Weak Convergence Methods for Nonlinear Partial Differential ...

Using that Ψ ′ = F ′ Φ ′ we arrive at
∫
(h(y) − g(y)F ′ (y))Φ ′ (y) dy = 0.
Since this equation holds true for any convex Φ and thus any increasing Φ ′ , by
linearity it also holds true for differences of increasing functions and hence for any
smooth function. (Every w ∈ C 1 can be written as w(y) = w(0)+ ∫ y
∫ 0 (w′ ) + (s) ds−
y
0 (w′ ) − (s).) But then we must have
Now this proves that
h − g F ′ = 0.
(F(y) g(y) − y h(y)) ′ = F(y) g ′ (y) − y h ′ (y) + g(y) F ′ (y) − h(y)
where the last equality followed from
= F(y) g ′ (y) − y h ′ (y) = 0,
d(F(y) g ′ (y) − y h ′ (y)) = F(y) y dν (x,t) − y F(y) dν (x,t) = 0.
Since g and h vanish outside [α, β] we deduce
F(y) g(y) − y h(y) = 0.
Recalling that g < 0 on (α, β) and h −gF ′ = 0, i.e., h g = F ′ , we finally obtain
that
F(y) − y F ′ (y) = 0 ∀ y ∈ (α, β).
The only solutions of this differential equation are given by
F(y) = cy ∀ y ∈ (α, β)
for some constant c, which was to be proved.
In order to apply this theorem to conservation laws we will also need the
following lemma.
Lemma 3.5 If E 1 is compact in W −1,2 (Ω), E 2 bounded in M(Ω) and E 3 bounded
in W −1,∞ (Ω), then
(E 1 + E 2 ) ∩ E 3 is precompact in W −1,2
loc
(Ω).
Proof. Let (g (ν) ) be a sequence in E := (E 1 + E 2 ) ∩ E 3 . W.l.o.g. we may assume
that supp g (ν) ⊂ ˜Ω for some smoothly bounded subdomain ˜Ω ⊂ Ω.
Write g (ν) = g (ν)
1 + g (ν)
2 with g (ν)
i ∈ E i (i = 1, 2) and let v (ν)
i be the solution of
−∆v (ν)
i
= g (ν)
i in ˜Ω, v (ν) = 0 on ∂˜Ω.
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