# Identifying research supporting the United Nations Sustainable Development
Goals

Yury Kashnitsky
Elsevier, the Netherlands
<EMAIL_ADDRESS>
Guillaume Roberge
Elsevier, Canada
<EMAIL_ADDRESS>
Jingwen Mu
Auckland University, New Zealand
Kevin Kang
Auckland University, New Zealand
Weiwei Wang
Auckland University, New Zealand
Maurice Vanderfeesten
Vrije Universiteit, the Netherlands
Maxim Rivest
Anatomist.ai, Canada
Lennart Keßler
University of Duisburg-Essen, Germany
Robert Jaworek
Palacký University, Czech Republic
Maéva Vignes
University of Southern Denmark
Bamini Jayabalasingham
Elsevier, the USA
Finne Boonen
Elsevier, the Netherlands
Chris James
Elsevier, the Netherlands
Marius Doornenbal
Elsevier, the Netherlands
Isabelle Labrosse
Elsevier, Canada

###### Abstract

The United Nations (UN) Sustainable Development Goals (SDGs) challenge the
global community to build a world where no one is left behind. Recognizing
that research plays a fundamental part in supporting these goals, attempts
have been made to classify research publications according to their relevance
in supporting each of the UN’s SDGs. In this paper, we outline the methodology
that we followed when mapping research articles to SDGs and which is adopted
by Times Higher Education in their Social Impact rankings. We also discuss
various aspects in which the methodology can be improved and generalized to
other types of content apart from research articles. The results presented in
this paper are the outcome of the SDG Research Mapping Initiative that was
established as a partnership between the University of Southern Denmark, the
Aurora European Universities Alliance (represented by Vrije Universiteit
Amsterdam), the University of Auckland, and Elsevier to bring together broad
expertise and share best practices on identifying research contributions to
UN’s Sustainable Development Goals.

## Introduction

Numerous approaches to mapping research to the United Nations (UN) Sustainable
Development Goals (SDGs)111http://metadata.un.org/sdg/ have been documented
[1, 2, 3, 4, 5]. These approaches vary with regard to the framework used to
define inclusion and exclusion criteria, the methodology employed to retrieve
publications, and the publication database used. For example, the approach to
defining inclusion and exclusion criteria may be set conservatively to limit
publications to those documenting actions made to achieve the SDG targets, or
conversely, may be set using a more liberal approach, thereby including any
papers that increase knowledge on the overall topic. With regard to the
methodology employed to retrieve publications, publication sets for a specific
SDG can use a Boolean approach only or be complemented by machine learning
algorithms. The source of publications that the methodology is applied to can
also introduce variability, given the availability of many data sources,
ranging from open access, subscription-based, or a mixture of both.

To date, there is no broadly agreed-upon methodology for mapping research to
the SDGs and existing methods produce quite different results [2]. A common
approach to identifying research related to a topic is to use Boolean search
expressions. The Boolean method involves the use of keywords, either alone or
in combination, using conditional functions and applied to specified text
sections (title, abstracts, keywords, etc.) of scientific publications, and
results in the exclusive retrieval of articles within which the defined search
expressions were found. The authors of [2] applied the Boolean method, taking
an approach to limit their SDG publication sets to publications with a direct
contribution to targets and/or indicators, with efforts made to reduce the
impact of issues raised on the Boolean technique, resulting in a more
restrictive publication set. Bordignon’s strategy [3] aimed at reducing the
polysemy of terms by limiting keywords from Elsevier 2020 queries [1] to
relevant subject areas using the All Science Journal Classification (ASJC). A
text-mining tool (CorTexT) was then used to enrich those selected
publications. The Aurora European Universities Alliance [6] developed and
released their 169 target-level SDG queries222https://github.com/Aurora-
Network-Global/sdg-queries also using keyword combinations, boolean- and
proximity operators. The University of Auckland [7] developed queries informed
by the researchers within their network, this resulted in a localized version
able to count in more papers that are specific to Australian and New Zealand
research topics. In [4], the authors employ a two-step approach, involving
building SDG-specific terms obtained from many sources (policy reports,
publications, forums, etc), applying a selection process to the terms, and
then using the terms to identify citation-based communities of publications.
However, as described in [2], this process involves challenges related to the
interpretation of the themes and concepts of the SDGs, decisions around which
publications to designate as a "contribution" to the chosen interpretation of
the SDG, and the translation of concepts into a search query that will
accurately identify publications.

An alternative or complementary approach to query-based methods involves using
machine learning to map research articles to SDGs: either in a supervised
manner, i.e. performing classification, or unsupervised manner, i.e.
performing clustering. Supervised methods typically resort to the same SDG
queries to obtain a labeled dataset to train the model [8, 9]. Clustering is
typically done with paper text representations or citation graphs where the
resulting clusters are later mapped to SDGs either directly or via
intermediate clusters, e.g. “topics” [10, 11]. Refer to [12] for an overview
of some more methods of classifying documents into SDGs. However they all face
the same challenges noted above, and machine learning further introduces the
problem of interpretability of the model predictions or the clusters attained.

Since 2018, Elsevier has endeavored to map research to the SDGs, releasing in
2020 publicly available queries to facilitate transparency and reproducibility
[1]. Herein, we describe the approach taken to improve former attempts to map
research to the SDGs, taking feedback into account, resulting in the creation
of a more comprehensive query set with sub-queries addressing targets and
indicators and the application of a machine learning model to increase recall.
This methodology (“Elsevier 2021 SDG mapping” [13]) captures on average twice
as many articles as the 2020 version while keeping precision above 80%. Times
Higher Education (THE) are using Elsevier SDG mapping as part of their Social
Impact rankings [14]. “Elsevier 2022 SDG mapping” (to be released in late
2022) is the most up-to-date simplified version of the queries & ML model
differing from the 2021 version in Covid-related enhancement to SDG 3 queries
and queries designed for SDG 17 “Partnerships for the goals”.

To evaluate the approach, the output generated using the developed methodology
was compared to the results generated by Digital Science [10], Aurora European
Universities Alliance [15], and the University of Auckland [7].

##  1 Methodology

###  1.1 Developing SDG queries

The SDGs are goals to achieve rather than research topics, each SDG
encompassing many targets. Using Boolean search expressions to build SDG-
specific publication sets presents many challenges. We implemented a bottom-up
approach to the construction of each SDG-relevant publication set, whereby
several sub-queries were first constructed for each SDG target, and then
aggregated at the SDG level.

####  1.1.1 Building a query for each target within an SDG

Criteria for delineating the publication sets relevant to each SDG were
designed by a team consisting of a minimum of four analysts and were based on
an extensive literature review done by the team to gain an understanding of
the SDG. As a first step, the SDG was further subdivided into themes to
facilitate the creation of specific criteria linked to specific SDG targets.
The criteria defined for each theme aimed to specify topics of focus as well
as any requirements for “action terms” in association with the topics. For
example, for the topic of “poverty” the action term “alleviate”, or other
action terms holding similar meaning, might be deemed a requirement. To ensure
homogeneity in the approach, criteria developed by the team of analysts were
submitted to a review committee consisting of both those on the SDG team and
those external to the team. The review committee was responsible for reviewing
the criteria, recommending changes, and final approval of the criteria. Table
1 presents the criteria for SDG1 overall (SDG1-Main) and subcategories related
to SDG1. These criteria were defined for each SDG and theme related to the
SDG.

Table 1: An example of SDG 1 subtopics and associated SDG targets. Subset code | Criteria | Associated target
---|---|---
SDG1-Main |  Research focused on poverty and research as defined for any SDG1-subset below. “Action term” specified: the action term, “alleviate” was applied to make the topic term “poverty” more specific. |  Target 1.1: eradicate extreme poverty Target 1.2: reduce poverty by half All Targets associated with SDG1-Subsets
SDG1-Theme1 |  Research focused on social programs, including all articles discussing social security systems related to health, finance, and work. No “action terms” were required for inclusion of the topics above. |  Target 1.3: Implement nationally appropriate social protection systems
SDG1-Theme2 |  Research focused on microfinance, access to property, inheritance, natural resources, new technologies as they relate to facilitating access, equality and human rights. “Action term” specified: the action term, “access to” was applied. |  Target 1.4: equal rights to economic resources and basic services
SDG1-Theme3 |  Research focused on resilience, exposure and vulnerability to disasters (financial, climate-related, social..), particularly on understanding poor and vulnerable people and communities. No “action terms” were required for inclusion of the topics above. |  Target 1.5: build the resilience of the poor
SDG1-Theme4 |  Research focused on financial aid, policies, government support (such as food banks, support distribution strategies), strategies to eradicate poverty. No “action terms” were required for inclusion of the topics above. |  Target 1A: Ensure significant mobilization of resources from a variety of sources Target 1B: Create sound policy frameworks

Following the establishment of criteria defining the research areas of focus
relevant for each SDG (overall and per SDG-Theme), these criteria were used to
guide the development of queries to retrieve publication sets. Where possible,
the analyst responsible for query development was selected due to subject
matter expertise in the field. Otherwise, the process was informed by a
literature review. An iterative approach was taken to assess the precision
with which individual keywords and sets of keywords identified publications
that met the criteria. Keywords from the Elsevier 2020 [1] and Aurora European
Universities Alliance [6] queries were assessed first. Additional keywords
were identified using term-frequency and inverse-document frequency (TF-IDF)
analyses of text from titles, abstracts, and author keywords from publications
meeting the criteria. Additional efforts were taken to identify publications
that may have been excluded based on the developed query. Specifically, (1)
the query results were analyzed to identify specialized journals that would be
expected to include a high percentage of publications that fit the criteria,
and (2) the citation network of the publications retrieved using the query was
assessed to identify publications within the citation network of the results
(i.e. publications citing or cited by the publications retrieved by the query)
that were not retrieved by the query. Publications from these specialized
journals or the citation network that were not being retrieved by the query
were assessed to identify additional keywords to include in the query to
increase recall. Relevant exclusions were built into the queries to increase
precision and could result in the exclusion of specific terms using Boolean
operators or the exclusion of fields of science deemed to be outside the scope
of the criteria.

To facilitate the continuous evaluation of the query, publications were
manually reviewed to assess their fit against the criteria. An evaluation of a
minimum of 100 random publications by two independent analysts was done to
support the calculation of precision metrics for each query and a minimum
precision threshold of 90% was required for a query to be considered
acceptable. The recall was assessed against independent publication sets
developed by an analyst consisting of publications from specialized journals
identified to fit the criteria. As most specialized journals do not
exclusively focus their content on a single SDG, a minimum recall of 60% was
required for a query to be considered acceptable. In cases where no single
journal was specific enough for all publications within that journal to fit
the criteria set for an SDG or SDG-Theme, a publication set was constructed by
manually selecting publications from a journal with high relevance to the SDG
(or SDG-Theme), and recall was assessed against this set.

Below we refer to the mentioned recall evaluation dataset as to the Science-
Metrix recall dataset.

###  1.2 Machine learning applied to SDG classification

On top of the mapping produced by the queries described above, additional
articles are mapped to the SDGs by a machine learning model.

In a nutshell, the model is a logistic regression trained with TF-IDF
representations of titles, keywords, abstracts, and two more optional text
fields – main terms extracted from the full text, and subject areas of the
journal that published the paper. Thus, the model learns similar keyphrases
for each SDG and helps to improve the recall of the queries. To keep precision
high, we keep only those papers that are classified by the model with 95% or
higher predicted probability for some SDG.

In the “Elsevier 2021 SDG mapping” release [13] we specify the input data for
the model, the targets that it’s trained with, the technical details of the
model itself, and model performance. Also, to ease the interpretation of the
model classification outcomes, we share the SDG-specific key phrases learned
by the model, as well as sample articles classified by the model. Please refer
to the mentioned documentation for more details on the machine learning
component of our approach.

###  1.3 Combining the queries and the model

The end-to-end approach to map Scopus records to SDGs is two-staged:

  * •

first, the keyword SDG queries are run (orange in Fig. 1)

  * •

then, the ML model adds about 3.5% of papers (blue in Fig. 1) on top of what
is classified by the keyword queries. We only keep the most confident model
predictions by thresholding predicted scores at 0.95.

Figure 1: Distribution of the number of papers mapped by the queries (SM,
orange) and by the model (ML, blue), by SDG (ignoring SDG 17).

##  2 Results

###  2.1 Comparison between the SDG queries

Below we describe the SDG queries and validation datasets that we used for the
comparison in terms of precision, recall, and F1 scores.

####  2.1.1 Query models

Table 2 describes the different classification methods that we compared. These
can be either keyword queries (“Elsevier queries 2020”, “Aurora queries v5”,
“Auckland queries v2”, and "Bergen SDG queries") or machine learning models
(“Aurora ML v0.2”) or both (“Elsevier queries+ML 2021”, “Elsevier queries+ML
2022”).

Table 2: SDG classification methods (both keyword queries and ML models) used in the evaluation. Classification method |  Description |  Web location
---|---|---
Elsevier queries 2020 |  “Identifying research supporting the United Nations Sustainable Development Goals” [1] |  link
Elsevier queries+ML 2021 |  “Improving the Scopus and Aurora queries to identify research that supports the United Nations Sustainable Development Goals (SDGs) 2021” [13] |  link
Elsevier queries+ML 2022 |  A simplified version of “Elsevier queries+ML 2021” with Covid-related addendum to SDG 3 |  to be shared in late 2022
Aurora queries v5 (2020) |  "Mapping Research Output to the Sustainable Development Goals (SDGs)" [16] |  link
Auckland queries v2 |  To gain a better understanding of our research contribution, the University of Auckland SDG Keywords Dictionary Project seeks to build on the processes developed by the United Nations and THE in order to create an expanded list of keywords that can be used to identify SDG-relevant research |  link
Aurora ML v0.2 (2021) |  “AI for mapping multi-lingual academic papers to the United Nations’ Sustainable Development Goals (SDGs)” [15] |  link
Bergen SDG queries |  The Bergen approach created queries for Web of Science to retrieve SDG-related publications for a limited number of SDGs. The queries have been translated for Scopus and a sample of the results has been taken as positive examples. These have been supplemented by other publications which did not appear in the queries as negative examples. Two datasets were created, one based on the Action Approach queries and one based on the Topic Approach queries – referred to as Bergen TAA and Bergen TBA respectively. [2] |  link
Digital Science |  "Contextualizing Sustainable Development Research" [10] |  —

####  2.1.2 Validations sets; collection method, sizes, and quality

Table 3: SDG validation datasets used in the evaluation. Valid. set |  Description & method of collection |  Web location |  Size of the corpus |  Remarks on quality
---|---|---|---|---
Science Metrix recall dataset |  See Sec. 1.1.1 |  Shared via ICSR Lab, see Data availability |  465k |  The dataset is noisy in the sense that not all papers from an SDG-specific journal are relevant to the same SDGs. Hence we don’t aim at 100% recall w.r.t. to this dataset
Aurora Survey |  "Survey data of “Mapping Research output to the SDGs” by Aurora European Universities Alliance (AUR)" 244 senior researchers from different universities in Europe and US filled in a survey. They were only allowed to enter the survey if they were familiar with the SDG they had selected to evaluate. The first question was to provide a list of research papers they believe are relevant to that selected SDG. The second question was to handpick from a given set of 100 randomly drawn papers in the Aurora query result set, the papers they believe (based on reading the title, abstract, journal name, and authors) belong to the selected SDG. The suggested papers and the selected papers are included in the validation set. |  source |  6741 |  Bias: the researchers located at western European universities.
Aurora suggested papers |  The papers suggested by researchers, see "Survey data of “Mapping Research output to the SDGs”. |  source |  3964 |  The dataset is noisy in the sense that researchers are not specialists in SDGs. They might also have an incentive to cite their own research
Elsevier multi-label |  The dataset consists of 6000 papers annotated by 3 experts each. These papers come from 5 data sources to span as diverse as possible set of SDG-related papers |  Shared via ICSR Lab, see Data availability |  6000 |  Annotators are not as versed in SDGs as Science-Metrix analysts
Chilean multi-label |  The dataset is provided by Pontificia Universidad Católica (PUC) based in Chile and consists of about 1200 papers self-assessed by PUC researchers and labeled with 0, 1 or 2 SDGs. |  source |  1200 |  Biases: self-assessment, only Chilean researchers
OSDG Comunity Dataset |  A public dataset of thousands of text excerpts, which were validated by approximately 1,000 OSDG Community Platform (OSDG-CP) citizen scientists from over 110 countries, with respect to the Sustainable Development Goals. |  source |  32431 |  Crowd-sourced dataset, the annotators are not versed in SDGs

Table 3 provides details on the validation datasets used in the comparison. It
also mentions the associated limitations and biases. It is important to
mention that there’s no single best validation dataset to evaluate the output
of SDG classification. In particular, for the 6 datasets that we used in the
evaluation:

  * •

the Science-Metrix recall dataset described in Sec. 1.1.1 (referred to as
“sm_recall” in Table 4) served well for query development but it is noisy in
the sense that not all papers from an SDG-specific journal are relevant to the
same SDG. Hence we don’t aim at 100% recall w.r.t. to this dataset nor can
this dataset serve as a source of ground truth SDG labels;

  * •

The Aurora survey dataset (“au_survey” in Table 4) is fairly small and it has
an intrinsic bias: the survey was limited to researchers from western European
countries;

  * •

the Aurora suggested papers dataset (“au_sugg” in Table 4) is fairly small and
it has an intrinsic bias: researchers might have an intention to mention their
own research;

  * •

The Elsevier multi-label dataset (“els_2021” in Table 4) is labeled with a 3rd
party, and the annotators are not specialists in SDGs;

  * •

The Chilean multi-label dataset [17] (“chile” in Table 4) is very small and
has intrinsic biases: it’s limited to Chilean research and also researchers
performed self-assessment;

  * •

Similarly, the OSDG Community Dataset (“osdg” in Table 4) is crowd-sourced,
with annotators not being versed in SDGs.

####  2.1.3 Performance; query models measured against validation sets

Table 4 provides the evaluation results for 6 SDG classification methods
outlined in Table 2 and 6 evaluation datasets described in Table 3. Each cell
shows 2 values: micro-average F1-score and macro-average F1-score, in percent
(%).

The code reproducing the experiments presented in this subsection is found on
GitHub333https://github.com/Yorko/sdg_mapping_queries_n_ml_benchmarks. Refer
to Sec. Data availability for instructions on getting data should you wish to
reproduce the presented experiments.

Note that micro-averaging favors well-represented, frequent classes (like SDG
3 in our case) while high macro-averaged scores mean then the method works
fairly well across all SDGs because bad results for a single SDG affect macro-
averaged metrics much more than micro-averaged ones. By attending to both
micro- and macro-averaged F1 scores we try to assess both aspects: how well
the method is at classifying papers into frequent or rare classes.

Table 4: F1 scores with micro/macro average (percents, %) for 6 classification methods and 6 validation datasets. | Method \
---
dataset
sm_recall | au_survey | au_sugg | els_2021 | chile | osdg
auckland_v2 | 72/40 | 55/40 | 61/32 | 74/62 | 73/36 | 53/43
aurora_v5 | 35/22 | 63/41 | 19/14 | 45/38 | 30/14 | 36/22
aurora_ml | 62/33 | 58/42 | 9/4 | 46/43 | 65/37 | 62/49
els_2020 | 69/37 | 55/34 | 62/25 | 71/47 | 72/25 | 40/28
els_2021 | 76/48 | 51/39 | 43/28 | 75/67 | 53/34 | 47/37
els_2022 | 76/48 | 52/39 | 43/29 | 75/67 | 54/34 | 48/39

Although, there is no single best approach, Auckland queries v2 and Elsevier
queries+ML 2021/2022 stand out and perform better (in terms of F1 scores) than
Elsevier queries 2020, Aurora queries v.5 and Aurora ML model. These queries
also produce more stable results across datasets, unlike Aurora queries and ML
model that fail at generalizing to the Aurora suggested papers dataset. In a
direct comparison between Elsevier queries+ML 2022 and Auckland queries v2 the
latter wins: the Elsevier queries performed better at 2 datasets, Auckland
queries v2 were better at the other 4 datasets. As Auckland queries v2 are
largely based on Elsevier queries 2020, it’s interesting to scrutinize the
reason for their efficiency. Partly, it’s due to localization, see 3.1 but
extensive ablation studies is s subject of follow-up research.

Some notes and observations:

  * •

Aurora queries v5 generalize badly to all datasets except for the Aurora
survey dataset;

  * •

We notice a clear “overfitting” phenomenon: Elsevier queries+ML 2021/2022 are
best when validated against the Science-Metrix recall dataset (developed by
Elsevier) while Aurora queries v.5/Aurora ML model achieve the highest F1
scores against the Aurora survey dataset. A probable explanation is that the
datasets were crafted for a specific definition/operationalization of SDGs and
these definitions are undoubtedly different from one project to another;

  * •

Bergen SDG queries were excluded from these experiments as they only covered
(at the time experiments were run) SDG 1 (No poverty), SDG 2 (Zero hunger),
SDG 3 (Good health and well-being), SDG 7 (Affordable and clean energy), SDG
13 (Climate action), and SDG 14 (Life below water).

###  2.2 Direct comparison to the SDG mapping by Digital Science

Independently, Elsevier did a face-to-face comparison to the SDG mapping
solution provided by Digital Science [10]. The protocol of the experiments is
the same as described above. We only report these results separately as the
comparisons were done with subsets of records or SDGs for two reasons:

  * •

the Digital Science solution used the WebOfScience dataset rather than Scopus
hence the overlap with the datasets mentioned above was not as high as to
include the results in Table 4 (2-3k. records);

  * •

The comparison to Digital Science includes the datasets provided by Bergen
queries which are limited to SDGs 1, 2, 3, 7, 13, and 14.

Figure 2: micro-averaged F1 scores for Elsevier and Digital Science SDG
mappings reported for 6 datasets. Figure 3: F1 scores for Elsevier and Digital
Science SDG mappings reported for two datasets: Aurora suggested papers and
Elsevier multi-label SDG dataset.

From these comparisons, we conclude that Elsevier’s approach is on par with
the one by Digital Science in terms of precision, and considerably outperforms
Digital Science’s solution in terms of recall and hence in F1 scores. Fig 2
shows micro-averaged F1 scores achieved by both Elsevier queries+ML 2021 and
Digital Science’s methods w.r.t. to 6 datasets:

  * •

4 of them are the same as presented in Table 4;

  * •

plus Bergen SDG datasets [18] that accompany paper [2].

Additionally, Fig. 3 shows F1 scores per SDG reported for two datasets: Aurora
suggested papers and Elsevier multi-label SDG dataset.

##  3 Discussion

In previous sections, we described the methodology and evaluation results.
Below, we outline possible improvements to the SDG mapping approach, including
localization of the SDG queries, query generalization to non-English
languages, and extending the approach to non-article content.

###  3.1 Localization

Research activities do not stand alone; they are an integral part of the
geographical place they were initiated and the communities they serve. An
attempt to measure SDG-related research activities can be improved by infusing
the local context within which the research activities take place. A
localization approach can further foster understanding of, for example, the
degree to which the prevailing SDG mapping approaches capture SDG research in
the geographical region that may or may not have been described by keywords
and keyphrases with close semantic relatedness to the keyword-based queries,
e.g. Elsevier 2020 queries.

The University of Auckland’s approach is one such localization attempt based
on Elsevier’s earlier 2020 queries, a mixture of the UN official targets and
indicators, and the suggested search terms by the Sustainable Development
Solutions Network (SDSN). The n-gram model was applied to two samples of
Scopus publication metadata, i.e. a global publication sample and a University
of Auckland publication sample. The n-gram tokens were scored by a range of
factors, including counts and measures of frequency, and were then ranked by
those scores. Keywords with a high rank were then evaluated in more detail and
manually reviewed and improved for SDG alignments. Table 5 shows the number of
University of Auckland publications between 2009-2020 captured by the
University’s queries compared with those captured by Elsevier 2020 queries.

Table 5: Comparison of Auckland v2 queries and Elsevier 2020 queries. SDG | Auckland queries output | Elsevier 2021 output | Intersection
---|---|---|---
1 | 522 | 229 | 125
2 | 1975 | 420 | 264
3 | 16894 | 7966 | 6894
4 | 2484 | 1043 | 745
5 | 611 | 609 | 360
6 | 684 | 486 | 362
7 | 1152 | 1187 | 799
8 | 428 | 440 | 154
9 | 1044 | 1139 | 519
10 | 1528 | 977 | 500
11 | 1886 | 1462 | 779
12 | 921 | 438 | 158
13 | 1032 | 577 | 466
14 | 1390 | 744 | 552
15 | 1641 | 769 | 473
16 | 891 | 779 | 409

For 13 out of the 16 SDGs documented in Table 5, the Auckland queries capture
more SDG-related publications. In some cases, the number of publications
captured by the Auckland queries double that captured by the Elsevier 2021
approach. A significant proportion of the additional publications are captured
through localized keywords and search terms. For example, "Te Whāriki" – the
New Zealand national curriculum document for early childhood education – was
used as an SDG4 keyphrase under the Auckland approach. It retrieved 19 SDG4
papers published by the University of Auckland, of which only 6 papers were
counted by the Elsevier 2021 approach. In some other cases, the Auckland
queries also gave rise to additional keywords that are fit for the global
settings. For example, “marine biodiversity” as an Auckland keyphrase
retrieved 24 SDG14 papers published by the University of Auckland, of which 19
papers were counted by the Elsevier 2021 approach.

###  3.2 Multilingual queries

In CRISs
systems444https://en.wikipedia.org/wiki/Current_research_information_system
and repositories there are many more publications that are not included in
Scopus and are written in the local language of the country to serve a
different audience. We found out we could not simply replace the keywords in
the queries and have the search work the same in other languages, because of
the syntax and morphology rules. That’s why Aurora chose to train mBERT models
to classify SDGs. Due to the lack of non-English SDG labeled data, we used
only English training data, specifically paper abstracts.

During the evaluation, the models for SDGs 1 to 5 and 11 were applied to
classify 888 German paper titles. To have a qualitative benchmark, we
performed a manual SDG classification only on titles as well. In doing the
latter, we tended to take a strict approach and tried to stick very close to
the respective SDG indicators (e.g., non-assignment of SDG4 to publications on
teacher training in Germany, as the SDG indicators only refer to teacher
training in the Global South).

The manual classification resulted in 43 SDG-related publications, whilst the
ML models resulted in 58 publications SDG-related publications. The total
overlap between these two methods was 8 publications. It was mainly for SDG3 –
Good Health and Wellbeing (5/8) and can most likely be explained by great
similarities in terminology between English and German for issues such as
multiple sclerosis, psychotherapy, suicide, alcohol, and illegal drugs (in
German: Multiple Sklerose, Psychotherapie, Suizid, Alkohol, illegale Drogen).

At the current phase of evaluation, the multilingual ability of the ML models
for research output in German cannot be positively assessed. However, further
analysis including the abstracts of publications for the classification of the
ML models may offer improvements in classification quality.

###  3.3 Generalization to other types of content

In addition to SDG-related research outputs, higher education institutions
have a strong interest in understanding SDG-related educational activities, as
done in Aurora SDG Course Catalogue.555https://bit.ly/aurora-sdg-courses These
SDG labels have been added manually by the course coordinators, but such a
process is labor-intensive, and not sustainable since this needs to be done
year after year.

Similar to publication metadata (e.g. title, abstract, keywords), many course
catalogs and curriculum management systems capture metadata in a similar way
(e.g. title, course short description, course long description). Whether the
SDG research mapping techniques can be translated and applied to SDG course
mapping represents an interesting topic to many.

A study was conducted by the University of Auckland to apply the Auckland
queries to classify courses taught by the university. The mapping results
identified 792 SDG courses out of 2441 courses in total offered to students in
the academic year 2020. Compared with the frequency and distribution of
keywords in research mapping, course mapping demonstrated a higher
concentration of keywords used to convey the SDG topics. For example, the 24
University of Auckland courses related to SDG14 are fully captured by the top
ten keywords in the Auckland queries by frequency (i.e. marine; fisheries;
coastal management; pollut*; aquaculture; marine environment; fisheries
management; eutrophical*; aquatic ecosystem; alga*).

## Conclusion

In this paper, we outlined the methodology behind research mapping to United
Nations (UN) Sustainable Development Goals (SDGs), how it compares to other
existing methods and how well it performs with existing SDG validation
datasets. We discussed possible improvements to the existing approach:
localization of the queries, and generalization to other languages and data
types.

## Data availability

The data underlying the results presented in the study (including the
processed version of publicly available datasets listed in Table 3) are
available, for scholarly research and upon application, from Elsevier BV on
the ICSR Lab. ICSR Lab is intended for scholarly research only and is a cloud-
based computational platform that enables researchers to analyze large
structured datasets, including aggregated data from Scopus author profiles,
PlumX Metrics, SciVal Topics, and Peer Review Workbench.

## Acknowledgments

This work is partly an outcome of SDG Research Mapping
Initiative666https://www.elsevier.com/about/partnerships/sdg-research-mapping-
initiative that Elsevier initiated with the Aurora European Universities
Alliance, the University of Auckland, and the University of Southern Denmark.
We are also grateful to Scopus for providing data for the analysis.

## References

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  * [2] Caroline S. Armitage, Marta Lorenz, and Susanne Mikki.  Mapping scholarly publications related to the Sustainable Development Goals: Do independent bibliometric approaches get the same results?  Quantitative Science Studies, 1(3):1092–1108, 08 2020.
  * [3] Frederique Bordignon.  Dataset of search queries to map scientific publications to the UN sustainable development goals, February 2021.
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  * [5] Marcelo T. LaFleur.  Art is long, life is short: An SDG classification system for DESA Publications.  https://www.un.org/development/desa/publications/working-paper/wp159, 2019\.  [Online; accessed 14-Sep-2022].
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  * [7] Mu Jingwen and Wang Weiwei.  The University of Auckland SDG Keywords Mapping .  https://www.sdgmapping.auckland.ac.nz/, August 2022.  [Online; accessed 14-Sep-2022].
  * [8] Rui Zhang, Maéva Vignes, Ulrich Steiner, and Arthur Zimek.  Matching research publications to the united nations’ sustainable development goals by multi-label-learning with hierarchical categories.  In Geoff Webb, Zhongfei Zhang, Vincent S. Tseng, Graham Williams, Michalis Vlachos, and Longbing Cao, editors, Proceedings - 2020 IEEE 7th International Conference on Data Science and Advanced Analytics, DSAA 2020, pages 516–525, United States, October 2020. IEEE.  7th IEEE International Conference on Data Science and Advanced Analytics, DSAA 2020 ; Conference date: 06-10-2020 Through 09-10-2020.
  * [9] South African SDG Hub.  https://sasdghub.up.ac.za/en/, 2020.  [Online; accessed 14-Sep-2022].
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  * [12] Lukas Pukelis, Nuria Bautista Puig, Mykola Skrynik, and Vilius Stanciauskas.  Osdg – open-source approach to classify text data by un sustainable development goals (sdgs), 2020.
  * [13] Maxime Rivest, Yury Kashnitsky, Alexandre Bédard-Vallée, David Campbell, Paul Khayat, Isabelle Labrosse, Henrique Pinheiro, Simon Provençal, Guillaume Roberge, and Chris James.  Improving the Scopus and Aurora queries to identify research that supports the United Nations Sustainable Development Goals (SDGs) 2021, June 2021\.
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  * [15] Maurice Vanderfeesten and Robert Jaworek.  AI for mapping multi-lingual academic papers to the United Nations’ Sustainable Development Goals (SDGs), February 2022.
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  * [17] Patricio Cortés Rodríguez, Renzo Rondanelli Delpiano, Paola Santander Meneses, and Ricardo Vilches Vargas.  Conjunto de datos: Categorization of articles 2017 with authorship of pontificia universidad católica de chile, through the sdgs.  2021\.
  * [18] Caroline Armitage, Marta Lorenz, and Susanne Mikki.  Replication data for: Mapping scholarly publications related to the Sustainable Development Goals: Do independent bibliometric approaches get the same results?, 2020.


# Comparison results for Poisson equation with mixed boundary condition on
manifolds

Haiqing Cheng  School of Mathematical Sciences, Soochow University, Suzhou,
215006, China<EMAIL_ADDRESS>,  Tengfei Ma  Department of Mathematics,
Nanjing University, Nanjing, 210093, China<EMAIL_ADDRESS>and  Kui Wang
School of Mathematical Sciences, Soochow University, Suzhou, 215006, China
<EMAIL_ADDRESS>
###### Abstract.

In this article, we establish a $L^{1}$ estimate for solutions to Poisson
equation with mixed boundary condition, on complete noncompact manifolds with
nonnegative Ricci curvature and compact manifolds with positive Ricci
curvature respectively. On Riemann surfaces we obtain a Talenti-type
comparison. Our results generalize main theorems in [2] to Riemannian setting,
and Chen-Li’s result [8] to the case of variable Robin parameter.

###### Key words and phrases:

Talenti’s comparison, Robin boundary, Riemannian manifolds

###### 2020 Mathematics Subject Classification:

53C21, 53C42

The research of the third author is supported by NSFC No.11601359

##  1\. Introduction

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with nonempty smooth
boundary, $\Omega^{\sharp}\subset\mathbb{R}^{n}$ be a round ball with the same
volume as $\Omega$, $f(x)$ be a nonnegative function on $\Omega$ and
$f^{\sharp}$ be the Schwarz decreasing rearrangement of $f$ (see Definition
2.1). Assume $u(x)$ and $v(x)$ are solutions to

$-\Delta u(x)=f(x),\qquad x\in\Omega,$

and

$-\Delta v(x)=f^{\sharp}(x),\qquad x\in\Omega^{\sharp}$

with Dirichlet boundary, respectively. Talenti [19] proved that

(1.1) $\displaystyle u^{\sharp}(x)\leq v(x),\qquad x\in\Omega^{\sharp}.$

Moreover the equality occurs if and only if $f(x)$ is a radial function and
$\Omega$ is a round ball, see [16]. The key tools of the proof are Schwarz
symmetrization and isoperimetric inequalities on manifolds. Talenti’s
comparison (1.1) plays important roles in both partial differential equations
and geometry problems. There provide $L^{\infty}$ estimates for solutions to
PDEs and Faber-Krahn type inequality for the first Dirichlet eigenvalue, see
[8, 14, 16]. Talenti’s comparison was generalized to nonlinear elliptic and
parabolic equations with Dirichlet boundary condition (cf. [4, 5, 6, 20] and
references therein), to compact Riemannian manifolds with positive Ricci
curvature [11], and to noncompact manifolds with nonnegative Ricci curvature
and positive asymptotic volume ratio [8]. We also refer the reader to the
excellent books [15, 17] for related topics.

Recently, Alvino, Nitsch and Trombetti [3] studied the Poisson equation with
Robin boundary condition when Robin parameter is a positive constant. They
proved estimate (1.1) on planer domains, and a sharp $L^{1}$ estimate for
higher dimensions. These results were generalized to Riemannian manifolds [9],
to Robin boundary with variable Robin parameter [2], and to the torsion
problem for the Hermite operator [10, 12].

The purpose of the present paper is to study Talenti’s comparison and related
estimates for solutions to Poisson equation with mixed boundary condition on
manifolds. In particular, we generalize Alvino-Chiacchio-Nitsch-Trombetti’s
result [2] to compact manifolds with positive Ricci curvature, and to
noncompact manifolds with nonnegative Ricci curvature.

To state our results, we give some notations. Let $(M,g)$ be an
$n$-dimensional complete Riemannian manifold, which is either compact with
$\operatorname{Ric}\geq(n-1)\kappa$ for $\kappa>0$, or noncompact with
nonnegative Ricci curvature and positive asymptotic volume ratio. Denote by

(1.2)
$\displaystyle\theta=\begin{cases}\lim_{r\to\infty}\frac{|B(r)|}{\omega_{n}r^{n}},&\quad\kappa=0,\\\
\frac{|M|}{|M_{\kappa}|},&\quad\kappa>0,\end{cases}$

where $B(r)$ is a round geodesic ball of radius $r$ in $M$, $M_{\kappa}$ is
the $n$ dimensional space form of sectional curvature $\kappa$, $\omega_{n}$
is the volume of the unit ball in $\mathbb{R}^{n}$, and $|M|$ denotes the
volume of $M$. It follows from the Bishop-Gromov volume comparison that
$\theta\leq 1$. Let $\Omega\subset M$ be a bounded domain with nonempty smooth
boundary, $f(x)$ be a nonnegative smooth function not identically zero on
$\Omega$, and $\beta(x)$ be a positive smooth function on $\partial\Omega$. We
consider the following Poisson equation with mixed boundary condition

(1.3) $\displaystyle\begin{cases}-\Delta u(x)=f(x),&\quad x\in\Omega,\\\
\frac{\partial u}{\partial\nu}+\beta(x)u=0,&\quad
x\in\partial\Omega,\end{cases}$

where $\Delta$ denotes the Laplace-Beltrami operator and $\nu$ denotes the
outward unit normal to $\partial\Omega$. Since $\beta(x)>0$ and $f\geq 0$, it
then follows easily from maximum principle that the solution to (1.3) is
positive on $\Omega$, see Proposition 2.1. Let $\Omega^{\sharp}$ be a round
geodesic ball in $M_{\kappa}$ satisfying $\theta|\Omega^{\sharp}|=|\Omega|$,
$f^{\sharp}$ defined on $\Omega^{\sharp}$ be the Schwarz decreasing
rearrangement of $f$, and $v(x)$ be the solution to the Schwarz decreasing
rearrangement equation of (1.3), namely

(1.4) $\displaystyle\begin{cases}-\Delta v(x)=f^{\sharp}(x),&\quad
x\in\Omega^{\sharp},\\\ \frac{\partial v}{\partial\nu}+\bar{\beta}v=0,&\quad
x\in\partial\Omega^{\sharp},\end{cases}$

where $\bar{\beta}$ is a positive constant defined by

(1.5)
$\displaystyle\bar{\beta}=\frac{\theta|\partial\Omega^{\sharp}|}{\int_{\partial\Omega}\frac{1}{\beta(x)}\,dA}.$

Throughout the paper, $dA$ denotes the induced measure on boundary
$\partial\Omega$ or $\partial\Omega^{\sharp}$. Since the boundary value
problem (1.4) is radially symmetric, then the solution $v(x)$ is radially
symmetric as well. Moreover $v(x)$ is monotone decreasing along the radial
direction, see Proposition 2.2 below.

Our first result is concerning a $L^{1}$ comparison between $u(x)$ and $v(x)$.
Precisely we prove

######  Theorem 1.1.

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold, which is
either compact with $\operatorname{Ric}\geq(n-1)\kappa$ for $\kappa>0$, or
noncompact with nonnegative Ricci curvature and positive asymptotic volume
ratio. Let $\Omega\subset M$ be a bounded domain with nonempty smooth
boundary, $f(x)$ be a nonnegative smooth function not identically zero on
$\Omega$, and $\beta(x)$ be a positive smooth function on $\partial\Omega$.
Let $u(x)$ and $v(x)$ be the solutions to equations (1.3) and (1.4)
respectively. If $n\geq 3$, we assume further that for all measurable
$E\subset\Omega$, it holds

(1.6)
$\displaystyle\int_{E}f(x)\,dx\leq\frac{|E|^{\frac{n-2}{n}}}{|\Omega|^{\frac{n-2}{n}}}\int_{\Omega}f(x)\,dx.$

Then

(1.7)
$\displaystyle||u||_{L^{1}(\Omega)}\leq\theta||v||_{L^{1}(\Omega^{\sharp})}.$

######  Remark 1.2.

In this paper, we mainly focus on $L^{1}$ estimate (1.7) on manifolds and we
then set up Theorem 1.1 in smooth case. In fact, Theorem 1.1 remains valid
under the assumptions that $\Omega$ is a bounded domain with Lipschitz
boundary, $f(x)\in L^{2}(\Omega)$ and $\beta(x)$ is a positive measurable
function defined on $\partial\Omega$, see [2].

When $n=2$ and $f$ is a constant, we prove a Talenti-type comparison,
similarly as in [2].

######  Theorem 1.3.

Let $(M,g)$ be a complete Riemann surface, which is either compact with
$\operatorname{Ric}\geq\kappa>0$, or noncompact with $\operatorname{Ric}\geq
0$ and positive asymptotic volume ratio. Let $u(x)$ and $v(x)$ be the
solutions to equations (1.3) and (1.4) respectively. Suppose $f(x)\equiv 1$
and $\beta(x)>0$. Then

(1.8) $\displaystyle u^{\sharp}(x)\leq v(x),\quad x\in\Omega^{\sharp}.$

Moreover, the equality case of (1.8) occurs if and only if $M$ and $\Omega$
are isometric to $M_{\kappa}$ and $\Omega^{\sharp}$ respectively, and
$\beta(x)$ is a constant on $\partial\Omega$.

The rest of this paper is organized as follows. In Section 2, we recall the
Schwarz decreasing rearrangement and isoperimetric inequalities on manifolds,
and give some properties on solutions to Poisson equation with Robin boundary
condition. In Section 3, we prove Theorem 1.1 and Theorem 1.2.

##  2\. Preliminaries

###  2.1. Schwarz decreasing rearrangements.

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold with
$\operatorname{Ric}\geq(n-1)\kappa$, which is either noncompact with
$\kappa=0$ and positive asymptotic volume ratio, or compact with $\kappa>0$.
Let $M_{\kappa}$ be the $n$-dimensional space form of constant sectional
curvature $\kappa$. Let $\Omega$ be a bounded domain in $M$ and
$\Omega^{\sharp}$ be a round geodesic ball in $M_{\kappa}$ with volume
$|\Omega|/\theta$, where $\theta\in(0,1]$ is a constant defined by (1.2).

We recall the definitions and properties of the Schwarz decreasing
rearrangement of non-negative functions on manifolds, see also [8, Section 2]
and [11, Section 2].

######  Definition 2.1.

Let $h(x)$ be a nonnegative measurable function on $\Omega$. Denote by
$\Omega_{h,t}=\\{x\in\Omega:h(x)>t\\}$ and $\mu_{h}(t)=|\Omega_{h,t}|$, the
decreasing rearrangement $h^{*}$ of $h$ is defined by

(2.1) $\displaystyle
h^{*}(s)=\begin{cases}\operatorname{ess}\sup_{x\in\Omega}\,h(x),&\quad s=0,\\\
\inf\\{t\geq 0:\mu_{h}(t)<s\\},&\quad s>0,\end{cases}$

for $s\in[0,|\Omega|]$. The Schwarz decreasing rearrangement of $h(x)$ is
defined by

(2.2) $\displaystyle
h^{\sharp}(x)=h^{*}\big{(}\theta\omega_{n}r(x)^{n}\big{)},\quad
x\in\Omega^{\sharp},$

where $r(x)$ is the distance function from the center of $\Omega^{\sharp}$ in
$M_{\kappa}$, and $\omega_{n}$ is the volume of unit ball in $\mathbb{R}^{n}$.

It follows from (2.1) and (2.2) that

(2.3) $\displaystyle\mu_{h}(t)=\theta\mu_{h^{\sharp}}(t)$

for $t\geq 0$. The Fubini’s theorem gives that for $h\in L^{p}(\Omega)$ and
$p\geq 1$ it holds

$\displaystyle\int_{\Omega}h^{p}(x)\,dx=\int_{0}^{|\Omega|}(h^{*})^{p}(s)\,ds=\theta\int_{\Omega^{\sharp}}(h^{\sharp})^{p}(x)\,dx.$

For any nonnegative functions $f(x)$ and $g(x)$, the Hardy-Littlewood
inequality gives

(2.4)
$\displaystyle\int_{\Omega}f(x)g(x)\,dx\leq\int_{0}^{|\Omega|}f^{*}(s)g^{*}(s)\,ds,$

and taking $g(x)$ as the characteristic function of $\Omega_{h,t}$ in above
inequality yields

(2.5)
$\displaystyle\int_{\Omega_{h,t}}f(x)\,dx\leq\int_{0}^{\mu_{h}(t)}f^{*}(s)\,ds.$

###  2.2. Isoperimetric inequalities on manifolds.

To prove Theorem 1.1 and Theorem 1.3, we require the following isoperimetric
inequality on manifolds with Ricci curvature bounded from below.

######  Theorem 2.2.

Let $M$, $\theta$, $\Omega$ and $\Omega^{\sharp}$ be the same as in Theorem
1.1. It holds

(2.6) $\displaystyle|\partial\Omega|\geq\theta|\partial\Omega^{\sharp}|,$

where $|\partial\Omega|$ denotes the $(n-1)$-dimensional area of
$\partial\Omega$. Moreover the equality case of (2.6) occurs if and only if
$\Omega$ is isometric to $\Omega^{\sharp}$.

When $\kappa>0$, inequality (2.6) was known as Lévy-Gromov isoperimetric
inequality [13], see also [18, Theorem 2.1]. When $M$ is noncompact and
$\kappa=0$, inequality (2.6) was proved by Agostiniani, Fogagnolo and Mazzieri
[1] when $n=3$, and by Brendle [7] for all dimensions. Brendle proved in [7]
that the isoperimetric inequality (2.6) also holds true when $\Omega$ is an
$n$-dimensional compact minimal submanifold of $M$ of dimension $n+2$ with
nonnegative sectional curvature as well.

###  2.3. Poisson equation with Robin boundary

In this subsection, we collect some known facts about solutions to Poisson
equation with Robin boundary condition.

######  Proposition 2.1.

Let $u(x)$ be the solution to (1.3). Assume $f(x)$ is nonnegative and not
identically zero on $\Omega$, and $\beta(x)>0$ on $\partial\Omega$. Then we
have

$\displaystyle u(x)>0$

for all $x\in\Omega$.

###### Proof.

Letting $\displaystyle\bar{u}(x)=\frac{1}{2}(u(x)-|u(x)|)$ and using
integration by parts, we estimate that

(2.7) $\displaystyle\int_{\Omega}\langle\nabla\bar{u},\nabla u\rangle\,dx=$
$\displaystyle\int_{\partial\Omega}\bar{u}\frac{\partial
u}{\partial\nu}\,dA-\int_{\Omega}\bar{u}\Delta u\,dx$ $\displaystyle=$
$\displaystyle-\int_{\partial\Omega}\beta
u\bar{u}\,dA+\int_{\Omega}\bar{u}f\,dx$ $\displaystyle\leq$ $\displaystyle 0,$

where we used equation (1.3) in the second equality and nonnegativity of $f$
in the inequality. Observing that

$\int_{\Omega}\langle\nabla\bar{u},\nabla
u\rangle\,dx=\int_{\Omega}|\nabla\bar{u}|^{2}\,dx,$

then we have

$\int_{\Omega}|\nabla\bar{u}|^{2}\,dx=0.$

So $\bar{u}(x)$ is a constant on $\Omega$ and inequality (2.7) holds as an
equality, hence $\bar{u}(x)=0$, equivalently $u(x)\geq 0$. Since $u(x)$ is a
supharmonic function by equation (1.3) and $f$ is not identically zero, then
we conclude from strong maximum principle that $u(x)>0$ on $\Omega$. $\square$

######  Proposition 2.2.

Let $v(x)$ be the solution to (1.4), rewritten as $\varphi(r(x))$. If $f(x)$
is non-negative and not identically zero on $\Omega$, and $\bar{\beta}>0$,
then

(2.8) $\displaystyle\varphi^{\prime}(r)<0$

for $r\in(0,R_{0})$. Where $\Omega^{\sharp}=B(R_{0})$, and
$R_{0}\leq\frac{\pi}{\sqrt{\kappa}}$ if $\kappa>0$ in view of the Myers
theorem.

###### Proof.

We rewrite $f^{\sharp}(x)$ as $h(r(x))$ and denote by

$\operatorname{sn}_{\kappa}(r)=\begin{cases}\frac{\sin\sqrt{\kappa}r}{\sqrt{\kappa}},&\kappa>0,\\\
r,&\kappa=0,\end{cases}$

then equation (1.4) is equivalent to

$\varphi^{\prime\prime}(r)+(n-1)\frac{\operatorname{sn}_{\kappa}^{\prime}(r)}{\operatorname{sn}_{\kappa}(r)}\varphi^{\prime}(r)=-h(r).$

Since $h(r)\geq 0$, we then have

$(\operatorname{sn}_{\kappa}^{n-1}(r)\varphi^{\prime}(r))^{\prime}=-\operatorname{sn}_{\kappa}^{n-1}(r)h(r)\leq
0$

for $r\in(0,R_{0})$. Since $h(r)$ is monotone nonincreasing in $(0,R_{0})$ and
not identically zero, then there exists a $\varepsilon_{0}<R_{0}$ such that
$h(r)>0$ for $r<\varepsilon_{0}$. Hence $\varphi^{\prime}(r)<0$ for all
$r\in(0,R_{0})$. $\square$

##  3\. Proofs of Theorem 1.1 and Theorem 1.3

In this section, we will prove the main theorems. For simplicity, we rewrite
$\Omega_{u,t}$ as $\Omega_{t}$, and $\Omega^{\sharp}_{v,t}$ as
$\Omega^{\sharp}_{t}$ for short. For $s\in(0,|\Omega^{\sharp}|)$, we denote

$\displaystyle a(s)=\frac{s^{\frac{n-1}{n}}}{|\partial B_{s}|},$

where $B_{s}$ is a round geodesic ball in $M_{\kappa}$ with volume $s$. It is
easily checked that $a(s)=n^{-1}\omega_{n}^{-1/n}$ if $\kappa=0$, and $a(s)$
is monotone increasing in $s$ if $\kappa>0$. Using isoperimetric inequality
(2.6), we prove the following lemma, which will be used later.

######  Lemma 3.1.

Under the hypotheses of Theorem 1.1, we have

(3.1) $\displaystyle\theta^{2}(\frac{\mu_{u}(t)}{\theta})^{\frac{2n-2}{n}}\leq
a^{2}\big{(}\frac{\mu_{u}(t)}{\theta}\big{)}\Big{(}-\mu_{u}^{\prime}(t)+\int_{\partial\Omega_{t}\cap\partial\Omega}\frac{1}{\beta(x)u(x)}\,dA\Big{)}\int_{0}^{\mu_{u}(t)}f^{*}(s)\,ds,$

and

(3.2)
$\displaystyle\mu_{v}^{\frac{2n-2}{n}}(t)=a^{2}\big{(}\mu_{v}(t)\big{)}\Big{(}-\mu_{v}^{\prime}(t)+\int_{\partial\Omega^{\sharp}_{t}\cap\partial\Omega^{\sharp}}\frac{1}{\bar{\beta}v(x)}\,dA\Big{)}\int_{0}^{\mu_{v}(t)}(f^{\sharp})^{*}(s)\,ds$

for a.e. $t>0$.

###### Proof.

By the Morse-Sard theorem, we have

$\displaystyle\partial\Omega_{t}=\\{x\in\Omega:u(x)=t\\}\cup\\{x\in\partial\Omega:u(x)\geq
t\\}$

for a.e. $t>0$. Let

$\displaystyle g(x)=\begin{cases}|\nabla
u|,&x\in\partial\Omega_{t}\cap\Omega,\\\ -\frac{\partial
u}{\partial\nu},&x\in\partial\Omega_{t}\cap\partial\Omega.\end{cases}$

Applying the divergence theorem and equation (1.3), we derive that

(3.3)
$\displaystyle\int_{\partial\Omega_{t}}g(x)\,dA=\int_{\partial\Omega_{t}}-\frac{\partial
u}{\partial\nu}\,dA=-\int_{\Omega_{t}}\Delta
u\,dx=\int_{\Omega_{t}}f(x)\,dx\leq\int_{0}^{\mu_{u}(t)}f^{*}(s)\,ds,$

where we have used (2.5) in the inequality. Using the Hölder inequality, we
estimate

(3.4) $\displaystyle|\partial\Omega_{t}|^{2}\leq$
$\displaystyle\int_{\partial\Omega_{t}}g(x)\,dA\int_{\partial\Omega_{t}}\frac{1}{g(x)}\,dA$
$\displaystyle\leq$
$\displaystyle\int_{0}^{\mu_{u}(t)}f^{*}(s)\,ds\Big{(}\int_{\partial\Omega_{t}\cap\Omega}\frac{1}{|\nabla
u|}\,dA+\int_{\partial\Omega_{t}\cap\partial\Omega}\frac{1}{\beta(x)u(x)}\,dA\Big{)}$
$\displaystyle=$
$\displaystyle\int_{0}^{\mu_{u}(t)}f^{*}(s)\,ds\Big{(}-\mu_{u}^{\prime}(t)+\int_{\partial\Omega_{t}\cap\partial\Omega}\frac{1}{\beta(x)u(x)}\,dA\Big{)},$

where we have used inequality (3.3) in the second inequality and the coarea
formula in the equality.

On the other hand, the isoperimetric inequality (2.6) yields

(3.5)
$\displaystyle|\partial\Omega_{t}|\geq\theta|\partial(\Omega_{t}^{\sharp})|=\frac{\theta}{a(\mu_{u}(t)/\theta)}(\frac{\mu_{u}(t)}{\theta})^{\frac{n-1}{n}}.$

Assembling inequalities (3.4) and (3.5), we get

$\displaystyle\frac{\theta^{2}}{a^{2}(\mu_{u}(t)/\theta)}\mu_{u}^{\frac{2n-2}{n}}(t)\leq\int_{0}^{\mu_{u}(t)}f^{*}(s)\,ds\Big{(}-\mu_{u}^{\prime}(t)+\int_{\partial\Omega_{t}\cap\partial\Omega}\frac{1}{\beta(x)u(x)}\,dA\Big{)},$

which proves inequality (3.1).

If $v(x)$ is the solution to equation (1.4), $v(x)$ is a radial function on
$\Omega^{\sharp}$ and decreasing along the radial direction by inequality
(2.8), and $\Omega^{\sharp}_{t}$ is a round ball. Therefore all previous
inequalities hold as equalities if we replace $u(x)$ by $v(x)$, hence

$\displaystyle\mu_{v}^{\frac{2n-2}{n}}(t)=$ $\displaystyle
a^{2}(\mu_{v}(t))\Big{(}-\mu_{v}^{\prime}(t)+\int_{\partial\Omega^{\sharp}_{t}\cap\partial\Omega^{\sharp}}\frac{1}{\bar{\beta}v(x)}\,dA\Big{)}\int_{0}^{\mu_{v}(t)}(f^{\sharp})^{*}(s)\,ds$

for all $t>0$. Thus we complete the proof of the lemma. $\square$

######  Lemma 3.2.

Suppose $u$ and $v$ are solutions to (1.3) and (1.4). Then both $u$ and $v$
attain their minima on $\partial\Omega$ and $\partial\Omega^{\sharp}$.
Moreover if we denote by $u_{0}$ and $v_{0}$ the minima of $u$ and $v$
respectively, then

(3.6) $\displaystyle u_{0}\leq v_{0}.$

###### Proof.

Recall that $f$ is nonnegative, then $-\Delta u\geq 0$ and $-\Delta v\geq 0$.
Therefore $u$ and $v$ attain their minima on $\partial\Omega$ and
$\partial\Omega^{\sharp}$. Moreover $u(x)>u_{0}$ for $x\in\Omega$,
$v(x)>v_{0}$ for $x\in\Omega^{\sharp}$ unless $f$ is identically zero.

Noting that $v(x)=v_{0}$ on $\partial\Omega^{\sharp}$ and using integration by
parts we compute that

$\displaystyle v_{0}|\partial\Omega^{\sharp}|^{2}=$
$\displaystyle\int_{\partial\Omega^{\sharp}}\frac{1}{\bar{\beta}}\,dA\int_{\partial\Omega^{\sharp}}\bar{\beta}v(x)\,dA$
$\displaystyle=$
$\displaystyle\int_{\partial\Omega^{\sharp}}\frac{1}{\bar{\beta}}\,dA\int_{\Omega^{\sharp}}-\Delta
v\,dx$ $\displaystyle=$
$\displaystyle\frac{1}{\theta}\int_{\partial\Omega}\frac{1}{\beta(x)}\,dA\int_{\Omega^{\sharp}}f^{\sharp}(x)\,dx$
$\displaystyle=$
$\displaystyle\frac{1}{\theta^{2}}\int_{\partial\Omega}\frac{1}{\beta(x)}\,dA\int_{\Omega}-\Delta
u(x)\,dx$ $\displaystyle=$
$\displaystyle\frac{1}{\theta^{2}}\int_{\partial\Omega}\frac{1}{\beta(x)}\,dA\int_{\partial\Omega}\beta(x)u(x)\,dA,$

where we have used equations (1.3), (1.4), and equality (1.5). By the Hölder
inequality, it holds

$(\int_{\partial\Omega}\sqrt{u(x)}\,dA)^{2}\leq\int_{\partial\Omega}\frac{1}{\beta(x)}\,dA\int_{\partial\Omega}\beta(x)u(x)\,dA,$

then we have

$\displaystyle
v_{0}|\partial\Omega^{\sharp}|^{2}\geq\frac{1}{\theta^{2}}(\int_{\partial\Omega}\sqrt{u(x)}\,dA)^{2}\geq\frac{u_{0}}{\theta^{2}}|\partial\Omega|^{2}\geq
u_{0}|\partial\Omega^{\sharp}|^{2},$

where we used isoperimetric inequality (2.6) in the last inequality. Thus we
have $u_{0}\leq v_{0}$, proving (3.6). $\square$

Now we turn to prove Theorem 1.1.

###### Proof of Theorem 1.1.

Set $\tilde{\mu}_{u}(t)=\frac{\mu_{u}(t)}{\theta}$. Dividing inequality (3.1)
by $\tilde{\mu}_{u}(t)^{\frac{2}{n}-1}$ and integrating over $[0,\tau]$, we
obtain

$\displaystyle\theta^{2}\int_{0}^{\tau}\tilde{\mu}_{u}(t)\,dt\leq$
$\displaystyle\int_{0}^{\tau}\left(a^{2}(\tilde{\mu}_{u}(t))\tilde{\mu}_{u}(t)^{\frac{2}{n}-1}\left(\int_{0}^{\mu_{u}(t)}f^{*}(s)\,ds\right)\left(-\mu_{u}^{\prime}(t)\right)\right)\,dt$
$\displaystyle+\int_{0}^{\tau}a^{2}(\tilde{\mu}_{u}(t))\tilde{\mu}_{u}(t)^{\frac{2}{n}-1}\int_{0}^{\mu_{u}(t)}f^{*}(s)\,ds\left(\int_{\partial\Omega_{t}\cap\partial\Omega}\frac{1}{\beta(x)}\frac{1}{u(x)}\,dA\right)dt$
$\displaystyle\leq$
$\displaystyle\theta^{2}\int_{0}^{\tau}\left(a^{2}(\tilde{\mu}_{u}(t))\tilde{\mu}_{u}(t)^{\frac{2}{n}-1}\int_{0}^{\tilde{\mu}_{u}(t)}(f^{\sharp})^{*}(s)\,ds\right)(-d\tilde{\mu}_{u}(t))$
$\displaystyle+\theta
a^{2}(|\Omega^{\sharp}|)|\Omega^{\sharp}|^{\frac{2}{n}-1}\int_{0}^{|\Omega^{\sharp}|}(f^{\sharp})^{*}(s)\,ds\int_{0}^{\tau}\left(\int_{\partial\Omega_{t}\cap\partial\Omega}\frac{1}{\beta(x)}\frac{1}{u(x)}dA\right)\,dt,$

where we have used

$\mu_{u}(t)^{\frac{2}{n}-1}\int_{0}^{\mu_{u}(t)}f^{*}(s)\,ds\leq|\Omega|^{\frac{2}{n}-1}\int_{\Omega}f(x)\,dx=|\Omega|^{\frac{2}{n}-1}\int_{0}^{|\Omega|}f^{*}(s)\,ds,$

which follows from the assumption (1.6) and the fact $a(s)$ is monotone
increasing in $s$. Applying Fubini’s theorem, we estimate

(3.7)
$\displaystyle\int_{0}^{\tau}\int_{\partial\Omega_{t}\cap\partial\Omega}\frac{1}{\beta(x)u(x)}\,dA\,dt\leq\int_{0}^{\infty}\int_{\partial\Omega_{t}\cap\partial\Omega}\frac{1}{\beta(x)u(x)}\,dA\,dt=\int_{\partial\Omega}\frac{1}{\beta(x)}\,dA,$

then we conclude

(3.8) $\displaystyle\theta^{2}\int_{0}^{\tau}\tilde{\mu}_{u}(t)\,dt\leq$
$\displaystyle-\theta^{2}F(\tilde{\mu}_{u}(\tau))+\theta^{2}F(\tilde{\mu}_{u}(0))$
$\displaystyle+\theta
a^{2}(|\Omega^{\sharp}|)|\Omega^{\sharp}|^{\frac{2}{n}-1}\int_{0}^{|\Omega^{\sharp}|}(f^{\sharp})^{*}(s)\,ds\int_{\partial\Omega}\frac{1}{\beta(x)}\,dA,$

where $F(s)$ is defined by

$F(s):=\int_{0}^{s}a^{2}(\sigma)\sigma^{\frac{2}{n}-1}\int_{0}^{\sigma}(f^{\sharp})^{*}(r)\,drd\sigma.$

Substituting (1.5) into inequality (3.8) yields

(3.9)
$\displaystyle\int_{0}^{\tau}\tilde{\mu}_{u}(t)\,dt\leq-F(\tilde{\mu}_{u}(\tau))+F(\tilde{\mu}_{u}(0))+a^{2}(|\Omega^{\sharp}|)|\Omega^{\sharp}|^{\frac{2}{n}-1}\int_{0}^{|\Omega^{\sharp}|}(f^{\sharp})^{*}(s)\,ds\int_{\partial\Omega^{\sharp}}\frac{1}{\bar{\beta}}\,dA.$

For $\tau>v_{0}$, we have
$\Omega^{\sharp}_{\tau}\subset\subset\Omega^{\sharp}$, and then equality (3.2)
becomes to

$\displaystyle\mu_{v}^{\frac{2n-2}{n}}(t)=a^{2}\big{(}\mu_{v}(t)\big{)}\Big{(}-\mu_{v}^{\prime}(t)\Big{)}\int_{0}^{\mu_{v}(t)}(f^{\sharp})^{*}(s)\,ds,$

implying

(3.10)
$\displaystyle\int_{0}^{\tau}\mu_{v}(t)\,dt+F(\mu_{v}(\tau))=\int_{0}^{v_{0}}\mu_{v}(t)\,dt+F(\mu_{v}(v_{0})).$

Dividing equation (3.2) by $\mu_{v}(t)^{\frac{2-n}{n}}$ and integrating over
$[0,v_{0}]$ yields

$\int_{0}^{v_{0}}\mu_{v}(t)\,dt+F(\mu_{v}(v_{0}))=F(\mu_{v}(0))+a^{2}(|\Omega^{\sharp}|)|\Omega^{\sharp}|^{\frac{2}{n}-1}\int_{0}^{|\Omega^{\sharp}|}(f^{\sharp})^{*}(s)\,ds\int_{\partial\Omega^{\sharp}}\frac{1}{\bar{\beta}}\,dA,$

and substituting above equality to (3.10), we obtain for all $\tau>v_{0}$ that

(3.11) $\displaystyle\int_{0}^{\tau}\mu_{v}(t)\,dt+F(\mu_{v}(\tau))=$
$\displaystyle
F(\mu_{v}(0))+a^{2}(|\Omega^{\sharp}|)|\Omega^{\sharp}|^{\frac{2}{n}-1}\int_{0}^{|\Omega^{\sharp}|}(f^{\sharp})^{*}(s)\,ds\int_{\partial\Omega^{\sharp}}\frac{1}{\bar{\beta}}\,dA.$

Combining inequality (3.9) and equality (3.11), we get

(3.12)
$\displaystyle\int_{0}^{\tau}\tilde{\mu}_{u}(t)\,dt-\int_{0}^{\tau}\mu_{v}(t)\,dt\leq$
$\displaystyle-F(\tilde{\mu}_{u}(t))+F(\mu_{v}(t))+F(\tilde{\mu}_{u}(0))-F(\mu_{v}(0))$
$\displaystyle=$ $\displaystyle-F(\tilde{\mu}_{u}(t))+F(\mu_{v}(t)),$

where in the equality we used that

$\displaystyle\tilde{\mu}_{u}(0)=\frac{|\Omega|}{\theta}=|\Omega^{\sharp}|=\mu_{v}(0).$

Letting $\tau\to\infty$ in (3.12), we get

$\int_{0}^{\infty}\mu_{u}(t)\,dt\leq\int_{0}^{\infty}\theta\mu_{v}(t)\,dt,$

where we have used
$\displaystyle\lim_{\tau\to\infty}\mu_{u}(\tau)=\lim_{\tau\to\infty}\mu_{v}(\tau)=0$.
Hence (1.7) holds true. $\square$

Now we deal with the case $n=2$ and $f(x)\equiv 1$.

###### Proof of Theorem 1.3.

When $n=2$ and $f(x)\equiv 1$, inequality (3.1) gives

(3.13) $\displaystyle\theta\leq$
$\displaystyle\Big{(}-\mu_{u}^{\prime}(t)+\int_{\partial\Omega_{t}\cap\partial\Omega}\frac{1}{\beta(x)u(x)}\,dA\Big{)}a^{2}(\mu_{u}(t)/\theta)$
$\displaystyle\leq$
$\displaystyle\Big{(}-\mu_{u}^{\prime}(t)+\int_{\partial\Omega_{t}\cap\partial\Omega}\frac{1}{\beta(x)u(x)}\,dA\Big{)}a^{2}(|\Omega^{\sharp}|),$

where we have used the fact that $a(s)$ is monotone increasing in $s$ in the
last inequality. Integrating inequality (3.13) over $[0,\tau]$ yields

(3.14) $\displaystyle\theta\tau\leq$
$\displaystyle\Big{(}-\mu_{u}(\tau)+|\Omega|+\int_{0}^{\tau}\int_{\partial\Omega_{t}\cap\partial\Omega}\frac{1}{\beta(x)u(x)}\,dA\,dt\Big{)}a^{2}(|\Omega^{\sharp}|)$
$\displaystyle\leq$
$\displaystyle\Big{(}-\mu_{u}(\tau)+|\Omega|+\int_{\partial\Omega}\frac{1}{\beta(x)}\,dA\Big{)}a^{2}(|\Omega^{\sharp}|),$

where we used (3.7) in the last inequality.

Analogously, by using equality (3.2) we have

(3.15)
$\displaystyle\tau=\Big{(}-\mu_{v}(\tau)+|\Omega^{\sharp}|+\int_{\partial\Omega^{\sharp}}\frac{1}{\bar{\beta}}\,dA\Big{)}a^{2}(|\Omega^{\sharp}|)$

for $\tau\geq v_{0}$, then putting (3.14) and (3.15) together we get

$\mu_{u}(\tau)-\theta\mu_{v}(\tau)\leq 0.$

For $\tau<v_{0}$, it follows clearly that

$\displaystyle\mu_{u}(\tau)\leq|\Omega|=\theta|\Omega^{\sharp}|=\theta\mu_{v}(\tau),$

thus

$\mu_{u}(\tau)-\theta\mu_{v}(\tau)\leq 0$

holds for all $\tau>0$, hence inequality (1.8) holds.

If inequality (1.8) holds as an equality, then inequality (3.14) holds as an
equality as well, which implies $\Omega_{\tau}\subset\subset\Omega$, and
isoperimetric inequality (2.6) holds as an equality for $\Omega_{\tau}$. Hence
$\Omega_{\tau}$ is a geodesic ball in $M_{\kappa}$ for $\tau\geq v_{0}$. So
$\Omega$ is a round geodesic ball in $M_{\kappa}$, $\theta=1$, and $u(x)$ is a
radial function. Thus we conclude $\beta(x)$ is a constant on $\partial\Omega$
and $M$ is isometric to $M_{\kappa}$. We complete the proof of Theorem 1.3.
$\square$

## Acknowledgments

We thank Professor Ying Zhang for his encouragement and support, and are
grateful to Professors Daguang Chen and Carlo Nitsch for some helpful
discussions and comments. We also would like to thank the anonymous referees
for catching some typos, which improved the readability of this paper.

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