Outline

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

Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prerequisites ?	None
Corequisites ?	None
Prohibitions ?	None
Assumed knowledge ?	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.
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	Laurentiu Paunescu, laurentiu.paunescu@sydney.edu.au
Lecturer(s)	Florica-Corina Cirstea, florica.cirstea@sydney.edu.au

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.

For more information see guide to grades.

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.

Academic integrity

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.

Use of generative artificial intelligence (AI) and automated writing tools

You may only use generative AI and automated writing tools in assessment tasks if you are permitted to by your unit coordinator. If you do use these tools, you must acknowledge this in your work, either in a footnote or an acknowledgement section. The assessment instructions or unit outline will give guidance of the types of tools that are permitted and how the tools should be used.

Your final submitted work must be your own, original work. You must acknowledge any use of generative AI tools that have been used in the assessment, and any material that forms part of your submission must be appropriately referenced. For guidance on how to acknowledge the use of AI, please refer to the AI in Education Canvas site.

The unapproved use of these tools or unacknowledged use will be considered a breach of the Academic Integrity Policy and penalties may apply.

Studiosity is permitted unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission as detailed on the Learning Hub’s Canvas page.

Outside assessment tasks, generative AI tools may be used to support your learning. The AI in Education Canvas site contains a number of productive ways that students are using AI to improve their learning.

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
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.

Outcomes
Graduate qualities

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.

Graduate qualities

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

Learning outcomes	Graduate qualities
Learning outcomes

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.

Additional information

Disclaimer

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

To help you understand common terms that we use at the University, we offer an online glossary.

Current students

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Weekly schedule

Weekly schedule

Study commitment

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Disclaimer

On this page

Useful links

Media

Student links

About us

Connect

Current students

MATH4315: Variational Methods

Semester 2, 2020 [Normal day] - Camperdown/Darlington, Sydney

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Weekly schedule

Weekly schedule

Study commitment

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Disclaimer

On this page

Useful links

Media

Student links

About us

Connect