# Assessment of Fetal and Maternal Well-Being During Pregnancy Using Passive
Wearable Inertial Sensor

Eranda Somathilake  Dept. of Mechanical Eng.
University of Peradeniya
Peradeniya, Sri Lanka
<EMAIL_ADDRESS>Upekha Delay  Dept. Electrical & Electronic
Eng.
University of Peradeniya
Peradeniya, Sri Lanka
<EMAIL_ADDRESS>Janith Bandara Senanayaka  Samitha Gunarathne
Dept. Electrical & Electronic Eng.
University of Peradeniya
Peradeniya, Sri Lanka
<EMAIL_ADDRESS>Dept. Electrical & Electronic Eng.
University of Peradeniya
Peradeniya, Sri Lanka
<EMAIL_ADDRESS>Roshan Godaliyadda  Dept. Electrical & Electronic
Eng.
University of Peradeniya
Peradeniya, Sri Lanka
<EMAIL_ADDRESS>Parakrama Ekanayake  Janaka Wijayakulasooriya  Dept.
Electrical & Electronic Eng.
University of Peradeniya
Peradeniya, Sri Lanka
<EMAIL_ADDRESS>Dept. Electrical & Electronic Eng.
University of Peradeniya
Peradeniya, Sri Lanka
<EMAIL_ADDRESS>Chathura Rathnayake  Dept. of Obstetrics and Gynaecology
University of Peradeniya
Sri Lanka
<EMAIL_ADDRESS>
###### Abstract

Assessing the health of both the fetus and mother is vital in preventing and
identifying possible complications in pregnancy. This paper focuses on a
device that can be used effectively by the mother herself with minimal
supervision and provide a reasonable estimation of fetal and maternal health
while being safe, comfortable, and easy to use. The device proposed uses a
belt with a single accelerometer over the mother’s uterus to record the
required information. The device is expected to monitor both the mother and
the fetus constantly over a long period and provide medical professionals with
useful information, which they would otherwise overlook due to the low
frequency that health monitoring is carried out at the present. The paper
shows that simultaneous measurement of respiratory information of the mother
and fetal movement is in fact possible even in the presence of mild
interferences, which needs to be accounted for if the device is expected to be
worn for extended times.

###### Index Terms:

Fetal movement, breathing patterns, fetal health, accelerometers, deep
learning, Weiner filtering, wavelet transform, fast Fourier transform

##  I Introduction

Regular monitoring throughout the pregnancy allows early detection of well-
being problems that might arise and will aid their treatment, improving the
chance for the birth of a healthy baby. The health and condition of both the
mother and the baby is a clear indication of future complications or the well-
being of the fetus [1]. Fetal well-being can be monitored in different ways
[2, 3, 4], each with its own advantages and disadvantages. Hence, the
availability of multiple methods will provide the medical professionals with
better tools, which can be used in the different specific situations. Also,
the mood and health of the mother can have a dramatic impact on the
development of the baby, both in long and short terms [5]. This paper,
therefore, proposes a device to assess the condition of the fetus as well as
the mother, which is non-invasive, low cost, and simpler to use compared to
most of the existing methods available for fetal condition monitoring.

Fetal condition monitoring can be done in multiple ways and the area of focus
in this research is through fetal movement, which can be used as an indication
of future complications [6]. Fetal movement can be identified as a primary
indicator of fetal well-being. Reduction or absence of fetal movement is a
strong indication of fetal compromise [7, 8]. Fetal movement monitoring
methods can be divided into two approaches: active or passive. Active methods
directly observe the fetus using various imaging techniques while passive
methods measure the fetal movements indirectly by measuring other responses as
the movements of the fetus. CTG and Utrasound scanning are examples of active
methods, whereas use of sensors such as accelerometers or acoustic sensors are
examples of passive methods.

While most of the common active methods such as Ultrasound and CTG are used
extensively, they do possess some drawbacks that can make them undesirable in
certain situations. Although it is not proven clinically, the use of high
frequency audio waves which penetrates the uterus may cause harm to the fetus
[9]. Also, the equipments used for both CTG scanning and ultrasound scanning
are bulky and require trained professionals for operation and interpretation,
thus making them impractical to be used in a daily basis over an extended
period of time. Fetal state monitoring using these methods are only conducted
in clinical settings and most of the time is done after the mother’s
admittance to the hospital.It would, however, be beneficial if fetal movements
can be monitored domestically in an ambulatory setting, which will in turn
enable the assessments to be done more frequently and is ideal in times of a
pandemic, where the mothers can safely stay in their homes. In addition, it is
difficult to obtain images of obese mothers, which makes these methods less
viable to be used [10] even though they are more likely to encounter
complications [11]. The lack of experts and the expensive equipment required,
in rural areas, can be a major hindrance [12] in monitoring fetal movement.
Thus promoting the need for low cost and simpler methods.

Simpler passive methods, therefore, can be used as an alternative or along
with existing active methods to have a better assessment of fetal movement
throughout the pregnancy. The proposed device detects fetal movement through
an accelerometer. Obtaining the data through this device is more convenient,
non-invasive, and takes minimal time for setup. This device is designed to be
portable, ergonomic, and with modular electronics for easy repair.
Furthermore, it was observed that the accelerometer placed on the abdomen have
the ability to capture maternal respiratory motion data as well. It can,
therefore, be used to detect respiratory patterns [13], which can provide
valuable information such as respiratory rate and maternal energy expenditure
level to medical practitioners. Both respiratory rate [14] and maternal energy
expenditure [15] are key indicators of maternal well-being during pregnancy.
Hence, this device can be used as an overall condition monitoring device
rather than solely focusing on fetal movement.

The data obtained using the sensors consist of information of many processes
that happens in the body, namely fetal movements, maternal respiratory
patterns, and disturbances such as coughing, laughing, or walking. In this
study, the desired parameters were maternal respiratory motions and fetal
movements. Therefore, analyses were conducted to identify fetal movements and
to extract information on maternal respiratory data.

This device is designed to be used by lay users and the readings will not be
taken in a controlled environment. Hence, the device should have the
capability to identify and filter out unnecessary data. Therefore, in this
study, it is demonstrated that the artefacts introduced into the
accelerometric signal due to the walking during a session can be successfully
removed using a Weiner filter. Moreover, a peak detecting algorithm was used
to calculate the respiratory rate of the mothers, which can act as one of the
main well-being indicators for maternal health [16]. Furthermore, analyses
were conducted to classify the energy expenditure level of the mother’s body,
depending on the respiratory pattern. To accomplish that, feature extraction
was done using the discrete Wavelet Transform and then classified using a
Neural Network.

Fetal movement patterns can be complex in nature and hence machine learning
techniques were used to identify them [17, 18]. One approach was to use
Convolutional Neural Networks (CNN), where a scalogram was generated by taking
the wavelet transform of the signal and was then used to identify fetal
movement. The other approach was to use a recurrent neural network, where
Gated Recurrent Unit (GRU) cells were used so that the signal can be analysed
over long term dependencies to detect fetal movement.

##  II Background

Active methods such as Ultrasound scanning and CTG scanning observe the
movement of the fetus and give a physical representation of it. They are
therefore highly accurate and experts can assess fetal health fairly
accurately [1]. The negative aspects of these devices, as stated previously,
are them being too complex to use, bulky, and might have harmful effects on
both the mother and the baby. Furthermore, interpreting the data received from
these methods requires technical skill as well as time, which can act as a
hindrance to taking quick actions when necessary. Therefore, passive methods
that observe the surface of the uterus was considered in this study. Also,
active methods do not evaluate fetal movement constantly, but identification
of changes in fetal movement patterns over an extended time period can help
identify complications earlier [8].

Maternal perception of movement identification and monitoring is a widely used
and easy to implement method but it is highly subjective and inconsistent
[19]. Hence, there is a need for a viable fetal movement monitoring method
that eliminate these inaccuracies. One such method proposes a device that uses
acoustic sensors and accelerometers attached to a belt worn around the abdomen
[20]. Also, multiple accelerometer sensors were used in a different study [21,
22]. These multiple sensor approaches have yielded good results. However, as
stated in [20], acoustic sensors are too sensitive for this particular
problem. The use of multiple accelerometers, although is better than one, did
not seem to have a significant effect on movement detection and increasing the
sensors had a diminishing impact on the results while it led to an increase in
used equipment and complexity. Hence, in this study, a single sensor was used
and more attention was paid to optimising the post-processing of the data.
Most of the proposed devices mentioned above have performed well in an
experimental environment. However, their capability to be adapted into
practical situations where they are handled by users who are not technically
trained were not considered in most cases. The proposed device in this study
has been designed for practical implementation with ease of use and simplicity
in mind.

Since inertial sensors measure the physical movement of the surface of the
abdomen, in addition to capturing fetal movement data, they will
simultaneously record other bodily functions of the mother as well. Hence, the
device can observe multiple states of the mother. In this study, we have
focused on monitoring the breathing patterns of the mother using the same
inertial sensor. The use of accelerometers to measure breathing patterns is a
viable solution and it has been implemented to classify between different
breathing patterns [13]. A similar approach was considered in this study to
evaluate the breathing patterns of pregnant mothers to evaluate their health.

##  III Device implemented

The device implemented consists of an accelerometer and two buttons to get
external inputs. One input was used by the doctor who observes the fetus using
ultrasound scanning to be used as the ground-truth to train the neural
network, while the other button is used to record other interference that
might be detected by the accelerometer such as the mother laughing or
coughing. The same device was used in separate sessions to record breathing
patterns and this did not require any user input while recording. The
configuration of the components of the device is given in Figure 1.

Figure 1: Component configuration of the proposed device

The components are controlled by an Arduino Uno micro-controller, as stated,
the device consists of an accelerometer (MPU 9250) and two buttons. In
addition, there is a micro SD card module to store data, a real-time clock
(DS3231), and LEDs to indicate the current state of the device and button
presses. The accelerometer was attached to a belt which can be wound around
the mother’s uterus as shown in figure Figure 2. In this preliminary study,
the device was controlled by a PC to give better control over its operation.

Figure 2: Mother wearing the sensor

The sensor used here, MPU 9250, consists of a tri-axial accelerometer, tri-
axial gyroscope, and a compass. Only one axis of the tri-axial accelerometer
was used in this analysis. The sensor communicates with the Arduino using the
$I^{2}C$ interface up to a maximum sampling rate of 32 kHz, but the readings
obtained were at a sampling rate of around 100 Hz mainly due to the other
processes running in the micro-controller. The specified temperature range for
the operation of the sensor is from $-40\degree C$ to $85\degree C$, which
accommodates the operating temperatures for this application.

The device operates in two states, namely training state and prediction state:
in the training state, it records both the sensor readings and the user inputs
so that they can be used for creating the predictor models; in the prediction
state, it reads sensor readings and uses the predictor models to make
predictions, as depicted in Figure3.

Figure 3: Summarized description of the device

##  IV Data Analysis

###  IV-A Preliminary readings

Following the fabrication of the system, several preliminary tests were
conducted on the system to identify its performance in different environmental
conditions as well as how it responds to different behaviours of pregnant
mothers. Initially, the noise profile of the device is obtained. This is done
by taking readings from the device while it’s at a stationary position. The
time-domain accelerometric readings of a single axis, as well as the time-
frequency variation of the readings, can be observed in Figure 4a and Figure
4b. Moreover, the Probability Distribution Function (PDF) of the time domain
data and the Power spectrum of the data can be observed in Figure 4c and
Figure 4d, respectively. Furthermore, the kurtosis values of several samples
were evaluated and the mean of the values were 2.9854, which is approximately
3. Therefore, it can be observed that the stationery noise have a behaviour
similar to Gaussian White noise.

Figure 4: Different representation of the data obtained while the device is at
an stationery position

The environmental temperature in the region where the studies were conducted
varied from 16°C to 32°C. However, the usual body temperature is 37°C. Hence,
when the sensor is worn for an extended time, the temperature could vary from
16°C to 37°C. In order to observe the effect of temperature on the noise
features of the sensor, readings on different temperatures within the range of
16°C to 37°C were taken. The effect of the temperature can not be clearly
observed in the time domain. Therefore, the noise power level at each
temperature was calculated and plotted against the temperature to observe the
effects. It can be observed in Figure 5 that when the temperature is
increased, the noise power also increases gradually.

Figure 5: Variation of noise power in different Temperature levels

Then the device was worn by a non-pregnant subject and readings were taken to
observe the sensor response to different types of human behaviour. Initially,
the subject was advised to wear the belt and to stay seated in the Fowler’s
position for an extended time and the tri-axial accelerometric data were
recorded (see Figure 6). It can be observed that the subject’s breathing
pattern can also be observed very clearly from the accelerometric signals.
While X-Axis and Y-Axis show a slight pattern, the Z-Axis indicate a clear
pattern for the subject’s breathing. Hence, it can be concluded that this
body-worn accelerometer can also be utilized to monitor the subject’s
breathing patterns and further breathing anomalies. Furthermore, the effect of
the position of the subject on the readings was observed by taking reading
while the subject was in different positions such as the Fowler’s position,
Supine position, and lateral recumbent position. However, it was observed that
the initial position of the subject do not have a significant effect on the
data.

Figure 6: When the device is worn by a stationery subject

However, since the main aim of this research is to conduct a preliminary study
on the use of an accelerometric sensor-based system to monitor fetal and
maternal health in the home, the effect of maternal movements on the sensor
data was also observed. Initially, the effect on time-domain data when the
subject was changing from one position to another position was observed. The
subject was advised to change into different positions and the tri-axial
accelerometric data were observed. Different transitions have different
effects on each axis’s readings. However, in all cases, it was observed that
during the transition, there is a shift in the time domain data. It was also
observed that following the transition period, the time domain data have a
similar variation to time-domain data prior to transition. This can be
observed in Figure 7 where the Z-Axis variation when the subject is moving
from Fowler’s position to Supine position is depicted. Therefore, it can be
concluded that such small movements (in time scale) of the subject will not
have a significant effect on a session. The observed shift can be easily
eliminated by utilizing simple signal processing techniques in the pre-
processing stage. However, if a fetal movement is to coincide with a maternal
movement, the extraction of the fetal movement signal may be strenuous.
Nevertheless, due to the short duration of such activities, in practical
situations, the probability of such coinciding occurrences can be considered
to be very small.

Figure 7: Acceleration variation of the Z axis when the subject transit from
Fowler’s position to Supine position

Thereafter, the effect of frequent maternal movements such as walking and
talking were observed. The readings taken while the subject is walking can be
observed in Figure 8. It can be observed that taking steps have a direct
impact on the time domain data of all the axes. As discussed in the latter
part of this paper, filtering methods such as Wiener filters can be applied to
remove the imposed interference due to taking steps. However, it is advisably
for mothers to stay stationary during a session. Furthermore, time-domain data
obtained during the subject is speaking was compared with the data obtained
while not speaking. No noticeable differences were observed. Therefore, it was
concluded that speech has insignificant effect on the accelerometric data.

Figure 8: Tri-Axial acceleration variation when the subject is walking while
wearing the device

Subsequently, the time domain accelerometric variation of several maternal
motions such as cough, hiccup, yawn, and laugh were observed. These were
selected due to the fact that they may have a similar effect on the maternal
abdomen surface as of fetal movement. It can be observed in Figure 9 that
while cough, laugh, and hiccup have peaks similar to the fetal movement signal
in the time domain, yawns do not have a similar time-domain profile. Hence,
during analysis, more emphasis must be paid to extracting fetal movements
signals from data contaminated with a maternal laugh, hiccups, and cough.

Figure 9: Acceleration variation due to cough, Hiccup and yawn

###  IV-B Respiratory data analysis

When observing the accelerometric data, it was noted that maternal respiratory
movements are also present in the data. Furthermore, during the initial
analysis of the device, the respiratory motion was observed clearly as can be
seen in Figure 10. In Figure 10 the time domain data of the Z-Axis is given
and the general trend of the data is indicated in red colour.

Respiratory rate is one of the major well-being indicators of the human body.
While the respiratory rate is usually overlooked, documenting respiratory rate
can aid in predicting several serious clinical events [14]. Therefore, the
viability of using this device to identify the respiratory rate was also
studied.

In Figure 6, it can be observed that respiratory motion cause noticeable peaks
in the Z-Axis data in the time domain. Therefore, it was decided to monitor
these occurrences of peaks in the time domain in order to identify the
respiratory rate. Initially, a de-trending algorithm was applied to the data
to remove the mean as well as the trend in time domain data. In Figure 6 and
Figure 10, it can be observed that the respiratory motion have a magnitude of
approximately 50 AMU and the peaks are approximately 600 samples apart. This
reading is taken while the subject was wearing the device at a stationary
position. Further readings were taken after conducting two physical
activities, a 10 minute walk and a 10 minute run. In all these readings, the
magnitude of the respiratory motion was always less than 200 AMU and the peaks
were more than 400 samples apart. Using this information, a peak identifying
algorithm was developed and implemented and subsequently, the identified
number of peaks were divided by the time they occurred to obtain the
respiratory rate.

Figure 10: Z-Axis Accelerometric variation of maternal respiratory motion and
Fetal movement

Furthermore, the assessment of the energy expenditure of the human body is
considered to be of importance in sports as well as in other activities [23].
In [13], it is discussed how to estimate human body energy expenditure by
monitoring the respiratory patterns of the subject. They have utilized an
accelerometric sensor and have conducted studies on three levels of energy
expenditure; low EE, median EE, and high EE. In this study, it was studied how
this sensor system can be utilized in identifying these three types of energy
expenditure activities.

Initially, three types of tasks were selected for the three levels of energy
expenditure. Then an algorithm was implemented to identify the energy
expenditure level of the body based on the respiratory motion data. A three-
stage algorithm was utilized, where initially the data were prepossessed, then
features were constructed, and finally, the extracted features were utilized
in the classification of data. Initially, the tri-axial data were de-trended
and the mean was removed. Then, (1) was applied to the tri-axial data to
eliminate sensor rotation interference and to combine data of the three axes
[24].

$g(n)=\sqrt{x(n)^{2}+y(n)^{2}+z(n)^{2}}$ (1)

After prepossessing the data, the time domain signal was segmented into epochs
of 1000 samples with a 20% overlap. Then features were calculated for each
epoch. When selecting the features to best represent the data, available time-
domain features, as well as frequency domain features, were considered. More
attention was paid to selecting a set of features that are not redundant as
well as inclusive. It can be observed in Figure 10 that the time-domain data
are very noisy. Therefore, it was decided to utilize frequency domain
features.

When extracting features from the data in the frequency domain, initially, the
discrete wavelet transform was applied to the time domain data, and it was
segmented into four frequency bands [25]. Individual features from each band
were then computed. The features computed were; mean, standard deviation,
variance, and the skewness of each band. From this, for a single epoch, 16
features were calculated and these features were fed into the classifying
algorithm. However, in order to compare the performance of different types of
wavelets on accelerometric signal classification, several types of wavelets
(Daubechies wavelet db2, db4, db6, Symmlet wavelet sym2, and sym4) were used
and the accuracy of each one was compared.

The resulting features were fed into a simple standard neural pattern
recognition algorithm. The input was set to be the features constructed in the
previous step and the output was the three classes of activities. This network
is a feed-forward network and the training was done utilizing the scaled
conjugate gradient backpropagation. From the data set, 70% of the data were
used for training, 15% were used for validation, and the remaining 15% was
used for testing. The summary of the implemented algorithm is illustrated in
Figure 11.

Figure 11: Algorithm implemented to classify Energy Expenditure using
respiratory data

###  IV-C Fetal movement detection

The data set obtained using the mothers was used for the analysis involving
fetal movement detection. The analysis can be conducted in several different
ways as it has been stated from rudimentary methods where the signal is
observed for peaks to the use of machine learning algorithms. This paper
focuses on using machine learning techniques since they can be implemented in
small handheld devices with accurate results as it has been demonstrated in
many wearables that are currently in wide use [26].

The use of convolutional neural networks (CNNs) and recurrent neural networks
(RNNs) are considered here. CNNs have proven to have generally good results
for fetal movement identification as demonstrated in [17], here spectrogram
images obtained from the accelerometer readings were used as the input to the
CNN. In this paper, to explore a different perspective, wavelet transform was
used to generate the images. As stated in [27], wavelet transform suits better
for non-stationary signal analysis when compared with Fourier analysis. The
network used consisted of three convolution layers, each with 32 filters with
a kernel size of 3, each followed by a max-pooling layer and finally a dense
layer of 120 units.

Another approach that is considered here is the use of RNNs. RNNs have proven
to perform best with the use of spectrograms in [18] for fetal movement
detection. Here, as an extension to that research, the spectrograms were used
as the input to the network, but instead of using Long Short - Term Memory
(LSTM) networks, Gated Recurrent Unit (GRU) networks were considered, which is
almost similar in performance while being less computationally intensive [28].
The network implemented here consisted of 2 GRU layers along with dense, batch
normalisation, and a 1-dimensional convolution layer.

###  IV-D Interference removal

The objective of this research is to develop a device to be used at home by
mothers, without medical supervision. Hence, the posture and movements of the
mother will not be as restricted as they would be in a hospital environment.
This will require the device to have the ability to filter any interferences.

One of the common disturbances that can occur is due to walking, if the motion
that the accelerometer picks up due to the physical movement is considered as
a noise, this can be filtered to obtain the signal for breathing pattern
analysis. Both breathing and walking patterns are of a periodic nature, and
the Weiner filter would be a good choice to filter and obtain the pure
breathing signal Figure 16. The accelerometer readings obtained while seated
can be used as the reference signal.

##  V Results

###  V-A Respiratory Data Analysis

Initially an algorithm was implemented to identify the peaks of the repiratory
motion signal. The identified peaks can be observed in Figure 12. The
identified peaks are indicated using red arrowheads.

Figure 12: Peaks detected in the Z-Axis Accelerometric data of respiratory
motion

However, upon further investigation it was observed that since the target of
the algorithm is to detect peaks, it may identify peaks at occurrence of a
fetal movement as well. This can be observed in Figure 13. When considering
about the frequency of occurrence of fetal movement, the error introduced to
the respiratory rate due to fetal movement can be considered to be negligible.

Figure 13: Peaks detected when the signal contain fetal movement signal

Subsequently, it was attempted to estimate the energy expenditure level
utilizing respiratory data. During this, initially preprocessing and data
segmentation was conducted. Thereafter, discrete Wavelet Transform was
implemented on individual epochs to segment the time domain data into four
frequency bands and four features were extracted from each band. These
features were then used in the classification algorithm. When applying the
DWT, different types of Wavelet signals were used and their performance was
evaluated to identify the best wavelet to be utilized. Following the feature
extraction, classification was conducted and the True Positive rate for each
wavelet type is obtainded. The True positive rate obtained during training,
validation, testing are given in Table I.

TABLE I: True positive rate obtained for different Wavelet types Wavelet | Training | Validation | Testing | Total
---|---|---|---|---
Db2 | 99.8 | 96.9 | 90.6 | 98.1
Db4 | 98.7 | 100 | 96.9 | 98.6
Db6 | 99.3 | 100 | 93.8 | 98.6
Sym2 | 96.7 | 93.8 | 90.6 | 95.4
Sym 4 | 99.2 | 98.9 | 93.8 | 98.9

###  V-B Fetal movement detection

The wavelet transforms generated to be used as the inputs to the CNN showed
significant visual differences for the two instances of the presence and
absence of fetal movement Figure14, although this was not as evident in some
instances.

(a)

(b)

Figure 14: Wavelet scalograms generated from the data.(a) Scalogram with fetal
movement. (b) Scalogram without fetal movement

Although the network trained showed good results for the original data set
with higher accuracies (90%), significant performance degradation can be
observed if the validation of the results were done using a different mother.
Hence, it can be inferred that the fetal movement patterns are somewhat unique
to specific cases. Therefore, for better generalization, a significantly
larger dataset must be considered or the model should be fine-tuned so that it
fits each mother individually.

The same observation can be seen when an RNN was used, but with slightly
better accuracies, which may be due to the fact that it gave more emphasis on
the variation of the signal over time. A mean square error of 0.1 was observed
by the final model. Also, the performance of the model compared to the ground
truth is given in Figure15.

Figure 15: Comparison between predicted and ground truth values.

###  V-C Interference removal

Breathing patterns extracted from the accelerometer readings while walking did
correspond to the accelerometer signals that were obtained while seated
Figure16.

Figure 16: Comparison between filtered and reference data

The algorithms, although not ideal, have been able to extract relevant data by
filtering out the noise from walking, which has a much larger amplitude.

##  VI Conclusion

In conclusion, the simplicity of the device makes it cheap and easy to use and
therefore the mother can have an active role in her health monitoring and can
provide the medical experts with valuable information. This is mainly because
the device depends on different filtering and analysis techniques rather than
on complex hardware. Here, we have proposed the device to be used to monitor
the respiratory patterns of the mother and fetal movement. Both of these
metrics can provide valuable information when measured over a long period,
which is not possible in the existing methods and the doctors have had to
solely depend on the mother’s observations which can be inconsistent and
unreliable.

The device, along with the stated different analysis methods, can be
implemented to obtain reasonably accurate results for pregnant mothers outside
the hospital environment. It can give a rough estimate of the mother’s and
baby’s health. Moreover, experts can observe the data from the device for
further analysis as well. This device, therefore, allows a more personalised
and long term monitoring solution that, although not highly accurate, can be a
good addition to the existing health monitoring systems.

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# Newtheorem and theoremstyle test

Michael Downes
updated by Barbara Beeton  Shiguang Ma and Zijian Wang

#  $N$ -Laplacian and $N/2$-Hessian type equations with exponential reaction
term and measure data

Michael Downes
updated by Barbara Beeton  Shiguang Ma and Zijian Wang

###### Abstract

In this article, we will prove existence results for the equations of the type
$-\Delta_{N}u=H_{l}(u)+\mu$ and $F_{\frac{N}{2}}[-u]=H_{l}(u)+\mu$ in a
bounded domain $\Omega$, with Dirichlet boundary condition, where the source
term $H_{l}(r)$ takes the form $e^{r}-\sum_{j=0}^{l-1}\frac{r^{j}}{j!}$ and
$\mu$ is a nonnegative Radon measure.

##  1 Introduction

The following two types of equations appear in areas as quasi-regular
mappings, non-Newtonian fluids and reaction-diffusion problems, etc,

$\displaystyle-\Delta_{p}u$ $\displaystyle=F(u,x),$ (1) $\displaystyle
F_{k}[-u]$ $\displaystyle=F(u,x),$ (2)

where $\Delta_{p}u={\rm div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian
($p>1$), and $F_{k}[-u]$ is the $k$-Hessian $(k=1,2,\cdots,n)$ defined by

$F_{k}[-u]=\sum_{1\leq i_{1}<\cdots<i_{k}\leq
n}\lambda_{i_{1}}\cdots\lambda_{i_{k}},$

where $\lambda_{1},\cdots,\lambda_{n}$ are the eigenvalues of the Hessian
matrix $-D^{2}u$.

For $p$-Laplacian equation, we may refer to [HKM, IM, KM2, M1, MZ, S1, S2, SZ,
TW4] for the existence and regularity theory, estimates for supersolutions and
Wiener criterion, ect.

For $k$-Hessian equation, one may refer to [CNS, GT, GU, IV, KR, TR, TW1, TW2,
TW3, TW4, UR, LA] for related knowledge.

Here we are interested in quasilinear and fully nonlinear equations (1) and
(2) and the corresponding inequalities

$\displaystyle-\Delta_{p}u$ $\displaystyle\geq F(u,x),\,\,{\rm
and}\,\,F_{k}[-u]\geq F(u,x),\,\,u\geq 0\,\,\in\Omega.$

The inequalities can also be written as inhomogeneous equations with measure
data,

$-\Delta_{p}u=F(u,x)+\mu,\,\,F_{k}[-u]=F(u,x)+\mu,\,\,u\geq 0\,\,{\rm
in}\,\,\Omega,$ (3)

where $\mu$ is a nonnegative Borel measure on $\Omega$.

When $F(u,x)=u^{q}$, a major breakthrough on the existence of solutions to (3)
is made by [PV1]. The difficulties in studies of such equations and
inequalities lie in the competing nonlinearities. The argument of [PV1] relies
largely on nonlinear potential theory [KM2, LA]. Several necessary and
sufficient conditions are given in [PV1]. For example, let’s revise one of the
theorems in [PV1]. First, for $s>1,0\leq\alpha<\frac{N}{s}$ and
$0<T\leq\infty$, the $T$-truncated Wolff potential of a nonnegative Radon
measure $\mu$ is defined by

$W_{\alpha,s}^{T}[\mu](x)=\int_{0}^{T}(\frac{\mu(B_{t}(x))}{t^{N-\alpha
s}})^{\frac{1}{s-1}}\frac{dt}{t}.$

######  Theorem 1.1.

(Theorem 2.10 of [PV1])

The following two items are equivalent:

  1. 1.

there exists a nonnegative renormalized solution $u\in L^{q}(\Omega)$

$\begin{cases}-\Delta_{p}u=u^{q}+\varepsilon\omega,&{\rm in}\,\,\Omega,\\\
u=0&{\rm on}\,\,\partial\Omega\end{cases}$

for some $\varepsilon>0$;

  2. 2.

For all compact sets $E\subset\Omega,$

$\omega(E)\leq CCap_{G_{p},\frac{q}{q-p+1}}(E);$ (4)

  3. 3.

The testing inequality

$\int_{B}[W_{1,p}^{2R}\omega_{B}(x)]^{q}\leq C\omega(B)$

holds for all balls $B$ such that $B\cap{\rm supp}\omega\neq\emptyset;$

  4. 4.

There exists a constant $C$ such that

$W_{1,p}^{2R}(W_{1,p}^{2R}\omega)^{q}(x)\leq
CW_{1,p}^{2R}\omega(x),a.e.on\,\Omega.$

The equivalence conditions may be understood in the following way: the measure
can only concentrate in a relatively mild way (Condition 2) such that $u^{q}$
(or equivalently $(W_{1,p}^{2R}\omega)^{q}$) plays a minor role compared with
$\varepsilon\omega$(Condition 3 and $4$).

For generalizations, it is natural to consider the case when $F(u,x)$ is of
exponential type. In [NV], the authors considered the case when
$F(u,x)=H_{l}(\alpha u^{\beta}),\alpha>0,\beta\geq 1$, where

$H_{l}=e^{r}-\sum_{j=0}^{l-1}\frac{r^{j}}{j!}.$ (5)

Also we notice that, they consider $p$-Laplacian case $1<p<N$ and $k$-Hessian
case $1<k<\frac{N}{2}$. To get the existence of the solutions, the condition
of the type

$\|M_{p,2{\rm
diam}(\Omega)}^{\frac{(p-1)(\beta-1)}{\beta}}[\mu]\|_{L^{\infty}(\mathbb{R}^{N})}\leq
M$ (6)

is imposed, where

$M_{\alpha,T}^{\eta}[\mu](x)=\sup\\{\frac{\mu(B_{t}(x))}{t^{N-\alpha}h_{\eta}(t)}:0<t\leq
T\\},$

where

$h_{\eta}(t)=(-\ln t)^{-\eta}\chi_{(0,2^{-1}]}(t)+(\ln
2)^{-\eta}\chi_{[2^{-1},\infty)}(t).$

More general results are given in [NGU]. For example, the parabolic equation
is considered with the right hand side containing gradient terms. Also (6)
type conditions are required.

However, up to our knowledge, the following type equations are not studied

$-\Delta_{N}u=H_{l}(u)+\mu,\,\,F_{\frac{N}{2}}[-u]=H_{l}(u)+\mu,\,\,x\in\Omega,$

where $\Omega$ is a bounded domain and $\mu$ is a nonnegative Radon measure
and $H_{l}(u)$ is given by (5). The operators $\Delta_{N}$ and
$F_{\frac{N}{2}}$ are both borderline operators, which have fundamental
solutions of logarithm type. Our main results are list below.

######  Theorem 1.2.

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain, $l\in\mathbb{N}$ and
$l\geq N$ and $\mu$ is a nonnegative Radon measure supported in $\Omega$. Then
there exists $M>0$ depending only on $N,l$ such that if

$\mu(\Omega)\leq M,$

then the following Dirichlet problem

$\begin{cases}-\Delta_{N}u&=H_{l}(u)+\mu,\,\,{\rm in}\,\,\Omega,\\\
u&=0,\,\,{\rm on}\,\,\partial\Omega,\end{cases}$ (7)

admits a nonnegative renormalized solution $u$ which satisfies

$u(x)\leq C(N,p)W_{1,N}^{2{\rm diam}(\Omega)}[\bar{\mu}](x),\forall
x\in\Omega.$

Concerning the $k$-Hessian operator we recall some notions introduced by
Trudinger and Wang ([TW1, TW2, TW3]). For $k=1,\cdots,N$ and $u\in
C^{2}(\Omega)$ the $k$-Hessian operator $F_{k}$ is defined by

$F_{k}[u]=S_{k}(\lambda(D^{2}u)),$

where $\lambda(D^{2}u)=\lambda=(\lambda_{1},\cdots,\lambda_{N})$ denotes the
eigenvalues of $D^{2}u$, and

$S_{k}=\sum_{1\leq i_{1}<\cdots<i_{k}\leq
N}\lambda_{i_{1}}\cdots\lambda_{i_{k}}$

is the $k$th elementary symmetric polynomial of $\lambda$. For equations of
type (2), we always seek for $k$-admissible solutions, which satisfy

$\lambda(-\nabla^{2}u)\in\Gamma_{k}=\\{\lambda;S_{1}(\lambda)>0,\cdots,S_{k}(\lambda)>0\\}.$

For the existence of such solutions, we assume $\partial\Omega$ satisfies a
uniformly $(k-1)$-convex condition, that is

$S_{j}(\kappa)>0,\,\,{\rm on}\,\,\partial\Omega,\,\,{\rm
for}\,\,j=1,\cdots,k-1.$

where $\kappa=(\kappa_{1},\cdots,\kappa_{N-1})$ denote the principal
curvatures of $\partial\Omega$ with respect to its outward normal.

We refer to Definition 3 for the definition of $k$-subharmonic functions and
$\Phi^{k}(\Omega)$.

######  Theorem 1.3.

Let $l\in\mathbb{N}$ and $l>\frac{N}{2}$. Let $\Omega$ be a bounded uniformly
$(\frac{N}{2}-1)$ convex in $\mathbb{R}^{N}$. Let $\mu$ be a nonnegative Radon
measure, which has compact support in $\Omega$. Then there exists $M>0$, such
that if

$\mu(\Omega)\leq M$

then the following Dirichlet problem

$\begin{cases}F_{k}[u]&=H_{l}(u)+\mu,\,\,{\rm in}\,\,\Omega,\\\ u&=0,\,\,{\rm
on}\,\,\partial\Omega,\end{cases}$ (8)

admits a nonnegative solution $u\in\Phi^{k}(\Omega)$, continuous near
$\partial\Omega$, which satisfies

$u(x)\leq 2K_{2}W_{\frac{2N}{N+2},\frac{N+2}{2}}^{2{\rm
diam}(\Omega)}[\mu](x)+b,\forall x\in\Omega.$

######  Remark.

Here we note that a major difference between our results Theorem 1.2, 1.3 and
the results of Theorem 1.1, the results of [NV] is that, we do not need the
assumptions of the type (4) or (6). We only need $\mu(\Omega)$ to be small.

##  2 Estimates of the potentials

We use $B_{r}(a)$ to denote the ball of radius $r$ and with center $a$. We use
$\mu_{L}$ to denote the Lebesgue measure of $\mathbb{R}^{N}$. We use ${\rm
diam}(\Omega)$ to denote the diameter of , i.e.
$\sup\\{d(x,y);x,y\in\Omega\\}$. For a finite Radon measure supported on a
domain $\Omega$, we may sometimes consider it as a Radon measure on
$\mathbb{R}^{n}$ by extending it to be $0$ outside $\Omega$. If $D$ is a
subset of $\Omega$, we will use $\mu_{D}$ to represent $\mu$ restricted on
$D$.

We will usually use $C$ to denote a uniform positive constant and use
$C(R_{1},\cdots,R_{s})$ to denote a positive constant which depends on $R_{1}$
to $R_{s}$.

######  Definition 1.

Given a nonnegative Radon measure $\mu$ on $\mathbb{R}^{N}$, we define the
maximal function of $\mu$ as

$M_{\mu}(x):=\sup_{r>0}\frac{\mu(B_{r}(x))}{|B_{r}(x)|},\,\,x\in E.$

The following weak $(1,1)$ type Hardy-Littlewood maximal inequality about
measure is proved in [FO].

######  Lemma 2.1.

Suppose $\mu$ is a finite Radon measure on $\mathbb{R}^{N}$. There exists
$C(N)>0$ such that

$|\\{x\in\mathbb{R}^{N}:M_{\mu}(x)>\lambda\\}|\leq\frac{C(N)}{\lambda}\mu(\mathbb{R}^{N}),\,\,\forall\lambda>0.$

To prove our main theorem, we now generalize a result of Ma and Qing, [MQ,
Proposition 4.1].

######  Lemma 2.2.

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain with diameter $D$. And
let $\mu$ be a finite nonnegative Radon meaure support in $\Omega$. We let
$1<p\leq N,\alpha=\frac{N}{p}$. Then, for any $\delta\in(0,1)$

$\int_{\Omega}{\rm
exp}\left(\frac{N(1-\delta)W_{\alpha,p}^{R}[\mu](x,D)}{\mu(\Omega){}^{\frac{1}{p-1}}}\right)dx\leq\frac{c(N)|B(0,D)|}{\delta^{N+1}}.$

###### Proof.

The proof follows similarly from [MQ, Proposition 4.1]. For the convenience of
the readers, we present a proof here. To start, we choose $p-1<q<N$ and assume
$\mu(B(x,D))=1.$ Then

$\displaystyle W_{\alpha,p}^{D}[\mu](x)$
$\displaystyle\leq\int_{0}^{D}\mu(B(x,t))^{\frac{1}{q}}\frac{dt}{t}$
$\displaystyle=\mu(B(x,t))^{\frac{1}{q}}\log
t|_{0}^{D}+\int_{0}^{D}\log\frac{1}{t}d\mu(B(x,t))^{\frac{1}{q}}.$

Since

$\mu(B(x,t))\leq|B(0,t)|M_{\mu}(x)=\omega_{N}t^{N}M_{\mu}(x)$

we have

$\mu(B(x,t))^{\frac{1}{q}}\log t|_{0}^{D}=\log D$

almost everywhere.

Then by Jensen’s inequality

$\displaystyle\exp(W_{\alpha,p}^{D}[\mu](x))$ $\displaystyle\leq
D\int_{0}^{D}\frac{1}{t}d\mu(B(x,t))^{\frac{1}{q}}$ $\displaystyle\leq
D(\frac{1}{t}\mu(B(x,t))^{\frac{1}{q}}|_{0}^{D}+\int_{0}^{D}\mu(B(x,t))^{\frac{1}{q}}\frac{1}{t^{2}}dt).$

Since $q<N$, we know
$\frac{1}{t}\mu(B(x,t))^{\frac{1}{q}}|_{0}^{D}=\frac{1}{D}$ almost everywhere.

Then

$\displaystyle\exp(W_{\alpha,p}^{D}[\mu](x))$ $\displaystyle\leq
1+D^{\frac{N}{q}}\frac{1}{\frac{N}{q}-1}\omega_{N}^{\frac{1}{q}}M_{\mu}(x)^{\frac{1}{q}}.$

So for $\lambda\geq 2$, we have

$\displaystyle|\\{x\in\Omega;\exp(W_{\alpha,p}^{D}[\mu](x))\geq\lambda\\}|$
$\displaystyle\leq|\left\\{x\in\Omega;M_{\mu}(x)\geq\frac{(N-q)^{q}\lambda^{q}}{2^{q}q^{q}|B(0,D)|}\right\\}|$
$\displaystyle\leq\frac{C(N)2^{q}q^{q}|B(0,D)|}{(N-q)^{q}\lambda^{q}},$

where the last inequality is due to Lemma 2.1.

For $\delta\in(0,1)$, we fix some $q>\max\\{p-1,N(1-\frac{\delta}{2})\\}$ and
$q<N$

$\displaystyle\int_{\Omega}{\rm exp}(N(1-\delta)W_{\alpha,p}^{D}[\mu](x))dx$
$\displaystyle=\int_{0}^{+\infty}|\\{x\in\Omega;{\rm
exp}(W_{\alpha,p}^{\mu}(x,D))\geq t^{\frac{1}{N(1-\delta)}}\\}|$
$\displaystyle\leq\int_{2^{N(1-\delta)}}^{+\infty}\frac{C(N)2^{q}q^{q}|B(0,D)|}{(N-q)^{q}t^{\frac{q}{N(1-\delta)}}}dt+\int_{0}^{2^{N(1-\delta)}}|\Omega|dt$
$\displaystyle\leq C(N)\frac{|B(0,D)|}{\delta^{N+1}}.$

For the case $\mu(\Omega)\neq 1$, we consider $\frac{\mu}{\mu(\Omega)}$
instead and finally prove the claim.

∎

######  Lemma 2.3.

Let $2\leq p\leq N$ and $\alpha=\frac{N}{p}$. Let $\mu$ be a nonnegative Radon
measure supported on $\Omega$ satisfies

$\mu(\Omega)\leq 1$

and set $\bar{\mu}=\frac{\mu_{L}}{\mu_{L}(B_{10{\rm diam}(\Omega)})}+\mu$. We
fix $R\in(\frac{1}{10}{\rm diam}(\Omega),10{\rm diam}(\Omega))$. Then there
exist uniform positive constants $C_{1},\delta_{0}$ such that

$\|W_{\alpha,p}^{R}[\exp(\delta_{0}W_{\alpha,p}^{R}[\bar{\mu}])]\|_{L^{\infty}(\mathbb{R}^{N})}\leq
C_{1},$ (9)

and

$W_{\alpha,p}^{R}[\exp(\delta_{0}W_{\alpha,p}^{R}[\bar{\mu}])]\leq
C_{1}W_{\alpha,p}^{R}[\bar{\mu}]$ (10)

in $\Omega$.

###### Proof.

First we have

$\bar{\mu}(\Omega)\leq 2.$

For $0<r<R$, and $y\in\Omega$, we have

$\displaystyle W_{\alpha,p}^{R}[\bar{\mu}](y)$
$\displaystyle=W_{\alpha,p}^{r}[\bar{\mu}](y)+\int_{r}^{R}\bar{\mu}(B_{t}(y))^{\frac{1}{p-1}}\frac{dt}{t}$
$\displaystyle\leq W_{\alpha,p}^{r}[\bar{\mu}](y)+C(\log R-\log r)$
$\displaystyle\leq W_{\alpha,p}^{r}[\bar{\mu}](y)+C\log\frac{1}{r}+C(R).$

From $\exp((a+b)/2)\leq(\exp(a)+\exp(b))/2$, for some
$\theta\in(0,2^{-\frac{1}{N-1}}]$ to be fixed later, we have

$\displaystyle\exp(\theta
N(1-\delta)2^{-2}W_{\alpha,p}^{R}[\bar{\mu}(y)])\leq$
$\displaystyle\frac{1}{2}\exp(\frac{1}{2}N(1-\delta)W_{\alpha,p}^{r}[\bar{\mu}](y))$
$\displaystyle+C(R)\exp(\theta C\log\frac{1}{r})$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\exp(\frac{1}{2}N(1-\delta)W_{\alpha,p}^{r}[\bar{\mu}](y))$
$\displaystyle+C(R)r^{-\theta C_{2}}.$

Then for $r>0$, from Lemma 2.2, we have

$\displaystyle\int_{B_{r}(x)}\exp(\frac{1}{2}N(1-\delta)W_{\alpha,p}^{r}[\bar{\mu}](y))dy\leq$
$\displaystyle\int_{B_{r}(x)}\exp(\frac{N(1-\delta)W_{\alpha,p}^{r}[\bar{\mu}](y)}{\bar{\mu}(B_{2r}(x))^{\frac{1}{p-1}}})dy$
$\displaystyle\leq$ $\displaystyle Cr^{N}.$

Therefore by choosing $\theta>0$ small such that $\theta C_{2}\leq\frac{1}{2}$
we have

$\displaystyle W_{\alpha,p}^{R}[\exp(\theta
N(1-\delta)2^{-2}W_{\alpha,p}^{R}[\bar{\mu}][y])](x)$
$\displaystyle\leq\int_{0}^{R}(Cr^{N}+Cr^{-\theta
C_{1}}r^{N})^{\frac{1}{p-1}}\frac{dr}{r}$ $\displaystyle\leq C.$

Then we may choose $\delta_{0}=\frac{N(1-\delta)}{8C_{2}}$ and get (9).

Also note that

$\displaystyle W_{\alpha,p}^{R}[\bar{\mu}](x)$
$\displaystyle=\int_{0}^{R}\bar{\mu}(B(x,t))^{\frac{1}{p-1}}\frac{dt}{t}$
$\displaystyle\geq\int_{0}^{R}(\frac{\mu_{L}(B(x,t))}{\mu_{L}(B_{10{\rm
diam}(\Omega)})})^{\frac{1}{p-1}}\frac{dt}{t}$ $\displaystyle\geq C>0.$

Then (10) follows.

∎

Then we have

######  Theorem 2.4.

Let $2\leq p\leq N$ and $\alpha=\frac{N}{p}$. Suppose $\mu$ is a finite
nonnegative Radon measure on $\Omega$. Assume $R\in[\frac{1}{10}{\rm
diam}(\Omega),10{\rm diam}(\Omega)]$ and $K>0$ are positive real numbers. We
fix an integer $l>p-1$. Suppose $\\{u_{m}\\}$ is a sequence of nonnegative
measurable functions on $\mathbb{R}^{N}$ which satisfies

$\displaystyle u_{0}$ $\displaystyle\leq KW_{\alpha,p}^{R}[\mu],$ (11)
$\displaystyle u_{m+1}$ $\displaystyle\leq
KW_{\alpha,p}^{R}[H_{l}(u_{m})+\mu],\forall m\in\mathbb{N}.$ (12)

Then there exists $M>0$ depending only on $N,l,K,\delta_{0},C_{1}$ such that
if

$\displaystyle\mu(\Omega)\leq M,$ (13) $\displaystyle\bar{\mu}=$
$\displaystyle M\frac{\mu_{L}}{\mu_{L}(B_{10{\rm diam}(\Omega)}(0))}+\mu,$
(14)

there holds

$H_{l}(2KW_{\alpha,p}^{R}[\bar{\mu}])\in L^{s}(\Omega),\,\,{\rm for}\,\,{\rm
some}\,\,s>1$ (15)

and

$u_{m}\leq 2KW_{\alpha,p}^{R}[\bar{\mu}],\forall m\in\mathbb{N}.$ (16)

###### Proof.

First prove that, we may choose $M>0$ small such that when $\mu$ and
$\bar{\mu}$ satisfy (13)(14), then

$\displaystyle H_{l}(2KW_{\alpha,p}^{R}[\bar{\mu}])\in L^{s}(\Omega),$
$\displaystyle\,\,{\rm for}\,\,{\rm some}\,\,s>1,$ $\displaystyle
W_{\alpha,p}^{R}[H_{l}(2K\cdot W_{\alpha,p}^{R}[\bar{\mu}])]$
$\displaystyle\leq W_{\alpha,p}^{R}[\bar{\mu}].$

By Lemma 2.3, there exists $C>0$ and $\delta_{0}>0$ independent of $\bar{\mu}$
such that $\exp(\delta_{0}W_{\alpha,p}^{R}[M^{-1}\bar{\mu}])$ is integrable in
$\Omega$ and

$W_{\alpha,p}^{R}[\exp(\delta_{0}W_{\alpha,p}^{R}(M^{-1}\bar{\mu}))]\leq
C_{1}W_{\alpha,p}^{R}[M^{-1}\bar{\mu}]\,\,{\rm in}\,\,\bar{\Omega}.$

Here in fact we choose $\delta_{0}$ even smaller, so that
$\exp(\delta_{0}W_{\alpha,p}^{R}[M^{-1}\bar{\mu}])\in L^{s}(\Omega)$ for some
$s>1$.

Since $l>p-1\geq 1$, then for all $t\geq 0$ and $0<\theta\leq 1$

$\displaystyle\theta^{-l}H_{l}(t)$
$\displaystyle=\theta^{-l}(e^{t}-\sum_{i=1}^{l-1}\frac{t^{j}}{j!})=\theta^{-l}\sum_{i=l}^{\infty}\frac{t^{j}}{j!}\leq\sum_{i=l}^{\infty}\frac{(\theta^{-1}t)^{j}}{j!}=H_{l}(\theta^{-1}t).$

It follows that

$\displaystyle
W_{\alpha,p}^{R}[M^{-\frac{1}{2}(\frac{l}{p-1}+1)}H_{l}(\delta_{0}M^{-\frac{1}{2}(\frac{1}{p-1}-\frac{1}{l})}W_{\alpha,p}^{R}[\bar{\mu}])]$
$\displaystyle\leq$ $\displaystyle
W_{\alpha,p}^{R}[H_{l}(\delta_{0}M^{-\frac{1}{p-1}}W_{\alpha,p}^{R}[\bar{\mu}])]$
$\displaystyle\leq$ $\displaystyle
W_{\alpha,p}^{R}[\exp(\delta_{0}W_{\alpha,p}^{R}[M^{-1}\bar{\mu}])]$
$\displaystyle\leq$ $\displaystyle
C_{1}M^{-\frac{1}{p-1}}W_{\alpha,p}^{R}[\bar{\mu}].$

Then

$W_{\alpha,p}^{R}[H_{l}(\delta_{0}M^{-\frac{1}{2}(\frac{1}{p-1}-\frac{1}{l})}W_{\alpha,p}^{R}[\bar{\mu}])]\leq
C_{1}M^{\frac{1}{2(p-1)}(\frac{l}{p-1}-1)}W_{\alpha,p}^{R}[\bar{\mu}].$

Then if we choose
$M=\min\\{(\frac{\delta_{0}}{2K})^{\frac{2(p-1)l}{l-p+1}},C_{1}^{-\frac{2(p-1)^{2}}{l-p+1}}\\}$,
we have $H_{l}(2KW_{\alpha,p}^{R}[\bar{\mu}])\in L^{s}(\Omega)$ for some $s>1$
and

$W_{\alpha,p}^{R}[H_{l}(2KW_{\alpha,p}^{R}[\bar{\mu}])]\leq
W_{\alpha,p}^{R}[\bar{\mu}].$ (17)

Now we will prove (16) by induction. From (11), (16) holds for $m=0$. Now we
assume (16) holds with $m=k$. Then from (12) and $(\ref{potential iteration
inequality})$

$\displaystyle u_{k+1}$ $\displaystyle\leq
KW_{\alpha,p}^{R}[H_{l}(u_{k})+\bar{\mu}]$ $\displaystyle\leq
KW_{\alpha,p}^{R}[H_{l}(2KW_{\alpha,p}^{R}[\bar{\mu}])+\bar{\mu}]$
$\displaystyle\leq
KW_{\alpha,p}^{R}[H_{l}(2KW_{\alpha,p}^{R}[\bar{\mu}])]+KW_{\alpha,p}^{R}[\bar{\mu}]$
$\displaystyle\leq 2KW_{\alpha,p}^{R}[\bar{\mu}].$

Then we completes the proof.

∎

##  3 Quasilinear equations with Dirichlet boundary conditions

In this section, we use the estimates in the last section to study the
existence of solutions to

$\begin{cases}-\Delta_{N}u&=H(u)+\mu,\,\,{\rm in}\,\Omega\\\ u&=0,\,\,{\rm
on}\,\partial\Omega.\end{cases}$ (18)

First we need some basic knowledges related to $p$-Laplacian equations.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$, $1<p\leq N$ and
$p^{\prime}=\frac{p}{p-1}$. We use $\mathfrak{M}_{b}(\Omega)$ to deonte the
set of all finite Radon measures on $\Omega$. If
$\mu\in\mathfrak{M}_{b}(\Omega)$, by Jordan decomposition, we denote $\mu^{+}$
and $\mu^{-}$ as the positive part and negative part of $\mu$ and use
$\mathfrak{M}_{b}^{+}$ to denote the set of nonnegative Radon measures . We
use $\mathfrak{M}_{0,p}(\Omega)$ to denote the space of measures in $\Omega$
which are absolutely continuous with repect to $c_{1,p}^{\Omega}$-capacity,
which for any compact subset $K\subset\Omega$, is defined by

$c_{1,p}^{\Omega}(K)=\inf\left\\{\int_{\Omega}|\nabla\phi|^{p}dx;\phi\in
C_{0}^{\infty}(\Omega),\phi\geq\chi_{K}\right\\},$

where

$\chi_{E}=\begin{cases}1,&x\in E,\\\ 0,&{\rm otherwise}.\end{cases}$

Also, we use $\mathfrak{M}_{s,p}(\Omega)$ to denote the space of measures in
$\Omega$ supported on a set of zero $c_{1,p}^{\Omega}$-capacity. We know, any
$\mu\in\mathfrak{M}_{b}(\Omega)$ can be decomposed as $\mu=\mu_{0}+\mu_{s}$ in
a unique way, such that $\mu_{0}\in\mathfrak{M}_{0,p}\cap\mathfrak{M}_{b}$ and
$\mu_{s}\in\mathfrak{M}_{s,p}\cap\mathfrak{M}_{b}$. And it is well known that
$\mu_{0}$ part can be written as $f-{\rm div}g$ where $f\in L^{1}(\Omega)$ and
$g\in L^{p^{\prime}}(\Omega,\mathbb{R}^{N})$, see [DMOP].

For $k>0$ and $s\in\mathbb{R}$, we set $T_{k}(s)=\max\\{\min\\{s,k\\},-k\\}.$
From [BBGGPV], if $u$ is a measurable function on $\Omega,$ finite almost
everywhere such that $T_{k}(u)\in W_{{\rm loc}}^{1,p}(\Omega)$ for any $k>0$,
there exists a measurable function $v(x):\Omega\to\mathbb{R}^{N}$ such that
$\nabla T_{k}(u)=\chi_{\\{|u|\leq k\\}}(x)v(x)$ almost everywhere in $\Omega$
and for all $k>0$. We define $\nabla u=v$. Now we recall the definition of a
renormalized solution given in [DMOP].

######  Definition 2.

Let $1<p\leq N$ and $\mu=\mu_{0}+\mu_{s}\in\mathfrak{M}_{b}(\Omega)$ . A
measurable function $u$ defined in $\Omega$ and finite almost everywhere is
called a renormalized solution of

$\begin{cases}-\Delta_{p}u&=\mu,\,\,{\rm in}\,\,\Omega,\\\ u&=0,\,\,{\rm
on}\,\,\partial\Omega,\end{cases}$ (19)

if $T_{k}(u)\in W_{0}^{1,p}(\Omega)$ for any $k>0$, $|\nabla u|^{p-1}\in
L^{r}(\Omega)$ for any $1<r<\frac{n}{n-1}$, and $u$ has the property that for
any $k>0$ there exist $\lambda_{k}^{+}$ and $\lambda_{k}^{-}$ belonging to
$\mathfrak{M}_{b}^{+}\cap\mathfrak{M}_{0}(\Omega)$, respectively concentrated
on $u=k$ and $u=-k$ with the property that
$\mu_{k}^{+}\to\mu_{s}^{+},\mu_{k}^{-}\to\lambda_{s}^{-}$ in the narrow
topology of measures (that is, with test funciton $\varphi\in
L^{\infty}(\Omega)\cap C(\Omega)$) and such that

$\int_{\\{|u|<k\\}}|\nabla u|^{p-2}\nabla u\nabla\phi
dx=\int_{\\{|u|<k\\}}\phi d\mu_{0}+\int_{\Omega}\phi
d\lambda_{k}^{+}-\int_{\Omega}\phi d\lambda_{k}^{-},$

for every $\phi\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$.

Thanks to [DMOP], we have the important stability result.

######  Theorem 3.1.

Let $1<p\leq N$, $\mu=\mu_{0}+\mu_{s}^{+}-\mu_{s}^{-}$ with $\mu_{0}=F-{\rm
div}g\in\mathfrak{M}_{0}(\Omega)$ and
$\mu_{s}^{+},\mu_{s}^{-}\in\mathfrak{M}_{s}(\Omega)\cap\mathfrak{M}^{+}(\Omega).$
Let $\mu_{n}=F_{n}-{\rm div}g_{n}+\rho_{n}-\eta_{n}$ with $F_{n}\in
L^{1}(\Omega)$, $g_{n}\in L^{p^{\prime}}(\Omega)$ and
$\rho_{n},\eta_{n}\in\mathfrak{M}_{b}^{+}\cap\mathfrak{M}_{s}$. Assume
$F_{n}\overset{L^{1}}{\rightharpoonup}F$ weakly,
$g_{n}\overset{L^{p^{\prime}}}{\to}g$ strongly and ${\rm div}g_{n}$ is bounded
in $\mathfrak{M}_{b}$; Assume also that
$\rho_{n}\to\mu_{s}^{+},\eta_{n}\to\mu_{s}^{-}$ in the narrow topology. If
$u_{n}$ is a sequence renormalized solutions to (18) with $\mu=\mu_{n}$, then,
up to a subsequence, it converges a.e.in $\Omega$ to a renormalized solution
$u$ of (19). Furthermore, $T_{k}(u_{n})$ converges to $T_{k}(u)$ in
$W_{0}^{1,p}(\Omega)$ for any $k>0$.

Also we need the following estimate proved in [PV1], which extends Theorem 1.6
of [KM2] slightly.

######  Theorem 3.2.

(Theorem 2.1 of [PV1])

Let $1<p\leq N$, $\Omega$ be a bounded domain of $\mathbb{R}^{N}$. Then there
exists a constant $K_{1}>0$, depending only on $p$ and $N$ such that if
$\mu\in\mathfrak{M}_{b}^{+}(\Omega)$ and $u$ is a nonnegative renormalized
solution of (19) with data $\mu,$ there holds

$\frac{1}{K_{1}}W_{1,p}^{\frac{d(x,\partial\Omega)}{3}}[\mu](x)\leq u(x)\leq
K_{1}W_{1,p}^{2{\rm diam}(\Omega)}[\mu](x),\forall x\in\Omega$

where the positive constant $K_{1}$ only depends on $n,p$.

The following result is also due to [PV1].

######  Theorem 3.3.

(Lemma 6.9 of [PV1])

Let $\mu,\nu\in\mathfrak{M}_{b}^{+}(\Omega)$ and $\mu(D)\geq\nu(D)$ for all
$D\subset\Omega$. Suppose that $u$ is a renormalized solution of

$\begin{cases}-\Delta_{p}u&=\mu\,\,{\rm in}\,\,\Omega,\\\ u&=0\,\,{\rm
on}\,\,\partial\Omega.\end{cases}$

Then there exists $v\geq u$ such that

$\begin{cases}-\Delta_{p}v&=\nu\,\,{\rm in}\,\,\Omega,\\\ v&=0\,\,{\rm
on}\,\,\partial\Omega,\end{cases}$

in the renormalized sense.

Now we can prove our main theorem.

###### Proof.

(of Theorem 1.2) We are going to define a sequence of nonnegative renormalized
solutions $\\{u_{m}\\}_{m\in\mathbb{N}}$ by

$\begin{cases}-\Delta_{N}u_{0}&=\mu,\,\,{\rm in}\,\,\Omega,\\\
u_{0}&=0,\,\,{\rm on}\,\,\partial\Omega,\end{cases}$

and

$\begin{cases}-\Delta_{N}u_{m+1}&=H_{l}(u_{m})+\mu,\,\,{\rm in}\,\,\Omega,\\\
u_{m+1}&=0,\,\,{\rm on}\,\,\partial\Omega.\end{cases}$

Choose $M$ as in Theorem 2.4, which depends on $N,l,K=K_{1},\delta_{0},C_{1}$,
where $K_{1}$ is the constant which appears in Theorem 3.2. We assume $\mu$ is
a nonnegative Radon measure with $\mu(\Omega)\leq M$.

By Theorem 3.2, as long as we can solve the above equations and get
$u_{m},\,\,m\geq 0$, we will have estimates (11)(12) in Theorem 2.4, with $K$
replaced by $K_{1}$.

Then from (15)(16), we know $H_{l}(u_{m})\in L^{1}$. Then from Theorem 3.3, we
can continue to solve the above equation and get $u_{m+1}$ and $u_{m+1}\geq
u_{m}$.

Since

$u_{m}\leq 2KW_{1,N}^{R}[\bar{\mu}],m\in\mathbb{N},$

$u_{m}$ converges, a.e. in $\Omega$, to some function $u$ with

$u\leq 2KW_{1,N}^{R}[\bar{\mu}],\forall x\in\Omega.$

From (15) we know

$H_{l}(u_{m})\to H_{l}(u),\,\,{\rm in}\,\,L^{1}(\Omega).$

Then by Theorem 3.1, we know $u$ is a renormalized solution of (3.17).

∎

##  4 $N/2$-Hessian equation with Dirichlet boundary condition

Assume $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with $C^{2}$
boundary. For $k=1,\cdots,N$ and $u\in C^{2}(\Omega)$ we define

$F_{k}[u]=S_{k}(\lambda(D^{2}u))$

where $\lambda(D^{2}u)=\lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{N})$
deontes the eigenvalues of the Hessian matrix of $D^{2}u$ and $S_{k}$ is the
$k$th elementary symmetric polynomial that is

$S_{k}(\lambda)=\sum_{1\leq i_{1}<\cdots<i_{k}\leq
N}\lambda_{i_{1}}\cdots\lambda_{i_{k}}.$

We assume that $\partial\Omega$ is uniformly $(k-1)$-convex.

######  Definition 3.

An upper-semicontinuous function $u:\Omega\to[-\infty,\infty)$ is $k$-convex
(or $k$-subharmonic) if, for any $D\Subset\Omega$ for every function $v\in
C^{2}(D)\cap C(\bar{D})$ satisfying $F_{k}[v]\leq 0$ in $\bar{D}$, the
following implication holds

$u\leq v\,\,{\rm on}\,\,\partial D\Longrightarrow u\leq v\,\,{\rm in}\,\,D.$

We denote by $\Phi^{k}(\Omega)$ the set of all $k$-subharmonic functions in
$\Omega$ which are not identically equal to $-\infty$.

The following weak convergence result for $k$-Hessian operators proved in
[TW2] is of fundamental importance in our study.

######  Theorem 4.1.

Let $\Omega$ be a bounded $(k-1)$-convex domain of $\mathbb{R}^{N}$. For each
$u\in\Phi^{k}(\Omega)$, there exists a nonnegative Radon measure $\mu_{k}[u]$
in $\Omega$ such that

  1. 1.

$\mu_{k}[u]=F_{k}[u]$ for $u\in C^{2}(\Omega)$;

  2. 2.

If $\\{u_{n}\\}$ is a sequence of $k$-convex functions which converges a.e. to
$u$, then $\mu_{k}[u_{n}]\to\mu_{k}[u]$ in the weak sense of measures.

Also we need the following results proved in [PV1].

######  Theorem 4.2.

Let $\mu$ be a nonnegative finite Radon measure such that

$\mu=\mu^{\prime}+f$

where $\mu^{\prime}$ is a nonnegative Radon measure compactly supported in
$\Omega,$ and $f\geq 0,f\in L^{s}(\Omega)$ with $s>\frac{N}{2k}$, if $1\leq
k\leq\frac{N}{2}$, and $s=1$ if $\frac{N}{2}<k\leq N$. Suppose that $-u$ is a
nonpositive $k$-subharmonic function in $\Omega$, that is continuous near
$\partial\Omega$ and solves the equation

$\begin{cases}F_{k}[-u]&=\mu\,\,{\rm in}\,\,\Omega,\\\ u&=0\,\,{\rm
on}\,\,\partial\Omega.\end{cases}$

Then there is a constant $K=K(N,k)>0$ such that

$u(x)\leq KW_{\frac{2k}{k+1},k+1}^{2{\rm diam}(\Omega)}\mu$

for every $x\in\Omega$.

######  Theorem 4.3.

Let $\Omega,\mu,\phi$ and $u$ be as in the above theorem. Suppose $\nu$ is
another measure with $\nu=\nu^{\prime}+g$, where $\nu^{\prime}$ is a
nonneative measure compactly supported in $\Omega$, and $g\geq 0,g\in
L^{s}(\Omega)$ with $s>\frac{N}{2k},$ if $1\leq k\leq\frac{N}{2}$ and $s=1$ if
$\frac{N}{2}<k\leq N$. Then there exists a function $\omega$ such that
$-\omega\in\Phi^{k}(\Omega)$, $\omega\geq u$ and

$\begin{cases}F_{k}[-\omega]&=\mu+\nu,\,\,{\rm in}\,\,\Omega\\\
\omega&=0,\,\,{\rm on}\,\,\partial\Omega.\end{cases}$

In the end we prove Theorem 1.3. In fact the proof is similar to the proof of
Theorem 1.2 except for that we replace Theorem 3.1, 3.2, 3.3 by Theorem 4.1,
4.2, 4.3. One difference is that we use
$W_{\frac{2N}{N+2},\frac{N+2}{2}}^{D}[\mu]$ instead of $W_{1,N}^{D}[\mu]$.
Another difference is that when we apply Theorem 4.2, 4.3, the measure $\mu$
needs to be sufficiently regular near the boundary. Notice that from Theorem
2.4, we have $H_{l}(u_{m})\in L^{s}(\Omega)$ in the iteration procedure.

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