18.156 | Spring 2004 | Graduate

Differential Analysis

Lecture Notes

The lecture notes were prepared by two former students in the class. Zuoqin Wang prepared lecture notes 0 through 11 in LaTeX, and Yanir Rubinstein prepared lectures 12 through 24 in TeX. They used Professor Viaclovsky’s handwritten notes in producing them.

LEC # TOPICS LECTURE NOTES
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

(PDF)
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

(PDF)
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

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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

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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 (PDF)
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

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