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Prof. Dr. Moritz Kaßmann Fakultät für Mathematik

Sommersemester 17 Universität Bielefeld

Partial Differential Equations Exercise sheet IV, 05/10/17

Exercise IV.1 (5 Points)

Prove the following convergence result.

Theorem. Let(u_n)be an increasing sequence of harmonic functions in a domainΩ⊂R^d. Assume that there is a point y∈Ωfor which the sequence is bounded. Then the sequence (un) converges uniformly on every bounded subdomainΩ⁰ bΩto a functionu, which itself is harmonic in Ω.

Exercise IV.2 (5 Points)

Define the Green function of ∆for the upper half spaceR^d+,d≥2 by G(x, y) = Φ(y−x)−Φ(y−x),e

whereΦ is the fundamental solution of the Laplace equation andex= (x₁, ..., xd−1,−x_d) is the reflextion ofx at the hyperplane {x_d= 0}. For y∈∂R^d₊ compute ^−∂G_∂ν (x, y) and use this to obtain a solution formula for the problem

−∆u= 0 inR^d₊, u(·,0) =g on ∂R^d+,

whereg∈C(R^d−1) is bounded. Furthermore, prove that the solution u satisfies

x→zlimu(x) =g(z)

for allz∈∂R^d₊.

Exercise IV.3 (2+3 Points)

LetΩ⊂R^d be a domain andG: Ω×Ω\diag→Rthe Green function to ∆inΩ.

(a) Find the Green function for the intervalΩ = (a, b)⊂R.

(b) Find the Green function for the unit square Ω = (0,1)×(0,1)⊂R².

Exercise IV.4 (3+2 Points)

(a) Prove that a bounded function u:R² →R, which is subharmonic onR², is constant.

Hint: Use the subharmonic functionw^ε=u(x)−εln(|x|)

(b) Give an example of a subharmonic functionv:R^d→R,d≥3, which is non-constant and bounded.

Hint: Consider the solution to∆u=f for an appropriate choice off.