60
10 Appendix
where < . , . > denotes the scalar product in L
2
, ω = (−Δ)
1
2 and γ is the constant
occurring in (5.11), one has
H (t) = 1
4 < ωχ, ωχ > + 1
4 < ζ, ζ > + 1
2 < χ, χ > + < ψ 0 + 1
2 ζ, ψ 0 + 1
2 ζ >
+ < ω −1 h − 1
2 ωχ, ω −1 h − 1
2 ωχ > +
[G(χ(s)) + γ|χ| 2 ]d s x ≥ 1
4 δ X
(10.19)
and
< ζ + ψ 0 , ζ + ψ 0 >≤ 2{< ψ 0 +
1
2
ζ, ψ 0 +
1
2
ζ > +
1
4
< ζ, ζ >} ≤ 2H (10.20)
Hence,
H (t) = H (0) + 2
γ +
1
2
t
0
dτ < χ(τ ) , ζ(τ ) + ψ 0 >≤
≤ H (0) + 2
γ +
1
2
2
t
0
dτ H (τ ),
so that, by (10.19) and by Gronwall’s lemma,
1
4
δ(t) X ≤ H (t) ≤ H (0) exp[4(γ +
1
2
)t].
10.5 Time-Independent Solutions Defining Physical Sectors
We briefly discuss the non-linear elliptic problem associated with the investigation
of time-independent solutions which define physical sectors (see the first reference
in Chap. 5, footnote 24). For simplicity, we discuss the case s ≥ 3. By the discussion
of Chap. 6, we have to impose the condition ∇ϕ ∈ L
2
(R
s
).
Proposition 10.1 Let us consider the non-linear elliptic problem (U ∈ C
2
)
Δϕ − U
(ϕ) = 0, ϕ ∈ H
1
loc (R
s
), ∇ϕ ∈ L
2
(R
s
), s ≥ 3;
(10.21)
then,
i) the function ˜
ϕ(r, ω) ≡ ϕ(x), x = r ω, r > 0, ω ∈ S
s−1 (the unit sphere of R
s ),
is continuous in r and it has a finite limit ˜
ϕ(∞, ω) as r → ∞, for almost all
ω ∈ S
s−1 , and the limit is independent of ω, briefly
lim
|x|→∞
ϕ(x) = ϕ ∞ ,
(10.22)