Hand in y our solutions un til W ednesda y , Decem b er 13, 2.15 pm (PO b o x of y our T A in V3-128)

Hand in y our solutions un til W ednesda y , Decem b er 13, 2.15 pm (PO b o x of y our T A in V3-128)

total points: 20

Prof. Dr. Moritz Kaßmann Fakultät für Mathematik

Wintersemester 17/18 Universität Bielefeld

Partial Differential Equations II Exercise sheet IX, December 6

Exercise IX.1 (4 points)

Assume q > ^d₂,R >0 and f ∈L^q(B_R). As discussed in the lecture (and below) it can be shown that functions u∈H¹(BR) satisfying −∆u≤f inBR in the weak sense, satisfy

sup

B_R/2

u≤cR^θ1 Z

BR

u²¹

2 +cR^θ2kfk_Lq(BR) (1)

for some constantsc, θ1, θ2 independent of uand R. Use scaling arguments and compute θ₁, θ₂.

For the remaining questions assume that Ω⊂ R^d is open, q > ^d₂, and f ∈ L^q(Ω). Let u7→Lu= div(a(·)∇u) denote a strictly elliptic differential operator. The idea of the two exercises is to use techniques that are very similar to the proof in the lecture. Apply the standard definition of weak subsolutions.

Exercise IX.2 (8 points)

Show that there is a constant c ≥ 1 such that, if 0 < ρ < σ, Bρ(x0) b Bσ(x0) ⊂ Ω, for every function u∈H¹(Ω) satisfying −Lu≤f inΩ and everyk ∈R, the following Caccioppoli inequality holds true:

Z

Aρ(x0;k)

|∇u|² ≤c(σ−ρ)⁻² Z

Aσ(x0;k)

(u−k)²+c

k²+kfk²_Lq(Ω)

|(A_σ(x0;k))|^1−1/q. (2)

Exercise IX.3 (8 points)

Assume that u∈H¹(Ω) is a function that satisfies (2) for every B_ρ(x₀) bB_σ(x₀)⊂Ω and every k∈R. Show that then (1) holds.

Hint: You may use without proof the auxiliary iteration lemma given in the lecture.