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Unit of study_

MATH4315: Variational Methods

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.

Code MATH4315
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
Prerequisites:
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None
Corequisites:
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None
Prohibitions:
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None
Assumed knowledge:
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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.

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.

Unit outlines

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