PDE/Analysis graduate courses

Question

I'm just starting my graduate studies in Analysis and PDE's and am a bit lost about what topics should I cover in order to do a good Phd program. I`ve already done the usual undergrad courses, plus Real and complex analysis (graduate level), functional analysis and measure theory.

So, if you guys can recommend me which courses I should do, (I can get my university to open new courses as needed), and which books I should study, it'd make me really happy

Answer 1

Some interesting courses that can be done with a standard PDE course: (with exemplary lecture notes so you can have a look into these)

• Calculus of Variations
  1. Finite dimensional optimization problems
  2. Calculus of variations with one independent variable
  3. Calculus of variations and elliptic partial differential equations
  4. Deterministic optimal control and viscosity solutions
• Nonlinear Evolution Equations
  1. The Contraction mapping Theorem
  2. Sobolev Spaces and Laplace’s Equation
  3. The Diffusion Equation
• Reaction-Diffusion Equations
• Interactions between Dynamical Systems and PDEs
  1. Implicit Functions and Lyapunov-Schmidt
  2. Crandall-Rabinowitz and Local Bifurcations
  3. Sturm-Liouville and Stability of Travelling Waves
  4. Exponential Dichotomies and Evans Function
• PDEs and Mathematical Modeling
  1. Continuum Mechanics
  2. Hydrodynamics
  3. Elasticity Theory
• Semi-Group Theory
• Variational Methods
  1. Sobolev Spaces
  2. Homogenization
  3. Monotone Problems
  4. The Bochner Integral
• Numerics of PDEs
  1. Finite Difference Methods
  2. Ritz-Galerkin Method
  3. Finite Element Methods
  4. Finite Volume Methods

And some Analysis courses:

• Fourier Analysis
  1. Laplace Transform
  2. Fourier Series
  3. Fourier Transform
  4. Schwartz Functions
• Distribution Theory
  1. Distributions
  2. Tempered Distributions
  3. Distributions with compact support
• Dynamical Systems
  1. Linear Systems and Stability
  2. Nonlinear Systems and Stability
  3. Bifurcation Theory
  4. Chaos Theory
• Differential Forms
  1. Differential Forms: Definition
  2. Hodge Star Operator
  3. Lemma of Poincare
  4. Stokes' Theorem
• Nonlinear Functional Analysis
  1. Analysis in Banach Spaces
  2. Brouwer Mapping Degree
  3. Leray-Schauder Mapping Degree

Answer 2

Graduate level course in Complex Analysis, Real Analysis and PDE's which usually cover the following textbooks:

1. Complex Analysis by Lars Ahlfors
2. Complex Analysis by Elias M. Stein & Rami Shakarchi
3. Real Analysis: Measure Theory, Integration, and Hilbert Spaces by Elias M. Stein & Rami Shakarchi
4. Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland
5. Partial Differential Equations by Lawrence C. Evans

Then a graduate level course in Functional Analysis.

Answer 3

Fixed Point Theory is an important part of analysis to cover. And if you want to mix analysis with a little bit geometry, you MUST check the two brilliant books by I. Chavel: eigenvalues in riemannian geometry and isoperimetric inequalities. They do reveal beatiful applications of PDE's to geometric problems.