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18.156

Differential Analysis II

3-0-9 units · Graduate

Second part of a two-subject sequence. Covers variable coefficient elliptic, parabolic and hyperbolic partial differential equations.

Spring 2025 · Projection Theory

The class studies projection theory, starting from the first questions and building up to recent developments. Projection theory studies how a set *X* behaves under different orthogonal projections. Questions of this type aren't usually emphasized in the graduate analysis curriculum, but they come up in many areas of math, including harmonic analysis, analytic number theory, additive combinatorics, and homogeneous dynamics. We will survey several applications of projection theory. For each topic, we will introduce and motivate the topic and see how it connects with projection theory. We will prove something about each topic but not necessarily the strongest results.

Spring 2016 · Differential Analysis II: Partial Differential Equations and Fourier Analysis

Topics

  • Review of Harmonic Functions and the Perspective We Take on Elliptic PDE
  • Finding Other Second Derivatives from the Laplacian
  • Korn’s Inequality I
  • Korn’s Inequality II
  • Schauder’s Inequality
  • Using Functional Analysis to Solve Elliptic PDE
  • Sobolev Inequality I
  • Sobolev Inequality II
  • De Giorgi-Nash-Moser Inequality
  • Nonlinear Elliptic PDE I
  • Nonlinear Elliptic PDE II
  • Barriers
  • Minimal Graphs
  • Leray-Schauder Approach to Nonlinear PDE
  • Gauss Circle Problem I
  • Gauss Circle Problem II
  • Fourier Analysis in PDE and Interpolation
  • Applications of Interpolation
  • Calderon-Zygmund Inequality I
  • Calderon-Zygmund Inequality II
  • Littlewood-Paley Theory
  • Strichartz Inequality I
  • Strichartz Inequality II
  • The Nonlinear Schrödinger Equation

Readings

Spring 2004 · Differential Analysis

Topics

  • Course Overview
  • Harmonic Functions and Mean Value Theorem
  • Definition of Green’s Function for General Domains
  • Weak Solutions
  • A Removable Singularity Theorem
  • Kelvin Transform I: Direct Computation
  • Weak Maximum Princple for Linear Elliptic Operators
  • Quasilinear Equations (Minimal Surface Equation)
  • If Delta u in L^{infty}, then u in C^{1,alpha}, any 0 < alpha < 1
  • If Delta u in C^{alpha}, alpha > 0, then u in C^{2}
  • Interior C^{2,alpha} Estimate for Newtonian Potential
  • Schwartz Reflection Reviewed
  • Global C^{2,alpha} Solution of Poisson’s Equation Delta u = f in C^{alpha}, for C^{2,alpha} Boundary Values in Balls
  • Interior Schauder Estimate
  • Global Schauder Estimate
  • Continuity Method
  • Elliptic Regularity: If f and Coefficients of L in C^{k,alpha}, Lu = f, then u in C^{k+2,alpha}
  • C^{k,alpha} Regularity up to the Boundary
  • Sobolev Imbedding Theorem p < n
  • Sobolev Imbedding for p > n, Hölder Continuity
  • Characterization of W^{1,p} in Terms of Difference Quotients (cont.)
  • Interior W^{k+2,2} Estimates for Solutions of Lu = f in W^{k,2}
  • Weak L^2 Maximum Principle
  • Cube Decomposition
  • W^{2,p} Estimate for N.P., 1 < p < infty

Readings

  • Jost, Jurgen. Partial Differential Equations. New York: Springer-Verlag, 2002. ISBN: 0387954287.
    Required
  • Gilbarg, David, and Neil Trudinger. Elliptic Partial Differential Equations of Second Order. 2nd ed. Berlin: Springer-Verlag, 1983.
    Recommended

Prerequisites & course information

Prereq: 18.155 G (Spring) 3-0-9 units

Sources