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1. Course Overview

Course Title: Variational Inequalities and Their Applications

Language: English

Instructor: Henrik Shahgholian (henriksh@kth.se)

This course is designed to introduce students to the powerful mathematical framework of variational inequalities and their wide-ranging applications in mathematical physics, engineering, finance, and biology. The core focus is on how these tools can be used to model and solve real-world problems that involve constraints, such as free boundaries or moving interfaces.

2. Course Goals

The primary goal of this course is to provide a solid understanding of how mathematics—particularly partial differential equations (PDEs) and functional analysis—can be applied to formulate problems in physics, mechanics, finance, biology, and industry.

Upon completion of this course, students will be able to:

Identify and formulate problems that can be modeled as variational inequalities.
Understand the fundamental mathematical tools and concepts behind these problems.
Gain insights into real-world applications and active research topics, particularly in the field of Free Boundary Problems.
Develop a foundation for further study and research, including potential master's theses ("Examensarbete").

3. Course Topics

The course is structured around key concepts and specific applications:

Core Concepts:

The Obstacle Problem: A central theme in the course, serving as the foundational example for variational inequalities.
Variational Inequalities: Introduction to the theoretical framework, including the projection theorem, existence and stability results, and comparison and maximum principles.
Mathematical Tools: A review of basic functional analysis, function spaces, Green's formula, and second-order elliptic partial differential equations.

Specific Applications (Examples of Free Boundary Problems):

Mechanics: Deformation of a membrane, bending of a plate, torsion of an elastic-plastic cylindrical bar.
Fluid Dynamics: Cavitation problems in hydrodynamic lubrication, flow with wake in a channel, filtration, and the dam problem.
Finance: American option theory.
Biology: Mathematical models for wound healing and tumor growth.
Engineering & Physics: The Stefan problem (phase transition), continuous casting, conservation laws in continuum physics, and shape optimization.

4. Prerequisites

Standard courses in analysis (e.g., multivariable calculus, real analysis, and basic PDE theory).
Prior knowledge of basic functional analysis is helpful but not strictly required.
Students are encouraged to contact the course leader for a personalized assessment of their background.

5. Course Literature

Primary Text:

Obstacle problems in mathematical physics by José-Francisco Rodrigues (1986); North-Holland, Mathematics Studies 134.

Note: This book is out of print, but copies of relevant parts will be made available to students.

Additional Materials:

Supplementary handouts and relevant research papers will be provided throughout the course.
Class Notes.

6. Examination

The course will be assessed through a two-part examination:

Homework Assignments: A set of take-home assignments designed to test both theoretical understanding and problem-solving skills.
Presentation: A final presentation where students will present a specific topic or a solution to a problem, demonstrating their understanding and communication skills.