OVERVIEW

Equazioni Differenziali 2 is devoted to the study of Sobolev spaces and to their use in the weak formulation and solution of elliptic partial differential equations (PDEs). The central idea of the course is to show how many differential problems can be reformulated abstractly as equations between function spaces, of the form (A : X -> Y), where the operator A encodes the PDE and the boundary conditions, while the spaces X and Y must be chosen appropriately. Sobolev spaces provide precisely the natural language for this reformulation and make it possible to apply the tools of functional analysis effectively to the study of existence, uniqueness and regularity of solutions. The course will take the Laplacian problem as a model and will then develop the theory for more general classes of second-order elliptic equations with boundary conditions.

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to provide an introduction on Sobolev spaces, and to provide variational interpretation of some partial differential equations, also studying the regularity of the solutions. As an application we will provide simple results of existence of regular solutions.

AIMS AND LEARNING OUTCOMES

To provide fundamental knowledge concerning the theory of weakly differentiable functions, including distributions and Sobolev spaces. To interpret weak solutions of partial differential equations as minimizers of variational problems and, conversely, through the Euler-Lagrange equations; to use this principle to deduce existence, uniqueness and regularity results for such equations. The twofold aim of the course is to provide fundamental tools and results in mathematical analysis in the framework of partial differential equations, as well as general principles of the calculus of variations.

Learning outcomes: understanding of the concepts and proofs presented during the lectures. Ability to develop proofs that are variants of those presented in the course, to construct examples and counterexamples, and to solve exercises related to the topics of the course.

PREREQUISITES

A basic knowledge of measure theory, L^p spaces and Banach spaces is recommended. It is not necessary to have attended the Equazioni Differenziali 1 course.

TEACHING METHODS

Traditional teaching (theoretical lessons on the blackboard).

Compensatory and dispensatory measures Disability/Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to facilitate the examination.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the teacher in agreement with the Referee.

To take advantage of the adaptations during the examination, fill in the Adaptation request form; the request will be automatically sent by the system to the teacher in charge of the teaching, to the Contact Person of your School/Area/Department and in copy to the Sector; you will also receive a copy of the request sent by e-mail.

The adjustments available to students are as follows:

• Additional time (+30% DSA)
• Additional time (+50% disability/invalidity)
• Additional time during oral exams to organise the answer
• Calculator (programmable and graphing calculators are not allowed)
• Conceptual Maps
• Tables and/or Forms
• Take the exam in written form
• Take the exam in oral form
• Tutor reader (for written tests only)
• Tutor-writer (for written tests only)

Your request for adaptations must be submitted at least 7 working days before the scheduled exam date.

All information for students with disabilities and DSA is available on the webpage: Services for students with disabilities or DSA | UniGe | University of Genoa

Reference for inclusion: Sergio Di Domizio - sergio.didomizio@unige.it

SYLLABUS/CONTENT

• Preliminaries and complements: normed spaces, dual spaces, weak topology, reflexive spaces, the Riesz representation theorem, the Lax-Milgram theorem, the direct method in the calculus of variations.
• Distributions: spaces of test functions, distributions, distributional derivatives, order of a distribution, convolution, compactly supported distributions, fundamental solutions.
• Sobolev spaces: definitions and first properties, approximation by smooth functions, extension theorems, Poincaré inequality, continuous and compact embeddings, dual spaces of negative order, trace spaces and trace theorems.
• The Laplacian problem: weak formulation, a priori estimates, existence of weak solutions, Dirichlet and Neumann boundary conditions, regularity of solutions.
• Second-order elliptic operators: definitions, existence results, regularity, maximum principles in the regular and general cases.
• Examples of elliptic problems: the Neumann problem, the p-Laplacian, the biharmonic equation, the elasticity system.
• Eigenvalue problems: Fredholm alternative, spectral problems for elliptic operators, adjoint operators and eigenvalues in Hilbert spaces, eigenvalues of the Laplacian.
• Elliptic operators in divergence form: formulation of the problem, weak maximum principle, uniqueness and existence of solutions

RECOMMENDED READING/BIBLIOGRAPHY

Evans, "Partial Differential Equations"

Gilbarg & Trudinger, "Elliptic Partial Differential Equations of Second Order".

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

Classes will be held according to the schedule available at the following page

https://corsi.unige.it/corsi/11907/studenti-orario

Class schedule

DIFFERENTIAL EQUATIONS 2

EXAMS

EXAM DESCRIPTION

Oral exam

ASSESSMENT METHODS

The oral exam will assess:

• the ability to identify suitable methods for solving the proposed problems;
• the ability to apply the identified methods;
• proof-writing and argumentative skills.

FURTHER INFORMATION

Ask the professor for other information not included in the teaching schedule.