
limitations of leveraging autobiographical memory to enhance personalized
mobile VR experiences. This understanding is valuable for developers and
researchers aiming to create immersive and emotionally resonant virtual
experiences.

The multidisciplinary nature of these studies, incorporating fields such as
neuroscience, artificial intelligence, and psychology, highlights the
potential for collaborative and interdisciplinary research. This can lead to
the development of more holistic and effective empathic systems that consider
both technological and human factors.

The significance of this research lies in its contribution to advancing
communication technologies for individuals with hearing impairments. By
exploring emotional expression in STT interfaces, the study sheds light on the
evolving landscape of tools and techniques. The research emphasizes the
importance of inclusivity and accessibility in communication technologies,
fostering a more empathetic and inclusive future.

###  4.1 Limitations

The limitations of this study were: (i) it was a rapid review whose purpose is
to survey existing work in emotional expressions, find existing gaps and
inspire further research. (ii) the narrative review did not follow a strict
rule expected of systematic literature review, hence could have missed some
relevant papers. However, the papers that were found eligible and used in this
review are the most relevant. (iii) the review is limited to peer-reviewed
papers that were in English.

###  4.2 Gaps, Challenges and Opportunities

  1. 1.

Innovations in emotive captioning and speech emotion recognition warrant
further exploration, particularly in educational setting where DHH students,
including other categories of students rely on STT as their main method of
communication during lectures.

  2. 2.

Explore the integration of cognitive empathy and improve emotion detection in
short conversations. The study’s promising results underscore the potential
for empathy-driven enhancements in medical chatbots [27].

  3. 3.

Prioritize the development of an automatic metric for assessing empathy in
text-based messages, eliminating reliance on human judges and enhancing result
comparability. Refining the definition of empathy to encompass additional
aspects like nationality and educational level is another crucial step. 2.
Further research should incorporate finer emotion categories and measure
emotion intensity. Improving the dialogue system’s adaptability to learn,
unlearn, and re-learn from user interactions, monitoring emotional states and
personal information, is essential. 3. Additionally, addressing ethical
concerns related to enhanced empathy and human-like qualities in chatbots,
including guidelines on handling user information and drawing conclusions,
should be a focus of future investigations [29].

  4. 4.

Explore alternative markers for empathic responses, involve different
character styles and modes of realism, and incorporate physiological or
behavioral measures of empathy. 2. Additionally, replicating the study in a
virtual reality context was suggested to further understand the role of
immersive environments in influencing empathic experiences with virtual
characters [36].

  5. 5.

Future work to incorporate video and audio data with EEG and physiological
signals for a comprehensive approach to emotion recognition in conversations.
The methods and dataset presented are anticipated to benefit various
intelligent user interface applications [32].

  6. 6.

Explore the system’s potential applications in educational settings,
particularly for training and educating medical professionals. Assess how the
system could be integrated into medical education curricula [34].

  7. 7.

Future research with a larger participant pool, the inclusion of higher
emotional intensity words, and exploration of additional physiological signals
like EEG and heart rate variability (HRV). The study’s implications extend to
designing text-based VR interfaces, passive learning, exposure treatment, and
personalized VR gaming experiences [35].

###  4.3 Next Steps

This study has revealed that research in this domain is greatly limited, and
this is evident in the few papers deemed eligible for inclusion (only 31
papers). Based on the available literature reviewed, there is limited study on
the subject of incorporating emotional expression in STT in AR/VR for use in
educational setting purposes. Therefore, it is necessary to conduct further
research to fill the identified gap. To do this, my next step will involve
working with the following research questions as my Ph.D. research work:

  1. 1.

Can the integration of transcribed text in an Augmented Reality with
corresponding visual elements such as pictures, emojis, and avatars enhance
the conveyance of emotional expression in speech-to-text conversion?

Strategy: Subsequently, conduct a user study to assess the effectiveness of
this combined approach.

  2. 2.

What is the optimal channel and format for delivering speech-to-text
transcriptions while preserving the speaker’s facial expressions?

Strategy: Investigate different text formatting options, including presenting
a few words per line, highlighting emphasized words, and summarizing content
while maintaining eye contact, to determine the most effective approach.

  3. 3.

How can the user experience in speech-to-text technology be enhanced to
minimize cognitive load for both the speaker and the user?

Strategy: Explore strategies and design elements that contribute to an
improved user experience in speech-to-text interactions.

  4. 4.

Can the substitution of emotionally recognizable words with equivalent emojis
and avatars, introduced dynamically in a live transcription, effectively
preserve and convey emotional expression?

Strategy: Examine the impact of replacing words with visual elements on
emotional expression in speech-to-text, particularly in a live transcription
setting.

##  5 Conclusion

In conclusion, the narrative literature review traverses various communication
enhancement technologies, showcasing advancements in tools and techniques.
From live transcription and closed captioning to innovations in AR, emotive
captioning, emotion recognition, and empathic machines, the review provides a
holistic perspective on addressing the diverse sensory and cognitive needs of
individuals with hearing impairments. The study underscores the ongoing
challenges and potential avenues for future research in creating empathic
applications for human-computer interaction and enhancing communication for
individuals with hearing impairments.

This narrative review explores communication enhancement technologies for DHH
individuals, including anyone who relies on language transcription services to
help with communication, with a focus on the psychophysiology of STT
interfaces in VR. The review highlights advancements in STT and closed
captioning, emphasizing their relevance in enhancing communication
accessibility, emphasizing how emotional expression is lost in the process of
transcribing speech into text.

The results from selected studies matter as they not only contribute to the
ongoing efforts to make communication technologies more inclusive for the DHH
community but also pave the way for emotionally rich interactions in various
contexts, from virtual environments to healthcare and education. The cited
studies collectively suggest a growing recognition of the importance of
emotional expression in human-computer interactions and the ongoing pursuit of
innovative solutions to address this crucial aspect.

The integration of empathy into machines has the potential to redefine how we
interact with technology, making these interactions more intuitive, adaptive,
and emotionally resonant. However, ethical considerations, user preferences,
and ongoing research are crucial in shaping the responsible and effective
deployment of such technologies.

The review has inspired future research recommendations based on the key
findings uncovered from the study. In conclusion, this synthesis provides a
comprehensive overview of the evolution of communication enhancement
technologies, spanning from live transcription and closed captioning to
innovations in AR, VR, and speech emotion recognition. The interdisciplinary
nature of the field is underscored, emphasizing the need for continued
research and development to create more inclusive and accessible communication
technologies for individuals with hearing impairments and the general language
transcription user. As we navigate this dynamic landscape, the synthesis
serves as a guide for future research directions and the development of
impactful technologies that cater to diverse sensory and cognitive needs.

## References

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# An elliptic problem in dimension $N$ with a varying drift term bounded in
$L^{N}$.

Juan Casado-Díaz Dpto. de Ecuaciones Diferenciales y Análisis Numérico
Universidad de Sevilla<EMAIL_ADDRESS>

###### Abstract

The present paper is devoted to study the asymptotic behavior of a sequence of
linear elliptic equations with a varying drift term, whose coefficients are
just bounded in $L^{N}(\Omega)$, with $N$ the dimension of the space. It is
known that there exists a unique solution for each of these problems in the
Sobolev space $H^{1}_{0}(\Omega)$. However, because the operators are not
coercive, there is no uniform estimate of the solutions in this space. We use
some estimates in [2], and a regularization obtained by adding a small
nonlinear first order term, to pass to the limit in these problems.

Keywords: asymptotic behavior, elliptic problem, drift term, varying
coefficients.

Mathematics Subject Classification: 35B27, 35B30.

##  1 Introduction.

For a bounded open set $\Omega\subset\mathbb{R}^{N}$, we are interested in
passing to the limit in a sequence on elliptic equations with a varying drift
term, whose coefficients are just bounded in $L^{N}(\Omega)^{N}$,
($L^{p}(\Omega)^{N}$, $p>2$ if $N=2$). The problem is written as

$\left\\{\begin{array}[]{l}\displaystyle-{\rm div}(A\nabla
u_{n}+E_{n}u_{n})=f_{n}\ \hbox{ in }\Omega\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle u_{n}\in H^{1}_{0}(\Omega),\end{array}\right.$ (1.1)

with $A\in L^{\infty}(\Omega)^{N\times N}$ satisfying the usual uniform
ellipticity condition, and $f_{n}$ bounded in $H^{-1}(\Omega)$. Thanks to
Sobolev’s inequality, the integrability assumption on $E_{n}$ is the weaker
one to get the first order term well defined in $H^{-1}(\Omega)$. It is known
that for every $n\in\mathbb{N}$ problem (1.1) has a unique solution ([3], [4],
[15]), however, the problem is known to be not coercive and thus, there is no
estimate for $u_{n}$ in $H^{1}_{0}(\Omega)$. We recall some of these results
in Section 2.

L. Boccardo found in [2] an estimate for $\ln(1+|u_{n}|)$ in
$H^{1}_{0}(\Omega)$ depending only on $\|f_{n}\|_{H^{-1}(\Omega)}$ and
$\|E_{n}\|_{L^{2}(\Omega)^{N}}$. As a consequence of this result we get that
the measure of the sets $\\{|u_{n}|>k\\}$ tends to zero when $k$ tends to
infinity uniformly in $n$. Thanks to this result L. Boccardo proved in [5]
that $f_{n}$ converging weakly to some $f$ in $H^{-1}(\Omega)$, $E_{n}$
converging weakly to $E_{0}$ in $L^{N}(\Omega)$ and $|E_{n}|^{N}$ equi-
integrable, imply that the solution $u_{n}$ of (1.1) converges weakly in
$H^{1}_{0}(\Omega)$ to the solution $u_{0}$ of

$\left\\{\begin{array}[]{l}\displaystyle-{\rm div}(A\nabla
u_{0}+E_{0}u_{0})=f\ \hbox{ in }\Omega\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle u_{0}\in H^{1}_{0}(\Omega).\end{array}\right.$

In the case where $|E_{n}|^{N}$ is not equi-integrable, we do not have an
estimate for $u_{n}$ in $H^{1}_{0}(\Omega)$ and then the proof is more
involved. The ideas in [2] imply (see [5]) that $\ln^{q}(1+|u_{n}|)$ is
bounded in $H^{1}_{0}(\Omega)$ for any $q>1$. Using also the approximation of
$u_{n}$ given by the solution of a perturbation of (1.1) by a nonlinear zero
order term (see (3.12) below), we still manage to pass to the limit in (1.1)
but, instead of the weak convergence in $H^{1}_{0}(\Omega)$, we only get

$\ln^{q}(1+|u_{n}|)\,{\rm sgn}(u_{n})\rightharpoonup\ln^{q}(1+|u_{0}|)\,{\rm
sgn}(u_{0})\ \hbox{ in }H^{1}_{0}(\Omega),\quad\forall\,q\geq 1.$

This is the main result of the paper, which is proved in Theorem 3.1.

Another problem related to (1.1) is given by its adjoint formulation

$\left\\{\begin{array}[]{l}\displaystyle-{\rm div}(A^{T}\nabla
u_{n})+E_{n}\cdot\nabla u_{n}=f_{n}\ \hbox{ in }\Omega\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr\displaystyle u_{n}\in
H^{1}_{0}(\Omega),\end{array}\right.$ (1.2)

Some results about the asymptotic behavior of the solutions of this problem
have been obtained in [5]. Namely, assuming the equi-integrability condition
on $|E_{n}|$ and reasoning by duality, we deduce from the results stated above
for problem (1.1) that the solutions of (1.2) are bounded in
$H^{1}_{0}(\Omega)$. Moreover, the results in [9] prove that $u_{n}$ is
compact in $W^{1,q}_{0}(\Omega)$ for $1\leq q<2$. Then, it is immediate to
pass to the limit in this problem. In the case where $|E_{n}|^{N}$ is not
equi-integrable, we do not have any estimate for $u_{n}$ and then we are not
able to passing to the limit in (1.2). However, adding a sequence of zero
order terms $a_{n}u_{n}$ with $a_{n}$ satisfying

$a_{n}\geq\gamma\ \hbox{ a.e. in }\Omega,$

for some $\gamma>0$, and assuming $f_{n}$ bounded in $L^{\infty}(\Omega)$, it
is proved in [5] that $u_{n}$ is bounded in $H^{1}_{0}(\Omega)\cap
L^{\infty}(\Omega)$. This allows us to pass to the limit in (1.2).

The homogenization of a sequence of elliptic PDE’s with a singular term of
first order has also been carried out in other papers. In this sense we refer
to [6] where it is considered problem (1.1) with the matrix function $A$ also
depending on $n$, the sequence $E_{n}$ converging weakly in
$L^{2}(\Omega)^{N}$, and ${\rm div}\,E_{n}$ converging strongly in
$H^{-1}(\Omega)$. The definition of solution in this case is related to the
definition of entropy or renormalized solution (see e.g. [1], [8], [14]). In
[10] and [13] it is considered the case of a first order term of the form

$E_{n}\nabla u_{n}+{\rm div}(E_{n}u_{n}).$

Since this term is skew-symmetric we can obtain an estimate in
$H^{1}_{0}(\Omega)$ which is independent of $E_{n}$ (we refer to [7] for a
related existence result). Assuming $E_{n}$ just bounded in
$L^{2}(\Omega)^{N}$ it is proved that the limit problem contains a new term of
zero order. This is related to the results obtained in [16] for the Stokes
equation with an oscillating Coriolis force. We also refer to [11], [12] for
related results in the case of the evolutive elastic system submitted to an
oscillating magnetic field. Now, the limit problem is nonlocal in general.

##  2 Some reminders about elliptic problems with a drift or convection term.

For a bounded open set $\Omega\subset\mathbb{R}^{N}$, $N\geq 2$, a matrix
function $A\in L^{\infty}(\Omega)^{N\times N}$, such that there exists
$\alpha>0$ satisfying

$A(x)\xi\cdot\xi\geq\alpha|\xi|^{2},\quad\forall\,\xi\in\mathbb{R}^{N},\
\hbox{ a.e. }x\in\Omega,$ (2.1)

two measurable functions $E:\Omega\to\mathbb{R}^{N}$, $a:\Omega\to\mathbb{R}$,
$a\geq 0$ a.e. in $\Omega$, and a distribution $f\in H^{-1}(\Omega)$, we
recall in this section, some results about the existence and uniqueness of
solution for problems

$\left\\{\begin{array}[]{l}{\cal A}u=f\ \hbox{ in }\Omega\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr\displaystyle u\in
H^{1}_{0}(\Omega),\end{array}\right.\qquad\left\\{\begin{array}[]{l}{\cal
A}^{\ast}u=f\ \hbox{ in }\Omega\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle u\in H^{1}_{0}(\Omega),\end{array}\right.$ (2.2)

with

${\cal A}u=-{\rm div}(A\nabla u+Eu)+au,\qquad{\cal A}^{\ast}u=-{\rm
div}(A^{T}\nabla u)+E\cdot\nabla u+au,$

Observe that due to Sobolev imbedding theorem, in order to have the terms
${\rm div}(Eu)$, $E\cdot\nabla u$ and $au$ in $H^{-1}(\Omega)$, when $u$ is in
$H^{1}_{0}(\Omega)$, we need to assume

$\left\\{\begin{array}[]{cc}E\in L^{N}(\Omega)^{N}&\hbox{ if }N>2\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle E\in L^{p}(\Omega)^{N},\
p>2&\hbox{ if }N=2,\end{array}\right.\qquad\left\\{\begin{array}[]{cc}a\in
L^{N\over 2}(\Omega)&\hbox{ if }N>2\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle a\in L^{q}(\Omega)^{N},\ q>1&\hbox{ if
}N=2,\end{array}\right.$ (2.3)

In this case, ${\cal A}$ and ${\cal A}^{\ast}$ are continuous linear operators
from $H^{1}_{0}(\Omega)$ into $H^{-1}(\Omega)$ and ${\cal A}^{\ast}$ is the
adjoint operator of ${\cal A}$.

Assuming the further assumption

$E\in L^{p}(\Omega)^{N},\quad p>N,$ (2.4)

the compact imbedding of $H^{1}_{0}(\Omega)$ into $L^{2p\over p-2}(\Omega)$
proves that for every $\varepsilon>0$, there exists $C_{\varepsilon}>0$ such
that

$\|u\|_{L^{2p\over p-2}(\Omega)}\leq\alpha\varepsilon\|\nabla
u\|_{L^{2}(\Omega)^{N}}+C_{\varepsilon}\|u\|_{L^{2}(\Omega)},\quad\forall\,u\in
H^{1}_{0}(\Omega).$ (2.5)

Thus, Hölder’s and Young’s inequalities imply

$\begin{array}[]{l}\displaystyle\langle{\cal
A}u,u\rangle_{H^{-1}(\Omega),H^{1}_{0}(\Omega)}=\langle{\cal
A}^{\ast},u\rangle_{H^{-1}(\Omega),H^{1}_{0}(\Omega)}=\int_{\Omega}\big{(}A\nabla
u\cdot\nabla u\,dx+uE\cdot\nabla u+au^{2}\big{)}dx\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr\displaystyle\geq\alpha\|\nabla
u\|_{L^{2}(\Omega)^{N}}^{2}-\|E\|_{L^{p}(\Omega)^{N}}\|\nabla
u\|_{L^{2}(\Omega)^{N}}\|u\|_{L^{2p\over p-2}(\Omega)}\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr\displaystyle\geq\alpha\Big{(}{1\over
2}-\varepsilon\|E\|_{L^{p}(\Omega)^{N}}\Big{)}\|\nabla
u\|^{2}_{L^{2}(\Omega)^{N}}-{C_{\varepsilon}^{2}\|E\|_{L^{p}(\Omega)^{N}}^{2}\over
2\alpha}\|u\|^{2}_{L^{2}(\Omega)}.\end{array}$

Therefore, taking $\varepsilon\|E\|_{L^{p}(\Omega)^{N}}<1/2$, we deduce from
Lax-Milgram theorem that replacing $a$ by $a+\gamma$, with

$\gamma\geq{C_{\varepsilon}^{2}\|E\|_{L^{p}(\Omega)^{N}}^{2}\over 2\alpha},$

there exists a unique solution for both problems in (2.2). The compactness of
$({\cal A}+\gamma I)^{-1}$, considered as an operator in $L^{2}(\Omega)$,
allows then to use Fredholm theory to deduce that the existence and uniqueness
of solutions for both problems in (2.2) and every $f\in H^{-1}(\Omega)$ is
equivalent to the uniqueness of solutions for one of them.

In [15], Theorem 8.1, it is proved that problem

$\left\\{\begin{array}[]{l}{\cal A}^{\ast}u=f\ \hbox{ in }\Omega\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle u-\phi\in
H^{1}_{0}(\Omega),\end{array}\right.$

with $\phi\in H^{1}(\Omega)$ satisfies the weak maximum principle so that the
second problem in (2.2) has at most one solution (in [15] it is assumed $E\in
L^{\infty}(\Omega)^{N}$, but it is immediate to check that the same proof
works for $E$ just satisfying (2.3)).

We observe however that although the above result implies the existence and
continuity of the operators ${\cal A}^{-1}$ and $({\cal A}^{\ast})^{-1}$, it
does not provide any estimate for the norm of these operators and then on the
solutions of both problems in (2.2).

If we assume $N>2$ and $E\in L^{N}(\Omega)^{N}$, the above reasoning fails
because (2.5) with $p=N$ does not hold in general. The existence of solutions
in this case can be obtained from the following result due to L. Boccardo
([2], [5]). We recall that the truncate function at height $k>0$ is defined as

$T_{k}(s)=\left\\{\begin{array}[]{cl}-k&\hbox{ if }s<-k\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr\displaystyle s&\hbox{ if }-k\leq s\leq k\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr\displaystyle k&\hbox{ if }s>k.\end{array}\right.$
(2.6)

######  Theorem 2.1.

For every $f\in H^{-1}(\Omega)$, $E\in L^{2}(\Omega)^{N}$, and $a\in
L^{1}(\Omega)$, $a\geq 0$ a.e. in $\Omega$, there exists an entropy solution
of

$\left\\{\begin{array}[]{l}\displaystyle{\cal A}u=f\ \hbox{ in }\Omega\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle u=0\ \hbox{ on
}\partial\Omega,\end{array}\right.$ (2.7)

in the following sense

$\left\\{\begin{array}[]{l}\displaystyle T_{k}(u)\in H^{1}_{0}(\Omega),\
\forall\,k>0,\quad\log(1+|u|)\in H^{1}_{0}(\Omega)\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr\displaystyle\int_{\Omega}\Big{(}(A\nabla u+Eu)\cdot\nabla
T_{k}(u-\varphi)+auT_{k}(u-\varphi)\Big{)}\,dx\leq\langle
f,T_{k}(u-\varphi)\rangle_{H^{-1}(\Omega),H^{1}_{0}(\Omega)},\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr\displaystyle\forall\,\varphi\in
H^{1}_{0}(\Omega)\cap L^{\infty}(\Omega).\end{array}\right.$ (2.8)

Moreover, there exists $C>0$, independent of $f$ and $E$ such that

$\|\log(1+|u|)\|_{H^{1}_{0}(\Omega)}\leq
C\big{(}\|f\|_{H^{-1}(\Omega)}+\|E\|_{L^{2}(\Omega)^{N}}\big{)}.$ (2.9)

If there exists $\gamma>0$ such that

$a\geq\gamma\ \hbox{ a.e. in }\Omega,$ (2.10)

and $f$ belongs to $L^{1}(\Omega)$, such solution is also in $L^{1}(\Omega)$
and satisfies

$\|u\|_{L^{1}(\Omega)}\leq{1\over\gamma}\|f\|_{L^{1}(\Omega)}.$ (2.11)

Let us prove that the function $u$ given by the previous theorem is in fact a
distributional solution of the first problem in (2.2) when $E$ and $a$ satisfy
(2.3). This is given by the following theorem

######  Theorem 2.2.

Assume that the functions $E$ and $a$ in Theorem 2.1 satisfy (2.3), then the
solution $u$ of (2.7) is in $H^{1}_{0}(\Omega)$ and satisfies the elliptic
equation in the distributional sense. Moreover, if there exists $\gamma>0$
such that (2.10) holds, then this solution is unique.

Proof. For $k,m>0$, we take $\varphi=T_{m}(u)$ in (2.8). This gives

$\int_{\\{m<|u|<k+m\\}}\hskip-15.0pt\big{(}(A\nabla u+Eu)\cdot\nabla
u\big{)}dx+\int_{\Omega}au\,T_{k}(u-T_{m}(u))dx\leq\langle
f,T_{k}(u-T_{m}(u))\rangle_{H^{-1}(\Omega),H^{1}_{0}(\Omega)}.$

Using in this equality that

$\left|\int_{\\{m<|u|<k+m\\}}\hskip-15.0ptEu\cdot\nabla
u\,dx\right|\leq\int_{\\{m<|u|<k+m\\}}\hskip-15.0pt|E|\big{(}m+|T_{k}(u-T_{m}(u))|\big{)}|\nabla
u|dx,$

combined with (2.1), Sobolev’s imbedding theorem and Hölder’s inequality we
deduce the existence of $C_{S}>0$, depending only on $N$ (and $|\Omega|$ if
$N=2$), such that

$\begin{array}[]{l}\displaystyle\|T_{k}(u-T_{m}(u))\|^{2}_{H^{1}_{0}(\Omega)}\big{(}\alpha-
C_{S}\|E\|_{L^{r}(\\{m<|u|\\})^{N}}\big{)}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\leq\big{(}\|f\|_{H^{-1}(\Omega)}+mC_{S}\|E\|_{L^{r}(\Omega)^{N}}\big{)}\|T_{k}(u-T_{m}(u))\|_{H^{1}_{0}(\Omega)},\end{array}$
(2.12)

with $r=p$ if $N=2$, $r=N$ if $N>2$.

By (2.9), we can take $m$ sufficiently large to get

$C_{S}\|E\|_{L^{r}(\\{m<|u|\\})}^{N}<\alpha.$

Letting $m$ fixed satisfying this inequality and taking $k$ tending to
infinity, we deduce that $u-T_{m}(u)$ belongs to $H^{1}_{0}(\Omega)$. Since
$T_{m}(u)$ also belongs to $H^{1}_{0}(\Omega)$, then $u$ belongs to
$H^{1}_{0}(\Omega)$.

Once we know that $u$ belongs to $H^{1}_{0}(\Omega)$, we can take $k$ tending
to infinity in (2.8) to deduce

$\int_{\Omega}\Big{(}(A\nabla
u+Eu)\cdot\nabla(u-\varphi)+au(u-\varphi)\Big{)}\,dx\leq\langle
f,u-\varphi\rangle_{H^{-1}(\Omega),H^{1}_{0}(\Omega)},$

for every $\varphi\in H^{1}_{0}(\Omega)\cap L^{\infty}(\Omega).$ Since
$H^{1}_{0}(\Omega)\cap L^{\infty}(\Omega)$ is dense in $H^{1}_{0}(\Omega)$,
the result holds for $\varphi$ just in $H^{1}_{0}(\Omega)$. Replacing
$\varphi$ by $u+\varphi$ we deduce that $u$ is a distributional solution of
${\cal A}u=f$ in $\Omega$.

The fact that the renormalized solutions are distributional solutions for $E$
satisfying (2.3) allows us to reason by linearity to deduce that they are
unique if and only if the unique entropy solution of (2.7) with $f=0$ is the
null function. When (2.10) holds this follows from (2.11). $\Box$

Theorem 2.1 proves that the first problem in (2.2) always has at least a
solution and it is unique if $a$ satisfies (2.10). As above, this allows us to
use Fredholm theory to deduce

######  Corollary 2.3.

Assume that $E$ satisfies (2.3), then for every $f\in H^{-1}(\Omega)$ both
problems in (2.2) have a unique solution. Moreover this solution depends
continuously on $f$.

##  3 Passing to the limit with a varying drift term.

In this section, for a bounded open set $\Omega\subset\mathbb{R}^{N},$ $N\geq
2$, a sequence of vector measurable functions $E_{n}:\Omega\to\mathbb{R}^{N}$
such that there exists $E_{0}:\Omega\to\mathbb{R}^{N}$ satisfying

$\left\\{\begin{array}[]{cc}E_{n}\rightharpoonup E_{0}\ \hbox{ in
}L^{N}(\Omega)^{N}&\hbox{ if }N>2\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle E_{n}\rightharpoonup E_{0}\ \hbox{ in
}L^{p}(\Omega)^{N},\hbox{ for some }p>2&\hbox{ if }N=2,\end{array}\right.$
(3.1)

we are interested in the asymptotic behavior of the solutions of

$\left\\{\begin{array}[]{l}\displaystyle-{\rm div}(A\nabla
u_{n}+E_{n}u_{n})=f_{n}\ \hbox{ in }\Omega\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle u_{n}\in H^{1}_{0}(\Omega),\end{array}\right.$ (3.2)

where $A\in L^{\infty}(\Omega)^{N\times N}$ satisfies (2.1) and $f_{n}\in
H^{-1}(\Omega)$ is such that there exists $f\in H^{-1}(\Omega)$ satisfying

$f_{n}\rightharpoonup f\ \hbox{ in }H^{-1}(\Omega).$ (3.3)

As we recalled in the previous section, problem (3.2) has a unique solution
but we do not know if it is bounded in $H^{1}_{0}(\Omega)$. Using the
estimates for the solutions of this problem given in [2] let us prove

######  Theorem 3.1.

Assume $\Omega$ a bounded open set of $\mathbb{R}^{N}$, $N\geq 2$, and $E_{n}$
a sequence of vector measurable functions in $\Omega$ such that there exists
$E_{0}$ satisfying (3.1). Then, for every sequence $f_{n}$ which satisfies
(3.3), the sequence of solutions $u_{n}$ of (3.2) satisfies

$T_{m}(u_{n})\rightharpoonup T_{m}(u_{0})\ \hbox{ in }H^{1}_{0}(\Omega),\
\forall\,m>0,$ (3.4) $\ln^{q}(1+|u_{n}|)\rightharpoonup\ln^{q}(1+|u_{0}|)\
\hbox{ in }H^{1}_{0}(\Omega),\ \forall\,q>0,$ (3.5)

with $u$ the unique solution of

$\left\\{\begin{array}[]{l}\displaystyle-{\rm div}(A\nabla
u_{0}+E_{0}u_{0})=f\ \hbox{ in }\Omega\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle u_{0}\in H^{1}_{0}(\Omega).\end{array}\right.$ (3.6)

Moreover, if one of the following assumptions hold:

  * •

$N=2$.

  * •

$N>2$, $|E_{n}|^{N}$ is equi-integrable,

then the convergence holds in the weak topology of $H^{1}_{0}(\Omega)$.

Proof.

Step 1. We start getting some estimates for the solutions of (3.2). They are
based on [2].

For $t\geq 2$, we use as test function in (3.2) $v=\phi(u_{n})$, with $\phi$
defined by

$\phi(s)=\int_{0}^{s}{\ln^{t}(1+|\rho|)\over(1+|\rho|)^{2}}d\rho,\quad\forall\,s\in\mathbb{R}.$

We get

$\int_{\Omega}\big{(}A\nabla u_{n}+E_{n}u_{n})\cdot\nabla
u_{n}{\ln^{t}(1+|u_{n}|)\over(1+|u_{n}|)^{2}}dx=\langle
f_{n},\phi(u_{n})\rangle.$

Using (2.1), Young’s and Hölder’s inequalities, and $\phi^{\prime}\in
L^{\infty}(\mathbb{R})$, we deduce the existence of $C>0$ depending on
$\alpha,t,N$ ($\alpha$, $t,$ $p$, $|\Omega|$ if $N=2$) such that

$\begin{array}[]{l}\displaystyle\int_{\Omega}\Big{|}\nabla\Big{(}\ln^{t+2\over
2}(1+|u_{n}|)\,{\rm sgn}(u_{n})\Big{)}\Big{|}^{2}dx={(t+2)^{2}\over
4}\int_{\Omega}|\nabla u_{n}|^{2}{\ln^{t}(1+|u_{n}|)\over(1+|u_{n}|)^{2}}dx\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\leq
C\int_{\Omega}|E_{n}|^{2}|u_{n}|^{2}{\ln^{t}(1+|u_{n}|)\over(1+|u_{n}|)^{2}}dx+C\|f_{n}\|_{H^{-1}(\Omega)}^{2}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\leq
C\int_{\Omega}|E_{n}|^{2}\ln^{t}(1+|u_{n}|)dx+C\|f_{n}\|_{H^{-1}(\Omega)}^{2}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\leq
C\left(\int_{\Omega}|E_{n}|^{r}dx\right)^{2\over
r}\left(\int_{\Omega}\ln^{tr\over r-2}(1+|u_{n}|)dx\right)^{r-2\over
r}+C\|f_{n}\|_{H^{-1}(\Omega)}^{2},\end{array}$ (3.7)

with

$r=\left\\{\begin{array}[]{cl}p&\hbox{ if }N=2\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr\displaystyle r=N&\hbox{ if }N>2.\end{array}\right.$ (3.8)

By Sobolev’s inequality, we also have

$\left(\int_{\Omega}\ln^{(t+2)\mu\over 2}(1+|u_{n}|)dx\right)^{2\over\mu}\leq
C_{S}\int_{\Omega}\Big{|}\nabla\Big{(}\ln^{t+2\over 2}(1+|u_{n}|)\,{\rm
sgn}(u_{n})\Big{)}\Big{|}^{2}dx,$

with $1\leq\mu$, if $N=2$ and $1\leq\mu\leq 2N/(N-2)$, if $N>2$. Choosing

$1<\mu:={2tr\over(t+2)(r-2)}<{2r\over r-2}$

and using Young’s inequality, we deduce (for another constant $C>0$), after
simplifying equal terms,

$\left(\int_{\Omega}\ln^{tr\over r-2}(1+|u|)dx\right)^{(t+2)(r-2)\over tr}\leq
C\left(\int_{\Omega}|E_{n}|^{r}dx\right)^{t+2\over
r}+C\|f_{n}\|^{2}_{H^{-1}(\Omega)}.$

Replacing this estimate in the right-hand side of (3.7), we finally get

$\int_{\Omega}\Big{|}\nabla\Big{(}\ln^{t+2\over 2}(1+|u_{n}|)\,{\rm
sgn}(u_{n})\Big{)}\Big{|}^{2}dx\leq
C\left(\int_{\Omega}|E_{n}|^{r}dx\right)^{t+2\over
r}+C\|f_{n}\|^{2}_{H^{-1}(\Omega)}.$ (3.9)

Since $t$ can be taken as large as we want, we conclude

$\ln^{q}(1+|u_{n}|)\,{\rm sgn}(u_{n})\ \hbox{ bounded in
}H^{1}_{0}(\Omega),\quad\forall\,q>0.$ (3.10)

In particular, $T_{m}(u_{n})$ is bounded in $H^{1}_{0}(\Omega)$, for every
$m\in\mathbb{N}$. These estimates and the Rellich-Kondrachov’s compactness
theorem prove the existence of a subsequence of $n$, still denoted by $n$, and
a measurable function $u$ such that for every $m,q>0$, we have

$T_{m}(u_{n})\rightharpoonup T_{m}(u)\hbox{ in
}H^{1}_{0}(\Omega),\quad\ln^{q}(1+|u_{n}|)\rightharpoonup\ln^{q}(1+|u|)\hbox{
in }H^{1}_{0}(\Omega).$ (3.11)

Now, we have to prove that $u=u_{0}$.

Step 2. Let us first consider the case $N=2$ or $N\geq 3$, $|E_{n}|^{N}$ equi-
integrable. It has been first carried out in [5]. Indeed, since for $N=2$, $p$
in (3.1) is any number bigger than 2, the problem reduces to assume
$|E_{n}|^{r}$ equi-integrable with $r$ given by (3.8). Taking as test function
in (3.2) $u_{n}-T_{m}(u_{n})$, we deduce

$\int_{\\{m<|u_{n}|\\}}\big{(}A\nabla u_{n}+E_{n}u_{n}\big{)}\cdot\nabla
u_{n}\,dx=\langle
f_{n},u_{n}-T_{m}(u_{n})\rangle_{H^{-1}(\Omega),H^{1}_{0}(\Omega)},$

which similarly to (2.12) implies

$\begin{array}[]{l}\displaystyle\|u_{n}-T_{m}(u_{n})\|^{2}_{H^{1}_{0}(\Omega)}\big{(}\alpha-
C_{S}\|E_{n}\|_{L^{r}(\\{m<|u_{n}|\\})^{N}}\big{)}\\\ \vskip 6.0pt plus 2.0pt
minus
2.0pt\cr\displaystyle\leq\big{(}\|f_{n}\|_{H^{-1}(\Omega)}+mC_{S}\|E_{n}\|_{L^{2}(\Omega)^{N}}\big{)}\|u_{n}-T_{m}(u_{n})\|_{H^{1}_{0}(\Omega)}.\end{array}$

Thanks to (3.10), we have

$\lim_{m\to\infty}\sup_{n\in\mathbb{N}}\big{|}\\{|u_{n}|>m\big{\\}}\big{|}=0,$

which combined with the equi-integrability of $|E_{n}|^{r}$ implies the
existence of $m\in\mathbb{N}$ such that
$\|E_{n}\|_{L^{r}(\\{m<|u_{n}|\\})^{N}}<\alpha/(2C_{S})$, for every
$n\in\mathbb{N}$. Therefore $u_{n}-T_{m}(u_{n})$ is bounded in
$H^{1}_{0}(\Omega)$. Since $T_{m}(u_{n})$ is bounded in $H^{1}_{0}(\Omega)$
too, we get $u_{n}$ bounded in $H^{1}_{0}(\Omega)$. Taking into account (3.4),
(3.5) we have that $u_{n}$ converges weakly to $u$ in $H^{1}_{0}(\Omega)$. By
the Rellich-Kondrachov’s compactness theorem, we can now easily pass to the
limit in (3.2) in the distributional sense to deduce that $u=u_{0}$, the
solution of (3.6).

Step 3. In this and the following step we assume $N\geq 3$, $|E_{n}|^{N}$ not
necessarily equi-integrable.

For $\delta>0$, we define $w_{n,\delta}$ as the solution of

$\left\\{\begin{array}[]{l}\displaystyle-{\rm div}\big{(}A\nabla
w_{n,\delta}+E_{n}w_{n,\delta}\big{)}+\delta|w_{n,\delta}|^{4\over
N-2}w_{n,\delta}=f_{n}\ \hbox{ in }\Omega\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle w_{n,\delta}\in H^{1}_{0}(\Omega).\end{array}\right.$
(3.12)

Using $w_{n,\delta}$ as test function in (3.12), we have

$\int_{\Omega}A\nabla w_{n,\delta}\cdot\nabla
w_{n,\delta}\,dx+\int_{\Omega}w_{n,\delta}E_{n}\cdot\nabla
w_{n,\delta}\,dx+\delta\int_{\Omega}|w_{n,\delta}|^{2N\over N-2}dx=\langle
f_{n},w_{n,\delta}\rangle.$ (3.13)

In the second term of this equality we use Young’s inequality with exponents
$2N/(N-2)$, $N$ and 2, to get

$\begin{array}[]{l}\displaystyle\left|\int_{\Omega}w_{n,\delta}E_{n}\cdot\nabla
w_{n,\delta}\,dx\right|=\left|\int_{\Omega}\Big{(}\delta^{N-2\over
2N}w_{n,\delta}\Big{)}\Big{(}{2^{1\over 2}\over\alpha^{1\over
2}\delta^{N-2\over 2N}}E_{n}\Big{)}\cdot\Big{(}{\alpha^{1\over 2}\over
2^{1\over 2}}\nabla w_{n,\delta}\Big{)}dx\right|\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr\displaystyle\leq{N-2\over
2N}\delta\int_{\Omega}|w_{n,\delta}|^{2N\over N-2}dx+{2^{N\over 2}\over
N\alpha^{N\over 2}\delta^{N-2\over 2}}\int_{\Omega}|E_{n}|^{N}dx+{\alpha\over
4}\int_{\Omega}|\nabla w_{n,\delta}|^{2}dx.\end{array}$

In the last term, Young’s inequality also gives

$\big{|}\langle
f_{n},w_{n,\delta}\rangle\big{|}\leq\|f_{n}\|_{H^{-1}(\Omega)}\|w_{n,\delta}\|_{H^{-1}(\Omega)}\leq{1\over\alpha}\|f_{n}\|^{2}_{H^{-1}(\Omega)}+{\alpha\over
4}\int_{\Omega}|\nabla w_{n,\delta}|^{2}dx.$

Using also (2.1) in the first term, we deduce from (3.12)

${\alpha\over 2}\int_{\Omega}\big{|}\nabla w_{n,\delta}\big{|}^{2}dx+{N+2\over
2N}\delta\int_{\Omega}|w_{n,\delta}|^{2N\over N-2}dx\leq{2^{N\over 2}\over
N\alpha^{N\over 2}\delta^{N-2\over
2}}\int_{\Omega}|E_{n}|^{N}dx+{1\over\alpha}\|f_{n}\|^{2}_{H^{-1}(\Omega)}.$
(3.14)

Thus, for every $\delta>0$, $w_{n,\delta}$ is bounded in $H^{1}_{0}(\Omega)$.
Thanks to Rellich-Kondrakov’s compactness theorem this allows us to pass to
the limit in (3.12) in the distributional sense, to deduce

$w_{n,\delta}\rightharpoonup w_{\ast,\delta}\ \hbox{ in }H^{1}_{0}(\Omega),$
(3.15)

with $w_{\ast,\delta}$ the solution of

$\left\\{\begin{array}[]{l}\displaystyle-{\rm div}(A\nabla
w_{\ast,\delta}+E_{0}w_{\ast,\delta})+\delta|w_{\ast,\delta}|^{4\over
N-2}w_{\ast,\delta}=f\ \hbox{ in }\Omega\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle w_{\ast,\delta}\in H^{1}_{0}(\Omega).\end{array}\right.$
(3.16)

We take $u_{0}$ the solution of (3.6). Taking
$T_{\rho}(w_{\ast,\delta}-u_{0})$, with $\rho>0$ as test function in the
difference of (3.16) and (3.6), we have

$\begin{array}[]{l}\displaystyle\int_{\\{|w_{\ast,\delta}-u_{0}|<\rho\\}}\hskip-10.0pt\big{(}A\nabla(w_{\ast,\delta}-u_{0})+E(w_{\ast,\delta}-u_{0})\big{)}\cdot\nabla(w_{\ast,\delta}-u_{0})\,dx\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle+\delta\int_{\Omega}\big{(}|w_{\ast,\delta}|^{4\over
N-2}w_{\ast,\delta}-|u_{0}|^{4\over
N-2}u_{0}\big{)}T_{\rho}(w_{\ast,\delta}-u_{0})\,dx=-\delta\int_{\Omega}|u_{0}|^{4\over
N-2}u_{0}T_{\rho}(w_{\ast,\delta}-u_{0})\,dx.\end{array}$

Using Young’s inequality in the second term of the first integral and

$\exists c_{N}>0:\quad c_{N}\big{|}|x|+|y|\big{|}^{4\over
N-2}|x-y|^{2}\leq\big{(}|x|^{4\over N-2}x-|y|^{4\over
N-2}y\big{)}(x-y),\quad\forall\,x,y\in\mathbb{R},$

in the second integral, we get

$\begin{array}[]{l}\displaystyle{\alpha\over
2}\int_{\\{|w_{\ast,\delta}-u_{0}|<\rho\\}}\hskip-16.0pt|\nabla(w_{\ast,\delta}-u_{0})|^{2}dx+c_{N}\delta\int_{\Omega}\big{(}|w_{\ast,\delta}|+|u_{0}|\big{)}^{4\over
N-2}|w_{\ast,\delta}-u_{0}||T_{\rho}(w_{\ast,\delta}-u_{0})|\,dx\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\leq{1\over
2\alpha}\int_{\\{|w_{\ast,\delta}-u_{0}|<\rho\\}}\hskip-16.0pt|E|^{2}|w_{\ast,\delta}-u_{0}|^{2}dx+\delta\int_{\Omega}|u_{0}|^{N+2\over
N-2}|T_{\rho}(w_{\ast,\delta}-u_{0})|\,dx.\end{array}$ (3.17)

This proves that for every $\rho>0$, $T_{\rho}(w_{\ast,\delta}-u_{0})$ is
bounded in $H^{1}_{0}(\Omega)$. Dividing by $\rho$ and taking the limit when
$\rho$ tends to zero, thanks to the Lebesgue dominated convergence theorem, we
get

$\int_{\Omega}\big{(}|w_{\ast,\delta}|+|u_{0}|\big{)}^{4\over
N-2}|w_{\ast,\delta}-u_{0}|dx\leq{1\over c_{N}}\int_{\Omega}|u_{0}|^{N+2\over
N-2}dx,$

and thus $w_{\ast,\delta}$ is bounded in $L^{N+2\over N-2}(\Omega)$. Hence,
for a subsequence of $\delta$ which converges to zero, still denoted by
$\delta$, there exits $w_{\ast}$ such that

$\left\\{\begin{array}[]{l}\displaystyle w_{\ast,\delta}\to w_{\ast}\ \hbox{
a.e. in }\Omega,\ \ w_{\ast,\delta}\rightharpoonup w_{\ast}\ \hbox{ in
}L^{N+2\over N-2}(\Omega)\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle T_{\rho}(w_{\ast,\delta}-u_{0})\rightharpoonup
T_{\rho}(w_{\ast}-u_{0})\hbox{ in }H^{1}_{0}(\Omega),\
\forall\,\rho>0.\end{array}\right.$ (3.18)

By the lower semicontinuity of the norm in $H^{1}_{0}(\Omega)$, this allows us
to pass to the limit when $\delta$ tends to zero in (3.17) to deduce

$\int_{\\{|w_{\ast}-u_{0}|<\rho\\}}\hskip-10.0pt|\nabla(w_{\ast}-u_{0})|^{2}dx\leq{1\over\alpha^{2}}\int_{\\{|w_{\ast}-u_{0}|<\rho\\}}\hskip-10.0pt|E|^{2}|w_{\ast}-u_{0}|^{2}dx.$

Dividing by $\rho^{2}$ and taking the limit when $\rho$ tends to zero, this
proves

${1\over\rho}T_{\rho}(w_{\ast}-u_{0})\rightarrow 0\ \hbox{ in
}H^{1}_{0}(\Omega)\ \hbox{ when }\rho\to 0.$

Combined with

${1\over\rho}T_{\rho}(w_{\ast}-u_{0})\rightharpoonup{\rm sgn}(w_{\ast}-u_{0})\
\hbox{ a.e. in }\Omega,$

we get

$w_{\ast}=u_{0}\ \hbox{ a.e. in }\Omega.$ (3.19)

Step 4. For $\rho>0$ we take, similarly to the Step 3,
$T_{\rho}(w_{n,\delta}-u_{n})$ as test function in the difference of (3.12)
and (3.2). We get

$\begin{array}[]{l}\displaystyle{\alpha\over
2}\int_{\\{|w_{n,\delta}-u_{n}|<\rho\\}}\hskip-12.0pt|\nabla(w_{n,\delta}-u_{n})|^{2}dx+\delta\int_{\Omega}|w_{n,\delta}|^{4\over
N-2}w_{n,\delta}T_{\rho}(w_{n,\delta}-u_{n})\,dx\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr\displaystyle\leq{1\over
2\alpha}\int_{\\{|w_{n,\delta}-u_{n}|<\rho\\}}\hskip-12.0pt|E_{n}|^{2}|w_{n,\delta}-u_{n}|^{2}dx.\end{array}$

Taking into account (3.11) and (3.15), and defining ${\cal E}$ by (it exists
for a subsequence)

$|E_{n}|^{2}\rightharpoonup{\cal E}\ \hbox{ in }L^{N\over 2}(\Omega),$

we can pass to the limit in $n$ in this inequality by semicontinuity and the
Rellich-Kondrachov’s compactnes theorem to deduce

${\alpha\over
2}\int_{\\{|w_{\ast,\delta}-u|<\rho\\}}\hskip-12.0pt|\nabla(w_{\ast,\delta}-u)|^{2}dx+\delta\int_{\Omega}|w_{\ast,\delta}|^{4\over
N-2}w_{\ast,\delta}T_{\rho}(w_{\ast,\delta}-u)\,dx\leq{1\over
2\alpha}\int_{\\{|w_{\ast,\delta}-u|\leq\rho\\}}\hskip-12.0pt{\cal
E}|w_{\ast,\delta}-u|^{2}dx.$

Now we pass to the limit when $\delta$ tends to zero thanks to (3.18) and
(3.19), to get

$\int_{\\{|u_{0}-u|<\rho\\}}\hskip-10.0pt|\nabla(u_{0}-u)|^{2}dx\leq{1\over\alpha^{2}}\int_{\\{|u_{0}-u|\leq\rho\\}}\hskip-10.0pt{\cal
E}|u_{0}-u|^{2}dx$

Dividing by $\rho^{2}$ and passing to the limit when $\rho$ tends to zero we
deduce as at the end of Step 3 that $u=u_{0}$. This finishes the proof. $\Box$

######  Remark 3.2.

One of the applications of Theorem 3.1 is the existence of solutions for some
control problems in the coefficients. In this way, combined with Fatou’s Lemma
it immediately proves the existence of solution for

$\begin{array}[]{c}\displaystyle\min\left\\{\int_{\Omega}\big{(}G(x,u)+\mu|E|^{p}\big{)}dx:\
E\in L^{p}(\Omega)^{N}\right\\}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\left\\{\begin{array}[]{l}-{\rm div}(A\nabla u+Eu)=f\
\hbox{ in }\Omega\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle u\in
H^{1}_{0}(\Omega),\end{array}\right.\end{array}$

with $p>2$ if $N=2$, $p\geq N$ if $N>2$, $f\in H^{-1}(\Omega)$ and
$G:\Omega\times\mathbb{R}\to\mathbb{R}$ such that

$G(.,s)\ \hbox{ measurable, }\forall\,s\in\mathbb{R},\quad G(x,.)\hbox{
continuous for a.e. }x\in\Omega,$ $\exists a\in\mathbb{R},b\geq 0,\ \hbox{
such that }\ G(x,s)\geq a-b|s|,\quad\forall\,s\in\mathbb{R},\ \hbox{ a.e.
}x\in\Omega.$

## Aknowledgments

This work has been partially supported by the project PID2020-116809GB-I00 of
the Ministerio de Ciencia e Innovación of the government of Spain.

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