Hand in y our solutions un til W ednesda y , Decem b er 6, 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 VIII, November 29
Every exercise is worth 10 points. You receive the maximal points (20 points) if you solve two exercises correctly.
The first two exercises concern the existence of minimizers of variational functionals. The third exercises proves Theorem 4.6 and establishes the Euler-Lagrange equation resp.
system.
Exercise VIII.1 (10 points)
LetΩ⊂Rd be open. Assume that F : Ω×Rd→R satisfies (i) F(·, v) is measurable for everyv∈Rd.
(ii) F(x,·) is convex for everyx∈Ω.
(iii) There are p >1,b >0, and a∈L1(Ω) such that F(x, v)≥ −a(x) +b|v|p for every v∈Rd and almost everyx∈Ω.
Letg∈W1,p(Ω)and M ={w+g|w∈W01,p(Ω)}. Prove that
v7→I(v; Ω) = Z
Ω
F(x, Dv(x))dx
attains its infimum in M.
Hint: You may use the following fact without a proof: Convex lower semicontinuous functions on convex subsets of separable reflexive Banach spaces are lower semicontinuous w.r.t. to weak convergence.
Exercise VIII.2 (10 points)
LetΩ⊂Rd be open and bounded. Assume thatF : Ω×RN ×RN d→Ris measurable in the first variable and continuous in the remaining variables, cf. Exercise VII.3. We assume that the functions
v7→F(x, v, z), z7→F(x, v, z)
are differentiable in the classical sense and satisfy, for someL≥1,p >1,
|DvF(x, v, z)| ≤L(1 +|z|)p−1, |DzF(x, v, z)| ≤L(1 +|z|)p−1
for almost every x ∈ Ω and every (v, z) ∈ RN ×RN d. Let u ∈ W1,p(Ω;RN), be a mimimizer of the functional
v7→I(v; Ω) = Z
Ω
F(x, v(x), Dv(x))dx .
Then u is a weak solution of
divDzF(x, u, Du) =DuF(x, u, Du) inΩ.
Exercise VIII.3 (10 points)
Assume α ∈ (0, d) and f : B1 → Rd, f(x) = |x|−αx. Show f ∈ W1,p(B1;Rd) for p∈[1, d/α).
Hint: You may or may not want to prove the following observation and to apply it to the aforementioned problem: AssumeΩ⊂Rdis open and bounded. Assume 1≤p≤q≤ ∞.
LetE ⊂Ωbe a set with the following property:
inf{kvkW1,q0
(Ω)|v∈Cc∞(Rd), v ≥1E,0≤v≤1}= 0. Letf ∈Lq(Ω)∩C1(Ω\E). AssumeDu∈Lp(Ω\E;Rd). Thenf ∈W1,p(Ω).
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