Unit of study_

# MATH4315: Variational Methods

## Overview

Variational and spectral methods are foundational in mathematical models that govern the configurations of many physical systems. They have wide-ranging applications in areas such as physics, engineering, economics, differential geometry, optimal control and numerical analysis. In addition they provide the framework for many important questions in modern geometric analysis. This unit will introduce you to a suite of methods and techniques that have been developed to handle these problems. You will learn the important theoretical advances, along with their applications to areas of contemporary research. Special emphasis will be placed on Sobolev spaces and their embedding theorems, which lie at the heart of the modern theory of partial differential equations. Besides engaging with functional analytic methods such as energy methods on Hilbert spaces, you will also develop a broad knowledge of other variational and spectral approaches. These will be selected from areas such as phase space methods, minimax theorems, the Mountain Pass theorem or other tools in the critical point theory. This unit will equip you with a powerful arsenal of methods applicable to many linear and nonlinear problems, setting a strong foundation for understanding the equilibrium or steady state solutions for fundamental models of applied mathematics.

### Unit details and rules

Unit code MATH4315 Mathematics and Statistics Academic Operations 6 None None None Assumed knowledge of MATH2X23 or equivalent; MATH4061 or MATH3961 or equivalent; MATH3969 or MATH4069 or MATH4313 or equivalent. That is, real analysis, basic functional analysis and some acquaintance with metric spaces or measure theory. Yes

### Teaching staff

Coordinator Laurentiu Paunescu, laurentiu.paunescu@sydney.edu.au Florica-Corina Cirstea

## Assessment

Type Description Weight Due Length
Final exam (Open book) Final exam
Take-home final exam
60% Formal exam period 2 hours
Outcomes assessed:
Assignment Assignment 1
Written work
20% Week 07 Two weeks
Outcomes assessed:
Assignment Assignment 2
Written work
20% Week 12 Two weeks
Outcomes assessed:
= Type C final exam

### Assessment summary

two assignments to provide written solutions to questions. Detailed information for each assessment can be found on Canvas.

### Assessment criteria

Result name

Mark range

Description

High distinction

85 - 100

Representing complete or close to complete mastery of the material.

Distinction

75 - 84

Representing excellence, but substantially less than complete mastery.

Credit

65 - 74

Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.

Pass

50 - 64

Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

### Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

This unit has an exception to the standard University policy or supplementary information has been provided by the unit coordinator. This information is displayed below:

Late submission of assignments will not be considered. The mark for each assignment will be recorded only if the assignment has been submitted before the deadline. The better mark principle will be applied for each assignment.

The Current Student website  provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

## Learning support

### Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

### Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

### Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 Introduction, definition and properties of weak derivatives, and distributions Lecture and tutorial (4 hr)
Week 02 Definition of Sobolev spaces and basic properties Lecture and tutorial (4 hr)
Week 03 Approximation by smooth functions, extension operators and density results Lecture and tutorial (4 hr)
Week 04 Sobolev inequalities including the Sobolev-Gagliardo-Nirenberg inequality and Morrey's inequality Lecture and tutorial (4 hr)
Week 05 General Sobolev inequalities and compact embeddings and applications to variational formulation of some boundary value problems Lecture and tutorial (4 hr)
Week 06 Lax-Milgram Theorem and Stampacchia Theorem, applications to prove the existence and uniqueness of weak solutions for linear elliptic problems Lecture and tutorial (4 hr)
Week 07 The maximum principle for the Dirichlet problem, eigenfunctions and spectral decomposition Lecture and tutorial (4 hr)
Week 08 Introduction to the variational approach to nonlinear elliptic problems, review of differential calculus: Frechet / Gateaux differentiability, examples of differentiable functionals in abstract and concrete spaces Lecture and tutorial (4 hr)
Week 09 Definition of weak solutions of nonlinear elliptic problems as critical points of energy functionals; the direct method in the calculus of variations to determine minimizers of energy functionals Lecture and tutorial (4 hr)
Week 10 The Lagrange Multiplier Theorem and applications to eigenvalue problems Lecture and tutorial (4 hr)
Week 11 The Mountain Pass Theorem and applications to nonlinear elliptic problems Lecture and tutorial (4 hr)
Week 12 The Pohozaev identity and review of variational methods for solving nonlinear elliptic problems Lecture and tutorial (4 hr)

### Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

## Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

• LO1. understand the basic theory of Sobolev spaces, including the definitions of weak derivatives, distributions, Sobolev spaces and their elementary properties.
• LO2. work with Sobolev spaces and Sobolev embeddings in relation to the variational formulation of elliptic boundary value problems, the maximum principle, eigenfunctions and spectral decomposition.
• LO3. understand and work with the definition of weak solutions for second order elliptic equations
• LO4. work with and apply the methods of linear functional analysis such as the Lax-Milgram Theorem to prove existence and regularity of weak solutions for linear elliptic equations.
• LO5. understand and apply a range of methods in the Calculus of Variations to obtain existence of weak solutions to nonlinear problems as minimizers (possibly under constraints), mini-max critical points or via the Mountain Pass Theorem .
• LO6. apply mathematical logic and rigor to solving linear and nonlinear elliptic problems and express mathematical ideas coherently using precise mathematical language.

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.

### Outcome map

GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

This is the first time this unit is offered.