RT
0 f∗(t, ξ(t)) dt if f∗(·, ξ(·))∈L1(0, T),
+∞ otherwise.
ii) The functional F is lower semicontinuous and convex on Lp(0, T;X), and there holdsF(v)>−∞ for all v ∈Lp(0, T;X).
iii) Let F be proper, and letv ∈dom(F)andξ ∈Lp∗(0, T;X∗). Then,ξ ∈∂F(v)⊂ Lp∗(0, T;X∗) if and only if ξ(t)∈∂f(t, v(t))⊂X∗ for a.e. t∈(0, T).
Proof. Assertions i) andii) follow from Kenmochi [99] and Rockafellar [140, Proposition 2 & Theorem 2] as well as Lemma 2.3.2, respectively. Assertion iii) follows from i), ii), Lemma 2.3.1, and the fact that
Z T 0
(f(t, v(t)) +f∗(t, ξ(t))− ⟨ξ(t), v(t)⟩) = 0 (2.3.7) if and only if
f(t, v(t)) +f∗(t, ξ(t))− ⟨ξ(t), v(t)⟩= 0 a.e. in (0, T),
which in turn follows from the fact that the integrand in (2.3.7) is by the Fenchel–
Young inequality always non-negative.
2.4 Mosco-convergence
In this section, we introduce the notion of the Mosco-convergence, which was originally introduced by the Italian mathematician Umberto Mosco [125] in order to study variational inequalities. Before we motivate the Mosco-convergence, we provide a definition.
Definition 2.4.1 A sequence of functionals fn : X → (−∞,+∞] converges to f :X → (−∞,+∞] in the sense of Mosco (we write fn −→M f) if and only if for all u∈X
a) f(u)≤lim infn→∞fn(un) for all un⇀ u in X, b) ∃ˆun→u in V such that f(u)≥lim supn→∞fn(ˆun).
We note that the implication in b) can be replaced by f(u) = limn→∞fn(ˆun) since the other direction of the inequality already follows from a). The existence of a strongly convergent sequence in b) is often referred to as the recovery sequence.
We note further that constant sequences of functions do not, in general, converge in the sense of Mosco since the functional is by a) assumed to be weakly lower semicontinuous. However, if we deal with functionals that are lower semicontinuous and convex, and thus weakly lower semicontinuous, then constant sequences converge in the sense of Mosco. The Mosco-convergence is related to the notion of Γ -convergence4 where the convergences ina) andb) in Definition 2.4.1 are assumed to hold with respect to the same topology, which usually is either the strong topology or the weak topology. The Γ-convergence gives a sufficient condition to conclude that a sequence of solutionsun to the minimization problems
v∈Xinf fn(v)
converge in a certain topology to a solution to a limiting minimization problem as n → ∞. The Mosco-convergence, which is a stronger notion of convergence, provides a sufficient condition to conclude that a sequence of subgradients converge to a subgradient of a limiting functional as we will see. Therefore, the Mosco -convergence and theΓ-convergence are very useful tools in, e.g., phase transitions, homogenization theory, dimension reduction, the formalization of the passage of a discrete model to a continuous model, etc., see [30, 114, 117–119, 147, 152]. We refer the interested reader to the monographsBraides [31] and Dal Maso [50] for an introduction toΓ-convergence.
In Lemma 2.2.2, we have seen that the lower semicontinuity and λ-convexity of a functional yields the strong-weak closedness of the graph of its subdifferential.
However, sometimes we do only have weakly convergent sequences un ⇀ u and ξn ⇀ ξ with ξn ∈∂f(un), n ∈N, at our disposal which is, in general, not enough to conclude ξ ∈ ∂f(u). However, for a proper, lower semicontinuous, and convex functional, a sufficient condition to make this conclusion is in fact given by the limsup estimate
lim sup
n→∞ ⟨ξn−ξ, un−u⟩ ≤0.
This holds even true for maximal monotone operators, see, e.g., Brézis, Crandall
& Pazy [33, Lemma 1.2], which in particular contain the set of subdifferential operators of proper, lower semicontinuous, and convex functionals, seeRockafellar [137, Theorem 4]. If we consider a sequence of functionals (fn)n∈Nso thatξn ∈∂f(un) is replaced byξn∈∂fn(un), our next question is: what type of convergence for the sequence (fn)n∈N to a functional f is sufficient to conclude ξ∈∂f(u). In fact, as we mentioned before, such a convergence is given by the Mosco-convergence.
Lemma 2.4.2 Let f, fn : X → (−∞,+∞] be proper, lower semicontinuous, and convex functionals for all n ∈ N, and denote with fn∗, f∗ : X∗ → (−∞,+∞] the
4The Γ-convergence has originally been introduced by the Italian mathematician Ennio De Giorgi[51–53, 55] in a series of articles, a couple of years after the introduction of theMosco -convergence, where he studiedGreenfunctions.
associated conjugate functionals. Moreover, let vn⇀ v in X and ξn ⇀ ξ in X∗ as n→ ∞ with ξn∈∂fn(un), n∈N such that
lim sup
n→∞ ⟨ξn−ξ, un−u⟩ ≤0.
If
fn−→M f or f(u) +f∗(ξ)≤lim inf
n→∞ (fn(un) +fn∗(ξn)), then u∈∂f(u) and
n→∞lim fn(un) =f(u), lim
n→∞fn∗(ξn) = f∗(ξ) or f(u) +f∗(ξ) = lim
n→∞(f(un) +f∗(ξn)), respectively.
Proof. We assume first thatfn −→M f. Letv ∈X, then by theMosco-convergence of (fn)n∈N there exists a strongly convergent sequence ˆvn →v inX asn → ∞such that f(v)≥lim supn→∞fn(ˆvn). With the liminf estimate a) for the Mosco-convergence, we obtain
f(u)−f(v)≤lim inf
n→∞ fn(un)−lim sup
n→∞ fn(ˆvn)
≤lim inf
n→∞ (fn(un)−fn(ˆvn))
= lim inf
n→∞ ⟨ξn, un−vˆn⟩
≤lim sup
n→∞ ⟨ξn, un−ˆvn⟩
=⟨ξ, u−v⟩ for all v ∈X,
whenceξ ∈f(u). Now, let ˆun →uinXasn→ ∞such thatf(u)≥lim supn→∞fn(ˆun).
Then, we obtain f(u)≤lim inf
n→∞ fn(un)
≤lim sup
n→∞ fn(un)
≤lim sup
n→∞ (⟨ξn, un−uˆn⟩+fn(ˆun))
≤lim sup
n→∞ (⟨ξn−ξ, un−uˆn⟩+⟨ξ, un−uˆn⟩+fn(ˆun))
≤lim sup
n→∞ ⟨ξn−ξ, un−uˆn⟩+ lim sup
n→∞ ⟨ξ, un−uˆn⟩+ lim sup
n→∞ fn(ˆun)
≤f(u),
and hence limn→∞fn(un) = f(u). Exploiting Lemma 2.3.1, we also obtain f∗(ξ) =⟨ξ, u⟩ −f(u)
= lim
n→∞(⟨ξn,uˆn⟩ −fn(ˆun))
≤lim inf
n→∞ sup
v∈X
(⟨ξn, v⟩ −fn(v))
= lim inf
n→∞ (⟨ξn, un⟩ −fn(un))
= lim inf
n→∞ fn∗(ξn)
≤lim sup
n→∞ fn∗(ξn)
= lim sup
n→∞ (⟨ξn, un⟩ −fn(un))
= lim sup
n→∞ (⟨ξn−ξ, un−u⟩+⟨ξ, u⟩ −fn(un))
≤lim sup
n→∞ ⟨ξn−ξ, un−u⟩+⟨ξ, u⟩ − lim
n→∞fn(un)
≤ ⟨ξ, u⟩ −f(u)
=f∗(ξ)
from which limn→∞fn∗(ξn) = f∗(ξ) follows. Now, we assume that f(u) +f∗(ξ) ≤ lim infn→∞(fn(un) +fn∗(ξn)). Then, with Lemma 2.3.1 and the Fenchel–Young inequality, we find
⟨ξ, u⟩ ≤f(u) +f∗(ξ)
≤lim inf
n→∞ (fn(un) +fn∗(ξn))
≤lim sup
n→∞ (fn(un) +fn∗(ξn))
≤lim sup
n→∞ ⟨ξn, un⟩
≤ ⟨ξ, u⟩
whenceξ ∈∂f(u) and f(u) +f∗(ξ) = limn→∞(fn(un) +fn∗(ξn)).
In view of Lemma 2.3.1, we obtain the same implication in the previous result by replacingfn −→M f with fn∗ −→M f∗. So it seems natural to assume that there is a relation between these two convergences. In fact, Attouch [17, Theorem 3.18, p. 295] has shown that they are equivalent if the underlying Banach space X is reflexive. Based on that, Stefanelli showed the following equivalence.
Lemma 2.4.3 Let X be a reflexive Banachspace and let f, fn:X →(−∞,+∞]
be proper, lower semicontinuous, and convex functionals for all n ∈N, and denote with f∗, fn∗ :X →(−∞,+∞] the associated conjugate functionals. Then, fn−→M f if and only if
a) f(u)≤inf{lim infn→∞fn(un) :un ⇀ u in X}, b) f∗(ξ)≤inf{lim infn→∞, fn∗(ξn) :ξn⇀ ξ in X∗}, c) (fn∗)n∈N is uniformly proper,
where point iii) means that there exists a bounded sequence (ξn)n∈N⊂X∗ such that ξn ∈dom(fn∗) for all n∈N.
Proof. This has been proven in Stefanelli [155, Lemma 4.1].
Lemma 2.4.3 gives a characterization of the Mosco-convergence in terms of a functional and its conjugate without assuming the existence of a recovery sequence, which makes it easier to verify in practice. The lemma also shows that the Mosco -convergence fn −→M f actually implies the liminf estimate for the sum fn+fn∗ in Lemma 2.4.2.
We want to employ the previous results in Chapter 3 where we study perturbed gradient systems and in Chapter 6 where we study nonlinearly damped inertial systems by choosing fn = Ψun. More precisely, we will choose fn = ΨUτn(t) where Uτn are the piecewise constant interpolations, see Section 3.4. In Chapter 3, we will obtain a strong convergence of the sequence (Uτn)n∈N uniformly on [0, T], which makes it reasonable to assume the Mosco-convergence of the sequence (Ψun)n∈N for strongly convergent sequences un →u. However, for nonlinearly damped inertial systems, we only obtain a pointwise weak convergence of (Uτn)n∈N so that assuming theMosco-convergence of the sequence (Ψun)n∈N is too restrictive and not necessary as we will see. Therefore, we will assume in Chapter 6 a liminf estimate for the sum ΨU
τn(t)+ΨU∗
τn(t) on suitable Bochner–Lebesgue spaces, which is already implied by theMosco-convergence. The following lemma shows that this will be sufficient in order to obtain the weak-weak closedness of the graph of the subdifferential.
Lemma 2.4.4 Let the functionals f, fn : [0, T]×X → (−∞,+∞] be given and fulfill the assumptions of Theorem 2.3.7, and let p ∈ (1,+∞). Furthermore, let (vn)n∈N ⊂ Lp(0, T;X) and (ξn)n∈N ⊂ Lp∗(0, T;X∗) with ξn ∈ ∂Fn(vn) such that vn ⇀ v in Lp(0, T;X) and ξn ⇀ ξ in Lp∗(0, T;X∗) as n → ∞ where Fn is the integral functional associated to fn. If
Z T
0
(f(t, v(t)) +f∗(t, ξ(t))) dt ≤lim inf
n→∞
Z T
0
(fn(t, vn(t)) +fn∗(t, ξn(t))) dt (2.4.1) and there holds
lim sup
n→∞
Z T 0
⟨ξn(t)−ξ(t), vn(t)−v(t)⟩dt ≤0, then ξ(t)∈∂f(t, v(t)) a.e. in (0, T) and
Z T 0
(f(t, v(t)) +f∗(t, ξ(t))) dt = lim
n→∞
Z T 0
(fn(t, vn(t)) +fn∗(t, ξn(t))) dt Proof. This immediately follows from Lemma 2.4.2 and Theorem 2.3.7.
Remark 2.4.5 It has been shown in Stefanelli [155, Lemma 4.1] that under the assumptions of Lemma 2.4.4, the convergence fn
−→M f impliesFn
−→M F, which in turn implies the liminf estimate (2.4.1).