UPPSALA UNIVERSITET

Matematiska institutionen Christer Kiselman

ÖVNINGAR, Blad 6

Teori för differentialekvationer 1998 02 10

6.1. Solve the Cauchy problem

\frac{∂^{2}u}{∂t^{2}} − c^{2} \frac{∂^{2}u}{∂x^{2}} = 0 when t > ax; u(x, t) = 0 and \frac{∂u}{∂t}(x, t) = x when t = ax,

where a is a constant ≠ ±1/c. Hint: Introduce new coordinates with the help of the Lorentz transformation

x′ = \frac{x − vt}{\sqrt{1 − v^{2}/c^{2}}} , t′ = \frac{t − vx/c^{2}}{\sqrt{1 − v^{2}/c^{2}}} ,

for a suitable v, or, even simpler,

x′ = x − vt, t′ = t − vx/c^{2}.

6.2. Solve the Dirichlet problem

\frac{∂^{2}u}{∂x^{2}} + \frac{∂^{2}u}{∂y^{2}} = 0 in the square 0 ⩽ x ⩽ π, 0 ⩽ y ⩽ π,

with the boundary values u(x, 0) = u(0, y) = u(π, y) = 0 and u(x, π) = sin mx.

6.3. Let g be a function which is continuous on R and satisfies g(x) ⩽ C/(1 + x^{2}). Define a function u on R^{2} as

u(x) = \frac{1}{2π} ∫_{R} log((x_{1} − y_{1})^{2} + x_{2}^{2})g(y_{1})dy_{1}.

Prove that Δu = 0 when x_{2} > 0 and that u_{x_{2}}(x_{1}, x_{2}) → g(x_{1}) as x_{2} → 0.

6.4. Decide whether the following statements are true or false.

a) If u is a harmonic function in the plane, then e^{u} is subharmonic.

b) If u is a harmonic function in the plane, then log(1 + u^{2}) is subharmonic.

6.5. The potential of a mass distribution with constant surface density on a sphere is defined by

u(x) = γ ∫_{|y|=r} \frac{1}{|x − y|}dS(y), x ∈ R^{3},

where γ is a positive constant. Determine u.

6.6. Prove that if u is a bounded solution to Δu = u in all of R^{n}, n ⩾ 2, then u must be zero. Hint: Use the translation invariance and the rotation invariance of the operator and then compare it with the solution of, e.g., v(x) = a + b|x|^{2}, where the constants a and b are chosen so that 0 ⩽ Δv ⩽ v.