C4.6 Fixed Point Methods for Nonlinear PDEs - Material for the year 2019-2020

2019-2020
Lecturer(s): 
Prof. Melanie Rupflin
General Prerequisites: 

Basic results on weak derivatives and Sobolev spaces either from B4.3 Distribution Theory or from C4.3 Functional Analytic Methods for PDEs. Some knowledge of functional analysis, mainly notions of Banach spaces, and weak convergence, is useful.

Course Term: 
Hilary
Course Lecture Information: 

16 lectures

Course Weight: 
1.00 unit(s)
Course Level: 
M

Assessment type:

Course 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.

Course 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. Recap of basic results on Sobolev spaces. 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)

Reading List: 
  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, onlinear 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).