Zentrum Mathematik Tehnishe Universität Münhen Prof. Dr. Bernd Shmidt 27. Otober 2009 Dr. Johannes Giannoulis Sheet 2 Partial Dierential Equations I Winter semester 2009/10 Exerise 5: Banah spae of harmoni funtions Let U ⊂ Rn be open and bounded. Show that ( u ∈ C(U ) | u harmoni in U , k·k∞ ) (with kuk∞ := sup |u(x)|) is a Banah spae. x∈U Hint: You may use the fat that (C(U ), k·k∞ ) is a Banah spae. Exerise 6: Neumann boundary-value problem for the Poisson equation Let U ⊂ Rn be open and bounded with a C 1 -boundary and outer unit normal ν , as well as f ∈ C(U ), g ∈ C(∂U). Show that if the Poisson equation with Neumann boundary onditions, −∆u = f in U , ∂ν u = g on ∂U has a solution u ∈ C 2 (U), then Z f dx + U Z g dS = 0. ∂U Exerise 7: Harnak's inequality Let U ⊂ Rn be open. Show that for every onneted V ⊂⊂ U there exists a C > 0 suh that sup u ≤ C inf u V V for all funtions u ≥ 0, whih are harmoni in U . Exerise 8: Faraday age The Poisson equation of eletrostatis ρ = −∆ϕ (1) desribes the relation between the eletri harge density ρ and the eletrostati potential ϕ. Suppose U ⊂ R3 , U open with C 1 -boundary, is a bounded region, surrounded by a onduting material ∂U . Let ρ ∈ Cc (R3 ) be a ompatly supported harge density, whih vanishes on U , i.e. supp (ρ) ⊂ R3 \U , and let ϕ ∈ C 2 (R3 ) be a orresponding potential, whih satises (1). Aording to the theory of eletrostatis, the potential ϕ has to be onstant on ∂U : ϕ|∂U = const. Show that this implies that the eletri eld vanishes in U : ∇ϕ|U = 0.