MATJ5107 MA1: Shape Optimization and Free Boundary Problems (JSS30) (1 cr)

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Study level:

Postgraduate studies

Grading scale:

Pass - fail

Language:

English

Responsible organisation:

Faculty of Mathematics and Science

Curriculum periods:

2020-2021, 2021-2022

Description

The first part of the course is an introduction to the field of shape optimization, based on a complete study of some examples. Such problems pop up naturally from different fields of applied sciences, as mechanics, biology or medicine. Some of them are of engineering type and have immediate applications, but some others have already a long history so that they are rather considered pure mathematical questions, as for instance the isoperimetric and the Faber-Krahn inequalities. Roughly speaking, there are three main steps in the analysis of such a problem: the mathematical modelling, the qualitative analysis of the model (existence of a solution, properties of the solution such as topological structure, regularity, symmetry, etc.) and numerical approximation of the solutions. The last step is crucial for the comprehension of a shape optimization problem, as analytical solutions are known in very few cases, only. In the second part of the course, I will describe some recent results obtained in the field of spectral shape optimization problems. In order to approach these problems I will detail some free boundary and free discontinuity techniques and show how to use them for obtaining information on existence, regularity and geometry of the optimal shapes.

Learning outcomes

Students are familiar with the most relevant shape optimization problems and understand their mathematical formulation. They are able to show the existence of the solution and some of its basic properties such as symmetry and regularity properties.

Description of prerequisites

The course requires basic notions of partial differential equations, functional analysis, Sobolev and BV spaces.

Literature

Bucur, Dorin; Buttazzo, Giuseppe, Variational methods in shape optimization problems. Progress in Nonlinear Differential Equations and their Applications, 65. Birkhäuser Boston, Inc., Boston, MA, 2005.
Henrot, Antoine; Pierre, Michel, Shape variation and optimization. A geometrical analysis. English version of the French publication [MR2512810] with additions and updates. EMS Tracts in Mathematics, 28. European Mathematical Society (EMS), Zürich, 2018.

Completion methods

Method 1

Evaluation criteria:

Exam

Select all marked parts

Parts of the completion methods

Participation in teaching (1 cr)

Type:

Participation in teaching