6.5 The time-discrete WIDE principle
6.5.3 Γ-convergence of discrete WIDE functionals
In order to conclude the proof of Lemma 6.4.1 we need to show that the time-discrete energy estimate in Proposition 6.5.4 passes to the limit as τ → 0 (for fixed ε > 0). To
6.5 The time-discrete WIDE principle
this aim, we check the discrete-to-continuous Γ-convergence Wε = Γ- limτ→0Wετ with respect to the weak topology on V(see [Bra02, Dal93] for relevant definitions and results on Γ-convergence).
For all vectors V ∈ HN+1, we indicate by vτ and vτ their backward constant and piecewise affine interpolants on the partition {iτ : i= 0, . . . , N}, respectively. Namely, we have vτ(0) =vτ(0) =V0 and
vτ(t)≡Vi,
vτ(t) =αi(t)Vi+ (1−αi(t))Vi−1
)
fort∈ (i−1)τ, iτ, i= 1, . . . , N, where we have used the auxiliary functions
αi(t) = (t−(i−1)τ)/τ fort∈](i−1)τ, iτ], i= 1, . . . , N.
With these definitions we reformulate the estimate in Proposition 6.5.4 as (ρ+ν)
Z T−2τ τ
u0ετ
2+E(uετ)dt≤C, (6.33) where uετ and uετ denote the piecewise affine and constant interpolants associated with the minimizer Uετ ∈Kτ(u0, u1) of the discrete WIDE functionalWετ, respectively.
As a first step in the proof of the Γ-convergence we introduce the space of piecewise affine functions with respect to the partition{iτ : i= 0, . . . , N}on [0, T] being a subspace of Vand the corresponding convex set Kbτ(u0, u1)
Vbτ ={u: [0, T]→Z∩X : u is piecewise affine} ⊂V, Kbτ(u0, u1) =nu∈Vbτ : u(0) =u0 and ρu≡ρu1 on [0, τ]o.
Hence, by identifying the discrete trajectories U ∈ Yτ with their piecewise affine inter-polants uτ ∈Vbτ we formulate the minimization of Wετ and Wε on the common space V by extending the WIDE functionals, i.e., we consider
Wε[u] =
(Wε[u] ifu∈K(u0, u1),
∞ otherwise, Wετ[u] =
(Wετ[U] ifu∈Kbτ(u0, u1),
∞ otherwise,
where U = (u(0), u(τ), . . . , u(T))∈Yτ for a piecewise affineu∈Vbτ.
As subtle detail note that for an arbitraryU ∈Yτ we have in generalUN−1, UN ∈/ Z∩X such that the corresponding piecewise affine interpolantuτ is in general not inV. However, from Remark 6.5.2 we know that the minimizers Uετ of Wετ satisfy UN−1ετ , UNετ ∈Z∩X so that we can neglect this subtlety.
Before we give the main result of this section we note the convergence of the (shifted) interpolants of the time-discrete weights eiε to their continuous counterpart. The proof is being omitted here.
Lemma 6.5.5 Let eτε andeτε denote the piecewise constant and affine interpolants of the
Figure 6.1: Interpolants: piecewise constant (solid), piecewise affine (dotted), piecewise quadratic (dashed)
discrete weights eiε, respectively. Then
eετ, eτε, eτε(·+τ), eτε(·+ 2τ)→t7→e−t/ε strongly in L∞(0, T), (6.34) the convergence of eετ being actually strong in W1,∞(0, T).
Proposition 6.5.6 (Discrete/continuous Γ-convergence) The time-discrete WIDE functionalsWετ converge in the sense of Mosco convergence to the continuous functionals Wε in V.
Before we prove the Proposition 6.5.6 let us finish the proof of Lemma 6.4.1.
Proof of Lemma 6.4.1: Proposition 6.5.4 yields that the minimizersuετ of the discrete functional Wετ fulfill estimate (6.33) and are hence weakly precompact in V. As Wετ Γ-converges toWε with respect to the same topology by Proposition 6.5.6 we can apply the Fundamental Theorem of Γ-convergence (see [Dal93, Ch. 7] and [Bra02, Sect. 1.5]), which yields thatuετ * uεweakly inV, whereuεis the unique minimizer ofWε. Finally, estimate (6.33) passes to the limit and we have proven Lemma 6.4.1.
Proof of Proposition 6.5.6: The proof is classically divided into (i) proving the Γ-liminf inequality and (ii) checking the existence of a recovery sequence (see [Dal93, Bra02]).
Ad (i). Assume to be given a sequence uτ ∈Ybτ such that uτ * uwith respect to the weak topology onVand lim infτ→0Wετ[uτ]<∞. Let us denote byueτ ∈H2(0, T;Z∩X)
6.5 The time-discrete WIDE principle
the piecewise quadratic interpolant ofUi=uτ(iτ),i= 0, . . . , N, defined by the relations ueτ(t) =uτ(t) for t∈[0, τ] and
ue0τ(t) =ατ(t)u0τ(t) + (1−ατ(t))u0τ(t−τ) for t∈[τ, T],
where we have used the notation ατ(t) =αi(t) for t∈ ](i−1)τ, iτ], i= 1, . . . , N. Hence, ueτ is defined such that its derivative is piecewise affine (see Figure 6.1). We preliminarily observe that
ue0τ(t) =u0τ(t−τ) +τ ατ(t)ue00τ(t) for almost every t∈]τ, T]. (6.35) Moreover, we check that
Wετ[uτ] = Z T
τ
eτεε2ρ
2 kue00τk2dt+ Z T−τ
τ
eτε(·+τ)εν
2 ku0τk2dt+ Z T−2τ
τ
eτε(·+ 2τ)E(uτ) dt.
Since by assumption lim infτ→0Wετ[uτ]<∞we can extract a not relabeled subsequence such that lim supτ→0Wετ[uτ]<∞ and use the convergences of the weightseτε in Lemma 6.5.5 to obtain
ρ Z T
τ
kue00τk2dt+ν Z T−τ
τ
ku0τk2dt+ Z T−2τ
τ
E(uτ) dt≤C.
Hence, by using the growth conditions (6.4) and by possibly further extracting a not relabeled subsequence (and considering standard projections for t > T −2τ) we have the weak convergence of the piecewise constant interpolant
uτ * u weakly in Lp(0, T;X), uτ * u weakly in L2(0, T;Z), (6.36) while for the piecewise affine interpolant we have
uτ * uweakly in H1(0, T;H). (6.37) Thus, applying the theorem by Arzelà-Ascoli we even have that uτ →uin C(0, T;H). In particular, an easy calculation shows that uτ −uτ →0 in L2(0, T;H) such that we arrive at
uτ →u in L2(0, T;H). (6.38)
Furthermore, there exists av such that for the piecewise quadratic interpolant we obtain ueτ * v weakly in H2(0, T;H), ρue0τ * ρv0 strongly in C(0, T;H). (6.39)
Indeed, we have thatv=u. In order to check this fixw∈L2(0, T;H) and compute that
where we have used the identity in (6.35), the convergence of the piecewise affine inter-polant (6.37), and the boundedness|ατ| ≤1 and ofue00 in L2(0, T;H). Hence, we have the convergenceρue0τ * ρu0 in L2(0, T;H) and v=u. In particular, owing to the convergence in (6.39) we have proved thatρu1=ρue0τ(0) =ρu0(0) andu∈K(u0, u1).
Eventually, we exploit the strong convergences in L∞(0, T) of the piecewise constant interpolants of the discrete weights in Lemma (6.5.5) and the convergences in (6.36)–
(6.39) in order to get by the weak lower semi-continuity of the L2-norm Z T
Due to (6.38) we can extract a (not relabeled) subsequence such thatuτ converges a.e. in Ω×[0, T]. Thus, together with uτ * u in L2(0, T;Z), the application of Fatou’s lemma
In particular, these lower estimates ensure Wε[u]≤lim inf
τ→0 Wετ[U] = lim inf
τ→0 Wετ[uτ], which is the desired Γ-lim inf inequality.
Ad (ii). In order to construct a recovery sequence for a given u∈K(u0, u1) we define first thebackward floating mean operator Mτ on L1(0, T;H) (also calledSteklov averaging operator, see [LSU68, Ch. 2 Sect. 4]) by setting
Mτ[u](t) =
In particular, using Lebesgue’s differentiation theorem we immediately check that for u ∈ Lq(0, T;H) (resp. Lq(0, T;Z), Lq(0, T;X)) we have the convergence Mτ[u] → u in Lq(0, T;H) for 1≤q <∞ (resp. Lq(0, T;Z), Lq(0, T;X)).
Letting an arbitrary u ∈ K(u0, u1) be fixed we define the discrete trajectory U =
6.5 The time-discrete WIDE principle
(U0, . . . , UN)∈Yτ by
U0=u0, ρU1 =ρ(u0+τ u1), Ui =Mτ[u](iτ) fori= 2, . . . , N.
We denote by uτ and uτ the piecewise affine and constant interpolants, respectively, as-sociated with U.
We aim to show thatuτ is a recovery sequence for u. Indeed, we clearly have that uτ
converges strongly tou in L2(0, T;Z)∩Lp(0, T;X), while uτ converges at least weakly to
Next, we exploit theλ-convexity of F and compute that Z T−2τ
In particular, by taking the lim sup as τ → 0 and recalling that uτ → u strongly in L2(0, T;Z)∩Lp(0, T;X) and the convergences (6.34), we have that Next, we deal with the second-order derivatives in time like we did in the first-order case
in (6.40). We compute
Finally, combining (6.40)–(6.42) we have proved that Wε[u] =
Namely,uτ is a recovery sequence foru.
Before closing this section let us stress that the obtained results can be adapted in order to encompass more general situations. In particular, we can consider unbounded domains (see [Ste11]) as well as different boundary conditions or the presence of additional source terms with no particular intricacy. Moreover, the WIDE approach can be applied to other classes of dissipative equations. For instance, one could recast the WIDE principle for the strongly damped wave equation
ρu00−ν∆u0−∆u+f(u) = 0,
suitably combined with boundary and initial conditions by replacing the dissipative term ενku0k2/2 with the H1-seminormενk∇u0k2/2 in the definition of the functionalWε.