Introduction to Partial Differential Equations

Assignments

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 B₁(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.]