Synopsis for C5.1b: Fixed Point Methods for Nonlinear Partial Differential Equations

Level: M-level Method of assessment: Written examination.
Weight: Half-unit (OSS paper code 2B75)

Recommended Prerequisites

C5.1a: Methods of Functional Analysis for PDEs. Some knowledge of functional analysis, in particular Banach spaces (as in B4) and compactness (as in Part A Topology), is useful.

Overview

This course gives an introduction to the techniques of nonlinear functional analysis with emphasis on the major fixed point theorems and their applications to nonlinear differential equations and variational inequalities, which abound in applications such as fluid and solid mechanics, population dynamics and geometry.

Learning Outcomes

Besides becoming acquainted with the fixed point theorems of Banach, Brouwer and Schauder, students will see the abstract principles in a concrete context. Hereby they also reinforce techniques from elementary topology, functional analysis, Banach spaces, compactness methods, calculus of variations and Sobolev spaces.

Synopsis

Examples of nonlinear differential equations and variational inequalities. Contraction Mapping Theorem and applications. Brouwer's fixed point theorem, proof via Calculus of Variations and Null-Lagrangians. Compact operators and Schauder's fixed point theorem. Applications of Schauder's fixed point theorem to nonlinear elliptic equations. Variational inequalities and monotone operators. Applications of monotone operator theory to nonlinear elliptic equations (p-Laplacian, stationary Navier-Stokes)

1. Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathematics (American Mathematical Society, 2004).
2. E. Zeidler, Nonlinear Functional Analysis I & II (Springer-–Verlag, 1986/89).
3. M. S. Berger, Nonlinearity and Functional Analysis (Academic Press, 1977).
4. K. Deimling, Nonlinear Functional Analysis (Springer–-Verlag, 1985).
5. L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute Lecture Notes (American Mathematical Society, 2001).
6. R.E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, vol.49 (American Mathematical Society, 1997).