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


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*Complex Analysis by Lars Ahlfors

*Complex Analysis by Elias M. Stein & Rami Shakarchi

*Real Analysis: Measure Theory, Integration, and Hilbert Spaces by Elias M. Stein & Rami Shakarchi

*Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland 

*Partial Differential Equations by Lawrence C. Evans


Then a graduate level course in Functional Analysis.
A: 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.
A: Some interesting courses that can be done with a standard PDE course: (with exemplary lecture notes so you can have a look into these)


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*Calculus of Variations


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*Finite dimensional optimization problems

*Calculus of variations with one independent variable

*Calculus of variations and elliptic partial differential equations

*Deterministic optimal control and viscosity solutions


*Nonlinear Evolution Equations


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*The Contraction mapping Theorem

*Sobolev Spaces and Laplace’s Equation

*The Diffusion Equation


*Reaction-Diffusion Equations

*Interactions between Dynamical Systems and PDEs


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*Implicit Functions and Lyapunov-Schmidt

*Crandall-Rabinowitz and Local Bifurcations

*Sturm-Liouville and Stability of Travelling Waves

*Exponential Dichotomies and Evans Function


*PDEs and Mathematical Modeling


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*Continuum Mechanics

*Hydrodynamics

*Elasticity Theory


*Semi-Group Theory

*Variational Methods


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*Sobolev Spaces

*Homogenization

*Monotone Problems

*The Bochner Integral


*Numerics of PDEs


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*Finite Difference Methods

*Ritz-Galerkin Method

*Finite Element Methods

*Finite Volume Methods



And some Analysis courses:


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*Fourier Analysis


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*Laplace Transform

*Fourier Series

*Fourier Transform

*Schwartz Functions


*Distribution Theory


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*Distributions

*Tempered Distributions

*Distributions with compact support


*Dynamical Systems


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*Linear Systems and Stability

*Nonlinear Systems and Stability

*Bifurcation Theory

*Chaos Theory


*Differential Forms


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*Differential Forms: Definition

*Hodge Star Operator

*Lemma of Poincare

*Stokes' Theorem


*Nonlinear Functional Analysis


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*Analysis in Banach Spaces

*Brouwer Mapping Degree

*Leray-Schauder Mapping Degree


