Chapter 5 : Laplace Equation 51

Assignment Sheet-4

(1) Let Ω ⊆ R2 be given by

Ω := (x, y) ∈ R2 : x2 + y2 > 1.

Let u ∈ C2(Ω) be such that ∆u = 0 in Ω, and lim|(x,y)|→∞ u(x, y) = 0. Show that

MaxΩ |u| = Max∂Ω |u|.

(2) Let Ω ⊆ R2 be a bounded domain. Let u ∈ C2(Ω) ∩ C(Ω) be a solution of

∆u+ a(x, y)ux + b(x, y)uy + c(x, y)u = 0,

where a, b, c ∈ C(Ω) and c(x, y) < 0 in Ω. Show that u = 0 on ∂Ω implies u = 0 in Ω.(Hint: Show that Maxu ≤ 0 and Minu ≤ 0.)

(3) Let Ω ⊆ R2 be given byΩ := (x, y) ∈ R2 : y > 0.

Let u ∈ C2(Ω)∩C(Ω) be a harmonic function in Ω. Further assume that u is boundedabove in Ω. Prove that

supΩ

u = sup∂Ω

u.

(Hint: Take for ε > 0 the harmonic function

u(x, y)− ε log(√x2 + (y + 1)2),

Apply the maximum principle to a region (x, y) ∈ R2 : x2 + (y + 1)2 < a2, y ≥ 0with large a. Let ε→ 0.)

(4) Let Ω ⊆ R2 be an open set and let u : Ω −→ R.(a) Let (a, b)R2 and let T : R2 → R2 be given by T (x, y) = (x + a, y + b). Define Ω

be given byΩ := (x− a, y − b) : (x, y) ∈ Ω.

Let T denote the restriction of T to Ω. Show that u ∈ C2(Ω) if and only ifuT ∈ C2(Ω) and that ∆(uT ) = (∆u)T . In particular,

∆u = 0 ⇐⇒ ∆(uT ) = 0.

(b) Laplace equation is invariant under any real orthogonal transformation. Let U :R2 → R2 be any real orthogonal transformation i.e.,

U(x, y) =MU

(xy

)∀(x, y) ∈ R2,

where MU is a real 2× 2 invertible matrix whose inverse is equal to its transpose.Define Ω := U−1(Ω) and let U be the restriction of U to Ω. Show that u ∈ C2(Ω)if and only if uU ∈ C2(Ω) and that ∆(uU) = (∆u)U . In particular,

∆u = 0 ⇐⇒ ∆(uU) = 0.

(5) Let a, b ∈ R be such that a < b. Let A,B ∈ R. Solve uxx + uyy = 0 in the annular

region 0 < a <√x2 + y2 < b with the boundary conditions u = A on r = a and

u = B on r = b. (Hint: Look for a solution depending only on r =√x2 + y2.)

Sivaji Ganesh Sista MA 515: Partial Differential Equations

52 5.3. Mean Value Property

(6) Solve uxx + uyy = 1 in√x2 + y2 < a with u vanishing on x2 + y2 = a2. (Ans:

u(x, y) = 14 (x

2 + y2 − a2).)

(7) Check the validity of the maximum principle for the harmonic function 1−x2−y2

1−2x+x2+y2

in the disk D = (x, y) : x2 + y2 ≤ 1. Explain.

(8) Let Ω ⊆ R2 be a bounded domain. u ∈ C2(Ω) ∩ C(Ω) be such that ∆u ≥ 0 in Ω.Prove that its maximum value is attained on the boundary ∂Ω. What can you sayabout the minimum value?

(9) Show that there is no maximum principle for the wave equation.

(10) Solve

ut = k uxx x > 0, t > 0,

u(0, t) = 1 t ≥ 0,

u(x, 0) = 0 x ≥ 0.

(Ans: u(x, t) = 0 for x > t and u(x, t) = 1 for 0 < x < t.)

(11) Solve

utt + ut = uxx, 0 < x < π, t > 0,

u(0, t) = u(π, t) = 0, 0 ≤ t

u(x, 0) = sin(x), 0 ≤ x ≤ π

ut(x, 0) = 0, 0 ≤ x ≤ π

(Ans: u(x, t) = e−t/2(cos(

√32 t) +

1√3sin(

√32 t)

)sinx)

(12) Let u be a harmonic function on the whole plane such that u = 3 sin(2θ) + 1) on thecircle x2 + y2 = 2. Without finding the concrete form of the solution, find the valueof u at the origin.

(13) Solve ∆u = 0 in the disk D = (x, y) : x2 + y2 < a2 with the boundary conditionu = 1 + 3 sin θ on the circle r = a. (Ans: u(r, θ) = 1 + 3r

a sin θ.)

(14) Solve the equation ∆u = 0 in the domain r > a with the boundary condition u =1+ 3 sin θ on the circle r = a and with the condition that the solution is bounded forr → ∞. (Ans: u(r, θ) = 1 + 3a

r sin θ.)

(15) Let u be a solution of ut = uxx, 0 ≤ x ≤ l, 0 ≤ t <∞.

(a) Let M(T ) be the maximum of the function u on the rectangle (x, t) : 0 ≤ x ≤l, 0 ≤ t ≤ T. Is M(T ) decreasing or increasing as a function of T?

(b) Let m(T ) be the maximum of the function u on the rectangle (x, t) : 0 ≤ x ≤l, 0 ≤ t ≤ T. Is m(T ) decreasing or increasing as a function of T?

(16) The aim of this exercise is to show that the maximum principle does not hold for theequation ut = xuxx which has a variable coefficient.

(a) Verify that the function u(x, t) = −2xt− x2 is a solution. Find its maximum onthe rectangle (x, t) : −2 ≤ x ≤ 2, 0 ≤ t ≤ 1.

(b) Where exactly does our proof of the maximum principle fail in the case of thisequation?

(17) Exercises from Pinchover-Rubinstein Chapter 7: Exercise numbers are 7,8,10,11, 14-16,21,22.

MA 515: Partial Differential Equations Sivaji Ganesh Sista