Aim of the course is to give to the students some basic tools and topics in partial differential equations.

1. Introduction to Elliptic problems on graphs.
2. Elements of Graph Theory:
2.1 Generalities. Locally finite graphs.
2.2 Sobolev spaces on graphs.
2.3 Embedding results.
3. Variational methods:
3.1 A local minimum result for smooth functionals.
3.2 Multiple critical point results.
3.3 Infinitely many critical points.
4. Applications to the Dirichlet problems on graphs.

There are no mandatory prerequisites.

Knowledge and understanding:
At the end of the course the student will learn the basic notions on PDEs.

Applying knowledge and understanding:
At the end of the course the student will learn the methodologies on PDEs
and will be able to apply them to the study of various problems.

Making judgements:
At the end of the course the student will be able to apply the techniques of PDEs in order to solve new problems, also coming from real-world applications.

Communications skills:
At the end of the course the student will have the ability to express the fundamental notions of
PDEs using a rigorous terminology.

Learning skills:
During the course the student will learn the ability to study the notions of PDEs, also in order to use it in solving different kind of problems.

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Teaching

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

• G. Molica Bisci - V. Radulescu - R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162 Cambride University Press - ISBN: 9781107111943. Foreword by J. Mawhin.

• G. Molica Bisci - P. Pucci, Nonlinear Problems with Lack of Compactness, De Gruyter Series in Nonlinear Analysis and Applications, 36 (2021).

• G. Molica Bisci, Variational and Topological Methods for Nonlocal Fractional Periodic Equations, Recent developments in the Nonlocal Theory, De Gruyter (2018).

Assessment

The exam of Optimization Methods consists of an oral exam on the topics of the course.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.

Teaching

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

• G. Molica Bisci - V. Radulescu - R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162 Cambride University Press - ISBN: 9781107111943. Foreword by J. Mawhin.

• G. Molica Bisci - P. Pucci, Nonlinear Problems with Lack of Compactness, De Gruyter Series in Nonlinear Analysis and Applications, 36 (2021).

• G. Molica Bisci, Variational and Topological Methods for Nonlocal Fractional Periodic Equations, Recent developments in the Nonlocal Theory, De Gruyter (2018).

Assessment

The exam of Optimization Methods consists of an oral exam on the topics of the course.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.