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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.
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Description
It is the second semester of a two-semester, graduate-level sequence on Differential Analysis.
Course content
- 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
Instructor
Massachusetts Institute of Technology
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