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Hand in y our solutions un til W ednesda y , 05/10/17, 14:15 (PO b o x of y our T A in V3-128)

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Prof. Dr. Moritz Kaßmann Fakultät für Mathematik

Sommersemester 17 Universität Bielefeld

Partial Differential Equations Exercise sheet III, 05/03/17

Exercise III.1 (5 Points)

LetΩ⊂R^d be a bounded domain. Assume gn∈C(∂Ω) for n∈N and sup

∂Ω

|g_m−gn| →0 for m, n→ ∞.

Forn∈Nlet un∈C²(Ω)∩C(Ω)be harmonic in Ωwithun|_∂Ω=gn.

Show that the sequence (un) converges uniformly to a function u ∈ C(Ω) which is harmonic inΩ.

Exercise III.2 (4+1 Points)

a) Let B⁺ ={x∈R^d | kxk<1, x_d>0}. Letu∈C²(B⁺)∩C(B⁺) be harmonic in B⁺ withu= 0 on ∂B⁺∩ {x_d= 0}.Define a function v:B1(0)→Rvia

v(x) =

(u(x) for x_d≥0,

−u(x₁, ..., xd−1,−x_d) for xd<0.

Prove thatv is harmonic in B1(0).

b) Let H={x∈R^d |x_d>0}. Assume u∈C²(H)∩C(H) is bounded and satisfies

∆u= 0 inH and u|_∂H= 0.

Showu≡0.

Exercise III.3 (5 Points)

Let Ω⊂R^dbe a domain. Let u: Ω→Rbe a harmonic function in Ω. Prove that u is real analytic, i.e.,u is an infinitely differentiable function and the Taylor series of u at any pointx0∈Ωconverges pointwise to uin a neighbourhood of x0.

Exercise III.4 (5 Points)

GivenL >0, find an explicit solution for the following problem:

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

∆u= 0 in(0, L)×(0.L), u(0,·) =f in(0, L),

u(L,·) = 0 in(0, L), u(·,0) = 0 in(0, L), u(·, L) = 0 in(0, L), wheref(y) = sin(^nπ_Ly) for n∈N.

Hint: Determine all harmonic functions u of the form u(x, y) = v(x)w(y). Adjust u according to the boundary conditions.