### Partial Differential Equation Assignment 1

Assignment 1 as a (PDF)

### Problem 1

Prove that a Harmonic function with an interior maximum is constant.

### Problem 2

Write out the laplacian in planepolar coordinates.

### Problem 3

A Green’s function on ^{n} is a harmonic function on ^{n} \{0} which depends only on the radius (for example log r on ^{2}). Find nontrivial Green’s functions for all dimensions.

### Problem 4

The heat equation for a function u: × [0, ∞) is . Find all solutions of the form u = ƒ(t)g(x).

### Problem 5

Find all solutions u of the heat equation on [0, 1] × [0, ∞) with the u = 0 on ({0} ∪ {1}) × [0, ∞).

### Partial Differential Equation Assignment 2

Assignment 2 as a (PDF)

### Problem 1

Let u be a function on the ball B_{1}(0) ⊂ ^{2} with ∫_{B1(0)} |u|^{p} <]infty for some constant p > 2. Show that u is holder continuous. [Hint: Use Morrey on ∫ 1.|u|^{2} ]

### Problem 2

Let u: ^{n} → , and define OSC_{Br(x)}u = sup_{Br(x)}u - inf_{Br(x)}u. Show that if there is some constant 0 < γ < 1 with

osc_{Br(x)}u ≤ γ osc_{B2r(x)}u

for all x and all r then u is Holder continuous.

### Problem 3

Let L be a uniformly elliptic 2nd order operator in divergence form taking

Let u be a function with Lu ≥ 0, and Φ: → a function with Φ’, Φ" ≥ 0. Show that L(Φ(u))≥ 0.

### Problem 4

Let L be an operator as in question 3, and let u be an L harmonic function. Prove that |u|^{2} is holder contiuous. [This is likely to be difficult.]