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Last updated:

November 4, 2022


Unlimited Duration


This course includes:

Unlimited Duration

Badge on Completion

Certificate of completion

Unlimited Duration


The main goal of this course is to give the students a solid foundation in the theory of elliptic and parabolic linear partial differential equations.

It is the second semester of a two-semester, graduate-level sequence on Differential Analysis.

Course Curriculum

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

About the instructor

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Massachusetts Institute of Technology