Non-Local Functionals for Imaging
8.3 Existence of Minimising Points for Non-Local Functionals
8.3.2 Sequential Lower Semi-Continuity of a Non-Local Functional
Proof. A direct estimate shows that
X
X
f−(x,y,u(x),u(y))dx dy≤ γ1+2λ1upp+Cu2pp <∞ for every function u∈Lp(X ;Rn). The function f thus fulfils all three assump-tion in the Definiassump-tion 8.1 and therefore defines the non-local functional Jfp
on Lp(X ;Rn).
For p=∞, we get a similar result.
Proposition 8.6. Let f : X×X×Rn×Rn→Rbe a measurable function satisfying the symmetry condition (8.2). If for every M∈(0,∞)there exists a non-negative functionγM∈L1(X×X)such that
f(x,y,w,z)≥ −γM(x,y) (8.16) for almost all(x,y)∈X×X and all w,z∈Rnwith|w| ≤M and |z| ≤M, then f defines a non-local functionalJf∞on L∞(X ;Rn).
Proof. For an arbitrary function u∈L∞(X ;Rn), we choose M=u∞and find
X
X
f−(x,y,u(x),u(y))dx dy≤ γM1<∞,
as desired.
8.3.2 Sequential Lower Semi-Continuity of a Non-Local Functional
In the following, we investigate conditions on a function f such that it defines a non-local functional on Lp(X ;Rn), which is sequentially lower semi-continuous with respect to the weak topology if p∈[1,∞)and with respect to the weak-star topology if p=∞.
In the first step, we consider only lower semi-continuity with respect to the strong topology, which can be guaranteed under very mild conditions. Indeed, a sufficient criterion is already that f is lower semi-continuous with respect to the last two variables, see [9].
Proposition 8.7. LetJfpbe a non-local functional on Lp(X ;Rn), p∈[1,∞], with f additionally satisfying (8.15) if p∈[1,∞)and (8.16) if p=∞.
Then the functional Jfp is lower semi-continuous with respect to the strong topology on Lp(X ;Rn)if the map
f(x,y):Rn×Rn→R, f(x,y)(w,z) =f(x,y,w,z) is for almost all(x,y)∈X×X lower semi-continuous.
Proof. Let(uk)k∈N ⊂Lp(X ;Rn)be a sequence converging to u∈Lp(X ;Rn). We choose a subsequence(uk)∈Nof(uk)k∈Nsuch that
→∞limJfp(uk) =lim inf
k→∞ Jfp(uk)
holds and such that
→∞limuk(x) =u(x) for almost all x∈X.
Let now first p∈[1,∞). To be able to apply Fatou’s Lemma, we use the lower bound (8.15) for f and consider instead of f the function
(x,y,w,z)→ f(x,y,w,z) +γ(x,y) +λ(x)|z|p+λ(y)|w|p+C|w|p|z|p, which is for almost all(x,y)∈X×X and all w,z∈Rnnon-negative. Then we find with the lower semi-continuity of f(x,y)that
lim inf
k→∞ Jfp(uk) +γ1+2λ1upp+Cu2pp
≥
X
X
lim inf
→∞
f(x,y,uk(x),uk(y)) +γ(x,y) +λ(x)|uk(y)|p
+λ(y)|uk(x)|p+C|uk(x)|p|uk(y)|p dx dy
≥
X
X
f(x,y,u(x),u(y))dx dy+γ1+2λ1upp+Cu2pp.
Thus, lim infk→∞Jfp(uk)≥Jfp(u), and we conclude thatJfpis sequentially lower semi-continuous.
If p=∞, we choose M=sup∈Nuk∞and get from condition (8.16) a non-negative functionγM∈L1(X×X)such that the function
(x,y)→f(x,y,uk(x),uk(y)) +γM(x,y)
is for almost all (x,y)∈X×X and all ∈N non-negative. Therefore, Fatou’s Lemma implies as before that
lim inf
k→∞ Jf∞(uk) +γM1
≥
X
X
lim inf
→∞ (f(x,y,uk(x),uk(y)) +γM(x,y))dx dy
≥
X
X
f(x,y,u(x),u(y))dx dy+γM1.
So, lim infk→∞Jf∞(uk)≥Jf∞(u), and we conclude thatJf∞is sequentially lower
semi-continuous.
However, we need sequential lower semi-continuity of the functional Jfp with respect to a weaker topology, since coercivity only guarantees that a minimising sequence is bounded, but in the strong topology a bounded sequence does not need to have a convergent subsequence and the existence of such a subsequence is one of the essential steps in the proof of Proposition8.4.
Similar to the classical case of local functionals, it is the convexity of the func-tion f which provides us with the sequential lower semi-continuity ofJfp with respect to the weak or weak-star topology, respectively, see [9]. Similar results al-ready appeared in [2,14,16,17], from where we also took the idea of this proof.
Proposition 8.8. LetJfpbe a non-local functional on Lp(X ;Rn), p∈[1,∞], with f(x,y) being continuous for almost all(x,y)∈X×X . We further assume that there exist a constant C∈Rand functionsγ∈L1(X×X)andλ ∈L1(X)such that
|f(x,y,w,z)| ≤γ(x,y) +λ(x)|z|p+λ(y)|w|p+C|w|p|z|p (8.17) for almost all(x,y)∈X×X and all w,z∈Rnif p∈[1,∞)and that there exists for every M∈(0,∞)a functionγM∈L1(X×X)such that
|f(x,y,w,z)| ≤γM(x,y) (8.18) for almost all(x,y)∈X×X and all w,z∈Rnwith|w| ≤M and|z| ≤M if p=∞.
Then,Jfpis sequentially lower semi-continuous with respect to the weak topol-ogy on Lp(X ;Rn)if p∈[1,∞)and with respect to the weak-star topology if p=∞ if the function
Φx,ψ:Rn→R, Φx,ψ(w) =
X
f(x,y,w,ψ(y))dy (8.19) is for everyψ∈Lp(X ;Rn)for almost all x∈X convex.
Proof. For the proof of this proposition, we will use the notion of Young measures.
For a short introduction to the theory of Young measures, we refer to Chap. 8 in [10].
Let(uk)k∈N⊂Lp(X ;Rn)be a sequence converging to u∈Lp(X ;Rn)with respect to the weak topology on Lp(X ;Rn)if p∈[1,∞)and with respect to the weak-star topology if p=∞.
In particular, the sequence(uk)k∈Nis bounded in Lp(X ;Rn)and therefore, there exists a subsequence(uk)∈N of(uk)k∈N generating a Young measureν. That is, we have a mapν: X→M(Rn;R), x→νx, whereM(Rn;R)denotes the set of all signed Radon measures onRn, fulfilling thatνxis a probability measure for almost all x∈X , and that for allφ∈C0(Rn)the function X→R, x→
Since f(x,y) is lower semi-continuous for almost all (x,y)∈X×X and the functions
X×X→[0,∞), (x,y)→f−(x,y,uk(x),uk(y)), ∈N,
are uniformly integrable because of the conditions (8.17) and (8.18), we get from the fundamental theorem for Young measures (see e.g. Theorem 8.6 in [10]) that
lim inf
are again due to the conditions (8.17) and (8.18) for almost all x∈X and all w∈Rnuniformly integrable, the fundamental theorem for Young measures addi-tionally shows that
Using thatνxis by definition of a Young measure for almost all x∈X a proba-bility measure, we thus find with Jensen’s inequality that
Since(uk)∈N converges weakly in Lp(X ;Rn)if p∈[1,∞)and weakly-star in L∞(X ;Rn)if p=∞to u, we get from (8.20) that
Rnw dνx(w) =u(x) for almost all x∈X.
Exploiting the symmetry of f and then using the convexity ofΦy,ufor almost all y∈X , we can again apply Jensen’s inequality and get
X
Rn
X
f(x,y,u(x),z)dx dνy(z)dy
≥
X
X
f(x,y,u(x),
Rnz dνy(z))dx dy=Jfp(u). (8.24) Putting the inequalities (8.21), (8.23) and (8.24) together, we finally find that
lim inf
→∞ Jfp(uk)≥Jfp(u),
proving the sequential lower semi-continuity ofJfp. In fact, the convexity of the functionΦx,ψ for everyψ∈Lp(X ;Rn)for almost all x∈X is even necessary for the sequential lower semi-continuity of the func-tionalJfp. For a proof of this fact, we refer to [9].
We remark that the only argument in the proof of Proposition8.8, where we need the upper bound on f , is to show that the function ˜Φx,νdefined in (8.22) is for every generated Young measureν: X →M(Rn;R)for almost all x∈X convex. If we thus guarantee the convexity of ˜Φx,νby directly imposing convexity on the function
f(x,y), we can neglect the upper bound, see also [17].
Corollary 8.9. LetJfpbe a non-local functional on Lp(X ;Rn), p∈[1,∞], with f additionally fulfilling (8.15) if p∈[1,∞)and (8.16) if p=∞.
Then,Jfpis sequentially lower semi-continuous with respect to the weak topol-ogy on Lp(X ;Rn)if p∈[1,∞)and with respect to the weak-star topology if p=∞if the function f(x,y)is separately convex, i.e. if the functionRn→R, w→f(x,y)(w,z) is for all z∈Rnconvex, for almost all(x,y)∈X×X .
In general, however, the separate convexity of f(x,y) is not necessary for the se-quential lower semi-continuity of the functionalJfp. Though in the case n=1 (under some additional regularity assumptions on f ), there exists for every se-quentially lower semi-continuous non-local functionalJfp on Lp(X) a function
˜f defining the same non-local functionalJfpas f such that ˜f(x,y) is for almost all (x,y)∈X×X separately convex, see again [9].