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Browse materialsMIT 18.156 · Spring 2004

Differential Analysis

Art: Anonymous painter,15th century, Westerner and Arab practicing geometry 15th century manuscript, 15th… · Public domain

Instructor: Prof. Jeff Viaclovsky
Term: Spring 2004
Level: Graduate
What's inside: 1 lecture note · 25 other files · 3 web pages
Materials open: MIT OCW

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18.156 · Spring 2016 · Study here
18.156 · Spring 2004 · Study here (this version)

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Source: MIT OpenCourseWare 18.156 Differential Analysis, Spring 2004, Prof. Jeff Viaclovsky — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

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Lecture Notes

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MIT OpenCourseWare, Prof. Jeff Viaclovsky, 18.156 Differential Analysis, Spring 2004.

License: CC BY-NC-SA 4.0 · Edition: Spring 2004 · Open the original course on OCW ↗

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Differential Analysis

Before you start

What's here, and what isn't

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Materials by section

Lecture Notes

Assignments

Lecture 0: Course Overview Examples of Harmonic Functions Fundamental Solutions for Laplacian and Heat Operator

Lecture 1: Harmonic Functions and Mean Value Theorem Maximum Principle and Uniqueness Harnack Inequality Derivative Estimates for Harmonic Functions Green’s Representation Formula

Lecture 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

Lecture 3: Weak Solutions Further Properties of Green’s Functions Weyl’s Lemma: Regularity of Weakly Harmonic Functions

Lecture 4: A Removable Singularity Theorem Laplacian in General Coordinate Systems Asymptotic Expansions

Lecture 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

Lecture 6: Weak Maximum Princple for Linear Elliptic Operators Uniqueness of Solutions to Dirichlet Problem A Priori C⁰ Estimates for Solutions to Lu = f, c leq 0 Strong Maximum Principle

Lecture 7: Quasilinear Equations (Minimal Surface Equation) Fully Nonlinear Equations (Monge-Ampere Equation) Comparison Principle for Nonlinear Equations

Lecture 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)

Lecture 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)

Lecture 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

Lecture 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

Lecture 13: Interior Schauder Estimate

Lecture 14: Global Schauder Estimate Banach Spaces and Contraction Mapping Principle

Lecture 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

Lecture 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

Lecture 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

Lecture 18: Sobolev Imbedding Theorem p < n Morrey’s Inequality

Lecture 19: Sobolev Imbedding for p > n, Hölder Continuity Kondrachov Compactness Theorem Characterization of W^{1,p} in Terms of Difference Quotients

Lecture 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 L²

Lecture 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}

Lecture 22: Weak L² Maximum Principle Global a priori W^{k+2,2} Estimate for Lu = f, f in W^{k,2}, c(x) leq 0

Lecture 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.

Lecture 24: W^{2,p} Estimate for N.P., 1 < p < infty W^{2,p} Estimate for Operators L with Continuous Leading Order Coefficients