Laplace's Equation and the Maximum Principle · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Steady landscapes with no interior peaks

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

  1. Let u(x,y) = 6x + y - 3. What is the average of u over a circle centred at (0,2)?

    Answer: ______________

  2. Let u(x,y) = 7 - x + 2y. What is the average of u over a circle of radius 5 centred at (4,1)?

    Answer: ______________

  3. Let u(x,y) equal 2x minus y plus 5. What is the average of u over a circle centered at (2,3)? The radius does not matter.

    Answer: ______________

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

    • x squared plus y squared
    • x cubed
    • x times y

  5. Let u(x,y) = x^2 - y^2. What is the average of u over a small circle centred at (0,3)?

    Answer: ______________

  6. Let u(x,y) = x - y on the closed disc of radius 2 about the origin, rim included. What is the maximum value of u, to two decimal places?

    Answer: ______________

  7. A nonconstant harmonic function on a disc takes the value negative 2 at an interior point and is larger everywhere else.

    Circle one:   True   False

  8. A harmonic function on a bounded region, continuous up to the boundary, has boundary values at most 4. Could it equal 6 somewhere inside?

    Circle one:   True   False

  9. Boundary data on a circle is 1 on the top half and 0 on the bottom half. What is true of the harmonic function inside?

    • It jumps from 1 to 0 across the middle line
    • It is smooth inside and equals 1/2 at the centre
    • It equals 1 everywhere above the centre

  10. A harmonic function in a disc extends continuously to boundary values between negative 3 and 2. What follows for the interior?

    • It must be constant inside
    • It stays between -3 and 2 inside
    • It may reach 3 near the center
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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. -1 · u is linear, so harmonic, and the average is u(0,2) = 0 + 2 - 3 = -1.
  2. 5 · u is linear, so harmonic; the average over any circle is the centre value 7 - 4 + 2 = 5, whatever the radius.
  3. 6 · A linear function is harmonic, so its average over any circle equals the centre value u(2,3) = 4 - 3 + 5 = 6.
  4. x times y · For xy, u_xx = 0 and u_yy = 0, so the Laplacian is 0.
  5. -9 · u is harmonic, so the average is u(0,3) = 0 - 9 = -9.
  6. 2.83 · The direction (1, -1) has length √2, so the rim maximum is 2√2 ≈ 2.83.
  7. False · The minimum principle forbids a strict interior minimum for a nonconstant harmonic function.
  8. False · No. By the maximum principle every interior value is at most the boundary maximum, 4.
  9. It is smooth inside and equals 1/2 at the centre · Averaging smooths the jump away inside, and the centre value is the plain average of the boundary: 1/2.
  10. It stays between -3 and 2 inside · The maximum and minimum both sit on the boundary, so they trap the interior between -3 and 2.
Worksheet · LightMySky