Laplace's Equation and the Maximum Principle · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Steady landscapes with no interior peaks

Mathematics · Differential Equations · ages 22-23
Name ______________________   Date ____________

  1. Let u(x,y) = 5x - 12y + 8. What is the average of u over a circle centred at the origin?

    Answer: ______________

  2. Let u(x,y) = 3x + 4. What is the average of u over a circle centred at (1,1)?

    Answer: ______________

  3. Which of these functions is harmonic on the whole plane?

    • x^2 + y^2
    • x^2 - y^2
    • x^3
    • x^2 y

  4. Two solutions of the Dirichlet problem share the same boundary data. The maximum principle forces them to be identical.

    Circle one:   True   False

  5. A harmonic function in a disc extends continuously to the boundary with values between 0 and 5. What follows for the interior?

    • u must be constant inside
    • u can exceed 5 at interior points
    • The 0 to 5 bounds hold inside by the max and min principles
    • u must attain 5 at the centre

  6. Two harmonic functions agree on the boundary of a disc. Their difference w is harmonic with zero boundary values. What must w be?

    • w is zero everywhere
    • w is a nonzero constant
    • w attains a strict interior maximum
    • w must change sign inside

  7. A nonconstant harmonic function on a disc takes the value 9 at an interior point and is smaller everywhere else.

    Circle one:   True   False

  8. Let u(x,y) = x^2 - y^2. What is the average of u over a small circle centred at (2,1)?

    Answer: ______________

  9. Boundary data on a circle is zero except for one sharp spike. What is true of the harmonic extension inside?

    • It stays smooth and finite at every interior point
    • It repeats the spike at the center
    • It becomes infinite near the spike side

  10. A harmonic function in a disc extends continuously to boundary values between 0 and 5. What follows for the interior?

    • It must be constant inside
    • It stays between 0 and 5 inside
    • It may exceed 5 near the center
LightMySky · lightmysky.comW1-mt_jSEkaTav1V-s1

Answer key

For grown-ups. Fold this page away before handing over the rest.

Steady landscapes with no interior peaks W1-mt_jSEkaTav1V-s1

  1. 8 · The function is harmonic and the mean value property returns its centre value u(0,0) = 8.
  2. 7 · A linear function is harmonic, and the mean value property says its average over any circle equals its centre value u(1,1) = 7.
  3. x^2 - y^2 · Only x^2 - y^2 has Laplacian 2 - 2 = 0; the others leave nonzero remainders.
  4. True · Their difference is harmonic with zero boundary values, hence zero everywhere.
  5. The 0 to 5 bounds hold inside by the max and min principles · Both the maximum and the minimum of a harmonic function sit on the boundary, so boundary values between 0 and 5 trap the interior between 0 and 5.
  6. w is zero everywhere · Since w is harmonic with zero boundary data, the maximum principle leaves w is zero everywhere as the only option.
  7. False · The maximum principle forbids a strict interior peak in a nonconstant harmonic function.
  8. 3 · Since u = x^2 - y^2 is harmonic, its average over the circle is the centre value u(2,1) = 3.
  9. It stays smooth and finite at every interior point · Interior values average the whole boundary, so averaging tames the spike.
  10. It stays between 0 and 5 inside · Both extremes sit on the boundary, so they trap the interior between them.
Worksheet · LightMySky