| LEC # | TOPICS | KEY DATES |
|---|---|---|
| 0 | Course Overview Examples of Harmonic Functions Fundamental Solutions for Laplacian and Heat Operator |
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| 1 | Harmonic Functions and Mean Value Theorem Maximum Principle and Uniqueness Harnack Inequality Derivative Estimates for Harmonic Functions Green's Representation Formula |
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| 2 | Definition of Green's Function for General Domains Green's Function for a Ball The Poisson Kernel and Poisson Integral Solution of Dirichlet Problem in Balls for Continuous Boundary Data Continuous + Mean Value Property <-> Harmonic |
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| 3 | Weak Solutions Further Properties of Green's Functions Weyl's Lemma: Regularity of Weakly Harmonic Functions |
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| 4 | A Removable Singularity Theorem Laplacian in General Coordinate Systems Asymptotic Expansions |
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| 5 | Kelvin Transform I: Direct Computation Harmonicity at Infinity, and Decay Rates of Harmonic Functions Kelvin II: Poission Integral Formula Proof Kelvin III: Conformal Geometry Proof |
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| 6 | Weak Maximum Princple for Linear Elliptic Operators Uniqueness of Solutions to Dirichlet Problem A Priori C^0 Estimates for Solutions to Lu = f, c leq 0 Strong Maximum Principle |
Homework 1 due |
| 7 | Quasilinear Equations (Minimal Surface Equation) Fully Nonlinear Equations (Monge-Ampere Equation) Comparison Principle for Nonlinear Equations |
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| 8 | If Delta u in L^{infty}, then u in C^{1,alpha}, any 0 < alpha < 1 If Delta u in L^{p}, p > n, then u in C^{1,alpha}, p = n/(1 - alpha) |
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| 9 | If Delta u in C^{alpha}, alpha > 0, then u in C^{2} Moreover, if alpha < 1, then u in C^{2,alpha} (Proof to be completed next lecture) |
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| 10 | Interior C^{2,alpha} Estimate for Newtonian Potential Interior C^{2,alpha} Estimates for Poisson's Equation Boundary Estimate on Newtonian Potential: C^{2,alpha} Estimate up to the Boundary for Domain with Flat Boundary Portion |
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| 11 | Schwartz Reflection Reviewed Green's Function for Upper Half Space Reviewed C^{2,alpha} Boundary Estimate for Poisson's Equation for Flat Boundary Portion Global C^{2,alpha} Estimate for Poisson's Equation in a Ball for Zero Boundary Data C^{2,alpha} Regularity of Dirichlet Problem in a Ball for C^{2,alpha} Boundary Data |
Homework 2 due |
| 12 | Global C^{2,alpha} Solution of Poisson's Equation Delta u = f in C^{alpha}, for C^{2,alpha} Boundary Values in Balls Constant Coefficient Operators Interpolation between Hölder Norms |
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| 13 | Interior Schauder Estimate | |
| 14 | Global Schauder Estimate Banach Spaces and Contraction Mapping Principle |
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| 15 | Continuity Method Can Solve Dirichlet Problem for General L Provided can Solve for Laplacian Corollary: Solution of C^{2, alpha} Dirichlet Problem in Balls for General L Solution of Dirichlet Problem in C^{2,alpha} for Continuous Boundary Values, in Balls |
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| 16 | Elliptic Regularity: If f and Coefficients of L in C^{k,alpha}, Lu = f, then u in C^{k+2,alpha} C^{2,alpha} Regularity up to the Boundary |
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| 17 | C^{k,alpha} Regularity up to the Boundary Hilbert Spaces and Riesz Representation Theorem Weak Solution of Dirichlet Problem for Laplacian in W^{1,2}_0 Weak Derivatives Sobolev Spaces |
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| 18 | Sobolev Imbedding Theorem p < n Morrey's Inequality |
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| 19 | Sobolev Imbedding for p > n, Hölder Continuity Kondrachov Compactness Theorem Characterization of W^{1,p} in Terms of Difference Quotients |
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| 20 | Characterization of W^{1,p} in Terms of Difference Quotients (cont.) Interior W^{2,2} Estimates for W^{1,2}_0 Solutions of Lu = f in L^2 |
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| 21 | Interior W^{k+2,2} Estimates for Solutions of Lu = f in W^{k,2} Global (up to the Boundary) W^{k+2,2} Estimates for Solutions of Lu = f in W^{k,2} |
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| 22 | Weak L^2 Maximum Principle Global a priori W^{k+2,2} Estimate for Lu = f, f in W^{k,2}, c(x) leq 0 |
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| 23 | Cube Decomposition Marcinkiewicz Interpolation Theorem L^p Estimate for the Newtonian Potential W^{1,p} Estimate for N.P. W^{2,2} Estimate for N.P. |
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| 24 | W^{2,p} Estimate for N.P., 1 < p < infty W^{2,p} Estimate for Operators L with Continuous Leading Order Coefficients |
