WEIGHTED INERTIA-DISSIPATION-ENERGY FUNCTIONALS FOR

WEIGHTED INERTIA-DISSIPATION-ENERGY FUNCTIONALSFOR SEMILINEAR EQUATIONS

MATTHIAS LIERO AND ULISSE STEFANELLI

Abstract. We address a global-in-time variational approach to semilinear PDEs of eitherparabolic or hyperbolic type by means of the so-called Weighted Inertia-Dissipation-Energy(WIDE) functional. In particular, minimizers of the WIDE functional are proved to converge,up to subsequences, to weak solutions of the limiting PDE. This entails the possibility ofreformulating the limiting di↵erential problem in terms of convex minimization. The WIDEformalism can be used in order to discuss parameters asymptotics via �-convergence and isextended to some time-discrete situation as well.

1. Introduction

This paper is concerned with the classical semilinear PDE

⇢utt + ⌫ut ��u+ f(u) = 0 in ⌦⇥ (0, T ), (1.1)

posed in a bounded smooth domain ⌦ ⇢ Rd up to some reference time T > 0. Here,the density ⇢ and viscosity ⌫ are nonnegative parameters such that ⇢ + ⌫ > 0, f = F 0 is ofpolynomial growth, and F is �-convex (see (2.2)). In particular, in the following we explicitlyconsider the case ⇢ > 0 and ⌫ > 0. Let us however mention that the theory includes thelimiting cases of the semilinear wave equation (⌫ = 0) and the semilinear heat equation (⇢ =0) as well. We complement equation (1.1) with homogeneous Dirichlet boundary conditions(for simplicity) and initial conditions u(x, 0) = u0(x), ⇢ut(x, 0) = ⇢u1(x).

The aim of this paper is to discuss a global-in-time variational approach to (1.1). Inparticular, for all " > 0 we shall be concerned with the functional

W"(u) =

Z T

0

Z

⌦

e�t/"

✓"2⇢

2|utt|2 +

"⌫

2|ut|2 +

1

2|ru|2 + F (u)

◆dxdt.

The latter is called Weighted Inertia-Dissipation-Energy (WIDE) functional as it features theweighted sum of the inertial term ⇢|utt|2/2, the dissipative term ⌫|ut|2/2, and the energetic

term |ru|2/2 + F (u).

2010 Mathematics Subject Classification. 49J45; 35M13.Key words and phrases. Semilinear equation, minimum principle, elliptic regularization, time

discretization.1

2 MATTHIAS LIERO AND ULISSE STEFANELLI

The main result of the paper consists in showing that the WIDE functional W" is uniformlyconvex for " small, hence admitting a unique minimizer u" with u(x, 0) = u0(x), ⇢ut(x, 0) =⇢u1(x), and that, up to not relabeled subsequences,

lim"!0

u" = u where u solves (1.1). (1.2)

The interest of this perspective resides in the possibility of connecting the di�cult semilinearPDE problem (1.1) with a comparably easier problem: the constrained minimization of theuniformly convex functional W". This possibility provides a novel variational insight to thedi↵erential problem by opening the way to the application of the tools of the calculus ofvariations to (1.1). For instance, we recall that the functional W" admits a unique minimizerwhereas no uniqueness is known for (1.1) under general nonlinearities f . In this regard,the WIDE functional approach can be expected to possibly serve as a variational selection

criterion in some nonuniqueness situation. This possibility has been already checked for aspecific ODE case in [11].

One has to mention that the WIDE variational program has already been successfullydeveloped in [23, 25] for the semilinear wave case ⌫ = 0 and in [17] for the semilinear heatcase ⇢ = 0. We aim here at combining the two techniques in order to deal with the mixedcase of equation (1.1), namely for ⇢ > 0 and ⌫ > 0.

For semilinear wave equations ⌫ = 0, the convergence (1.2) corresponds to a conjectureby De Giorgi [8] which is originally stated for T = 1. This conjecture has been checkedpositively by Serra & Tilli [23] and by the second author [25] (for T < 1) followingcompletely di↵erent approaches. Moreover, both for T finite and infinite, we have consideredsome ODE analogue of the De Giorgi conjecture in [11].

As for the semilinear parabolic case ⇢ = 0 one has to mention that the pioneering paperby Lions [12] as well as the monograph by Lions & Magenes [13]. As for the nonlinearcase, one has to recall the paper by Ilmanen [9] where the WIDE approach is used in orderto prove the existence and partial regularity of the so-called Brakke mean curvature flow ofvarifolds. In this regard, the reader is also referred to [24] for an application to mean curvatureflow of cartesian surfaces. The general case of abstract gradient flows of �-convex functionalsis discussed in [17] in the Hilbertian setting and then in [21, 22] for curves of maximal slopein metric spaces. Results and applications to rate-independent dissipative systems have beenpresented by Mielke & Ortiz [15] and then extended and coupled with time-discretizationin [16]. Two relaxation and scaling examples in Mechanics are provided by Conti & Ortiz[6], and some application to crack propagation is given by Larsen, Ortiz, & Richardson[10]. Moreover, the extension of the WIDE principle to doubly nonlinear parabolic equationsis discussed in [?, ?, 3]. Eventually, a similar functional approach (with " fixed though) hasbeen considered by Lucia, Muratov, & Novaga in connection with traveling waves inreaction-di↵usion-advection problems [14, 19, 20].

The aim of this paper is the extension of the analysis of [25] in order to take into accountdissipative e↵ects ⌫ > 0 as well. The outcome of this extension is a theory which is indeed

WIDE FUNCTIONALS FOR SEMILINEAR EQUATIONS 3

independent of the character of equation (1.1), provided either ⇢ or ⌫ is positive. This is aquite remarkable feature of the WIDE formalism which in principle could make it of use inrelation with a significant range of evolution problems. We exploit this fact in Subsection4.3 where the limits ⇢ ! 0 and ⌫ ! 0 are discussed by means of a �-convergence analysis.In order to illustrate the independence of the theory from the character of the equation weresolved in keeping track of the parameters ⇢ and ⌫ throughout the analysis.

After having introduced the relevant notion of weak solution, in Section 2 we state ourmain convergence result Theorem 2.1. Section 3 is focusing on the well-posedness of theminimization problem for the WIDE functional W" (Theorem 3.1) as well as on the relatedEuler-Lagrange equation (Lemma 3.2). Combined with initial and boundary conditions, thelatter corresponds to some weak form of the following

"2⇢utttt � 2"⇢uttt + (⇢�"⌫)utt + ⌫ut ��u+ f(u) = 0 in ⌦⇥ (0, T ), (1.3)

u(·, 0) = u0(·) in ⌦, ⇢ut(·, 0) = ⇢u1(·) in ⌦, u = 0 on @⌦⇥ (0, T ) (1.4)

"2⇢utt(·, T ) = 0, "2⇢uttt(·, T ) = "⌫ut(·, T ) in ⌦. (1.5)

Clearly equation (1.1) is nothing but the formal limit in (1.3) for " ! 0. In particular, theminimization of W" corresponds to an elliptic regularization in time of the original problem.Note that, as the above problem is of fourth order in time, the two extra final conditions

(1.5) arise and, at all levels " > 0, causality is lost. Owing to this fact, the convergence (1.2)is generally referred to as the causal limit for it results in restoring causality.

The proof of our main result is presented in Section 4 and rests upon the validity of an apriori estimate on the minimizers of the WIDE functional (Lemma 4.1). The proof of thisestimate requires the discussion of some time-discrete version of the WIDE principle whichmight be also of independent interest. We develop such a discretization in Section 5.

2. Main result

We shall start by recalling some assumptions and introducing our weak solution notionfor problem (1.3)-(1.5). Let ⌦ ⇢ Rd be a non-empty, open, and Lipschitz domain andf = F 0 2 C(R) be of polynomial growth. In particular, we ask for some constant C > 0 suchthat, for all v 2 R,

1

C|v|p F (v) + C and |f(v)|p0 C(1 + |v|p) (2.1)

where p � 2 and 1/p + 1/p0 = 1. Moreover, we assume that F is �-convex for some given� 2 R, i.e.,

v 7! F (u)� �

2|v|2 is convex. (2.2)

Equivalently, F is �-convex if and only if

F (✓u+ (1�✓)v) ✓F (u) + (1�✓)F (v)� �

2✓(1�✓)|u�v|2 8 u, v 2 R, 0 ✓ 1.


Note that the growth assumptions in (2.1) imply that F has at most p-growth.We define H = L2(⌦), X = Lp(⌦), and V = H1

0(⌦) so that V ⇢ H compactly and X ⇢ Hcontinuously and assume u0, u1 2 V \X. Let h·, ·i denote the duality pairing both on V 0⇥Vand X 0 ⇥X and by (·, ·) the usual scalar product on H. Moreover, | · | denotes the modulusas well as the norm on H and k · kB stands for the norm of the normed space B. We definethe (energy) functional E : V \X ! R as

E(u) =1

2

Z

⌦

|ru|2 dx+

Z

⌦

F (u) dx

and the operators A : V ! V 0 and B : X ! X 0 as

hAu, vi :=Z

⌦

ru·rv dx, hB(u), vi :=Z

⌦

f(u)v dx

so that hAu, ui = kuk2V and B(u) = f(u) almost everywhere. We have that

DE = A+B : V \X ! V 0 +X 0

being bounded. Finally, we introduce the spaces

U := H1(0, T ;H) \ L2(0, T ;V ) \ Lp(0, T ;X), V := {u 2 U : ⇢u 2 H2(0, T ;H)}and assume that B is weakly continuous, i.e., we have that

uk * u in U =) B(uk) * B(u) in U 0. (2.3)

A choice for the function f fulfilling these assumptions is f(u) = |u|p�2u + `(u) where` 2 C0,1(R).

The above assumptions will be tacitly assumed throughout the remainder of the paper.We are now in position to present the main result of this text which is proved in Subsection4.2.

Theorem 2.1 (WIDE principle). Let u"minimize the WIDE functional W" on the nonempty

and convex set K(u0, u1) := {u 2 V : u(0) = u0, ⇢ut(0) = ⇢u1}. Then, for some not relabeled

subsequence we have that u" * u in U , where u solves

⇢utt + ⌫ut +DE(u) = 0 a.e in (0, T ), u(0) = u0, ⇢ut(0) = ⇢u1. (2.4)

Before moving on let us stress that the latter result can be adapted in order to encompassmore general situations. In particular, we can consider unbounded domains (see [25]) as wellas di↵erent boundary conditions or the presence of additional source terms with no particularintricacy. Moreover, the WIDE approach can be applied to other classes of dissipative equa-tions including. For instance, one could recast the WIDE principle for the strongly damped

wave equation⇢utt � ⌫�ut ��u+ f(u) = 0,

suitably combined with boundary and initial conditions.


3. Well-posedness of the minimum problem

Let us start by checking that indeed W" admits a unique minimizer in the set K(u0, u1).In the convex case

�� := max{0,��} = 0

the existence of a (unique) minimizer is a direct consequence of the Direct Method. As forthe general nonconvex case �� > 0, existence and uniqueness of minimizers follow by letting" be small enough.

Theorem 3.1 (Well-posedness of minimum problem). For " small the WIDE functional W"

is uniformly convex with respect to the metric of H1(0, T ;H)\L2(0, T ;V ). In particular, W"

admits a unique minimizer u" 2 K(u0, u1).

Proof. As already mentioned the convex case �� = 0 is quite straightforward so let us assumefrom the very beginning that �� > 0 and decompose W" into the sum of a quadratic part Q"

and a convex remainder R" as follows

W"(u) =

Z T

0

e�t/"

2

�⇢"2|utt|2+⌫"|ut|2+kuk2V��|u|2

�dt+

Z T

0

Z

⌦

e�t/"G(u)dt dx

=: Q"(u) +R"(u)

with G(u) = F (u) + ��|u|2/2 convex by (2.2). In order to handle the quadratic part Q", wewill exploit the auxiliary function v(t) := e�t/(2")u(t) and readily check that

e�t/(2")ut(t) = vt(t) +1

2"v(t), e�t/(2")utt(t) = vtt(t) +

1

"vt(t) +

1

4"2v(t).

Note that, by possibly letting " be small, standard computations ensure that

e�T/"kuk2L2(0,T ;V ) kvk2L2(0,T ;V ) kuk2L2(0,T ;V ), (3.1)

"4e�T/"kuk2H2(0,T ;H) kvk2H2(0,T ;H) "�4kuk2H2(0,T ;H). (3.2)

Moreover, Q"(u) can be rewritten in terms of v as

Q"(u) =

Z T

0

✓⇢"2

2|vtt|2 +

⇢+"⌫

2|vt|2 +

⇢+4"⌫�16"2��

32"2|v|2 + 1

2kvk2V

◆dt

+

Z T

0

✓⇢"(vtt, vt) +

⇢

4(vtt, v) +

⇢+2"⌫

4"(vt, v)

◆dt

=

Z T

0

✓⇢"2

2|vtt|2 +

⇢+2"⌫

4|vt|2 +

⇢+4"⌫�16"2��

32"2|v|2 + 1

2kvk2V

◆dt

+⇢"

2

�|vt(T )|2�|vt(0)|2

�+⇢

4

⇣�vt(T ), v(T )

��vt(0), v(0)

�⌘

+⇢+2"⌫

8"(|v(T )|2�|v(0)|2) =: V"(v) + S"(v) (3.3)


where V" is the integral contribution whereas S" collects all boundary terms. By letting

" < max

⇢r⇢

16�� ,⌫

4��

�

the quadratic form V" is convex. Moreover, an elementary yet tedious calculation shows thatfor u 2 K(u0, u1) we have

S"(v) =3"⇢

8e�T/"|ut(T )|2 +

"⇢

8e�T/"|ut(T )�u(T )/"|2 + ⌫

4e�T/"|u(T )|2

� 3"⇢

8|u1|2 � "⇢

8|u1�u0/"|2 � ⌫

4|u0|2,

which is convex in u(T ), ut(T ) as well. Let now ✓ 2 [0, 1] and u, u 2 K(u0, u1) be given.Moreover, define v(t) := e�t/(2")u(t) and v(t) := e�t/(2")u(t). For all " small enough onededuces

Q"

�✓u+ (1�✓)u

�= V"

�✓v + (1�✓)v

�+ S"

�✓v + (1�✓)v)

�

✓V"(v) + (1�✓)V"(v) + ✓S"(v) + (1�✓)S"(v)

� ✓(1�✓)

2

Z T

0

⇢"2|vtt�vtt|2 +⇢+2"⌫

2|vt�vt|2 +

⇢+4"⌫�16"2��

16"2|v�v|2 + kv�vk2V dt

= ✓Q"(u) + (1�✓)Q"(u)

� ✓(1�✓)

2

Z T

0

⇢"2|vtt�vtt|2 +⇢+2"⌫

2|vt�vt|2 +

⇢+4"⌫�16"2��

16"2|v�v|2 + kv�vk2V dt.

By exploiting the first estimates in (3.1)-(3.2), we have proved that Q" is uniformly convexin with respect to the metric of H1(0, T ;H) \ L2(0, T ;V ) (or even H2(0, T ;H) if ⇢ > 0). AsW" = Q" + R" and R" is convex, the uniform convexity of W" and the existence of a uniqueminimizer u" 2 K(u0, u1) ensues. ⇤

3.1. Euler–Lagrange equation. Our analysis, in particular the derivation of a priori esti-mates, relies on the specific structure of the Euler–Lagrange equation forW". Let u" minimizeW" in K(u0, u1). By considering r 7! W"(u"+rv) for v 2 K(0, 0) we obtain that

0 =

Z T

0

e�t/"�"2⇢(u"

tt, vtt) + "⌫(u"t , vt) + hDE(u"), vi

�dt 8 v 2 K(0, 0). (3.4)

Hence, we have the following.


Lemma 3.2 (Euler-Lagrange equation). Let u"be the unique minimizer of the WIDE func-

tional W" in K(u0, u1). Then, u"solves

"2⇢u"tttt � 2"⇢u"

ttt + (⇢�"⌫)u"tt +DE(u") = 0 a.e. in (0, T ), (3.5)

u"(0) = u0, ⇢u"t(0) = ⇢u1, (3.6)

⇢u"tt(T ) = 0, "2⇢u"

ttt(T )� "⌫u"t(T ) = 0. (3.7)

4. Proof of the main result

The key step in the proof of Theorem 2.1 is to establish an integral energy estimate on u"

which is independent of ". Henceforth, the symbol C stands for any constant depending ondata and independent of ⇢, ⌫, and " (and, later, ⌧) and possibly changing from line to line.We shall prove the following lemma.

Lemma 4.1 (Estimate). Let u"minimize the WIDE functional W" over K(u0, u1). Then,

we have

(⇢+⌫)

Z T

0

|u"t |2dt+

Z T

0

E(u")dt C. (4.1)

Note that, owing to the growth conditions (2.1), the latter estimate entails in particularthat minimizers of W" on K(u0, u1) are uniformly bounded in U . This provides the necessarycompactness in order to prove our main result Theorem 2.1.

4.1. A formal argument. Let us however provide here a formal argument toward Lemma4.1. In particular, let us assume that the minimizers u" of the WIDE functional W" onthe convex set K(u0, u1) are actually strong solutions of the Euler-Lagrange equation (3.5)-(3.7). Note that this smoothness assumption is presently not justified. In particular, theargument below will be made rigorous by means of a time-discretization technique in Section5. Although quite technical, we believe this procedure to bear some interest in itself for itprovides the complete analysis of a time-discrete version of the WIDE principle.

Let us focus on the (more di�cult) case ⇢ > 0 only. Test (3.5) on the function t 7! v(t) :=(1+T�t)(u"

t(t)�u1) and take the integral on (0, T ). By recalling that

8g 2 L1(0, T ),

Z T

0

(1+T�t)g(t)dt =

Z T

0

g(t)dt+

Z T

0

✓Z t

0

g(s)ds

◆dt.


we easily compute thatZ T

0

"2⇢(u"tttt, v)dt =

Z T

0

"2⇢(u"tttt, u

"t�u1)dt+

Z T

0

Z t

0

"2⇢(u"tttt, u

"t�u1)dsdt

=(1+T )"2⇢

2|u"

tt(0)|2 �"2⇢

2|utt(T )|2 + "2⇢(u"

ttt(T ), u"t(T )�u1)

+ "2⇢(u"tt(T ), u

"t(T )�u1)� 3"2⇢

2

Z T

0

|u"tt|2dt (4.2)

where we have used integration by parts several times. Analogously, we obtain

�Z T

0

2"⇢(u"ttt, v)dt = 2"⇢

Z T

0

|u"tt|2dt+ 2"⇢

Z T

0

Z t

0

|u"tt|2ds dt

� 2"⇢(u"tt(T ), u

"t(T )� u1)� "⇢|u"

t(T )� u1|2,Z T

0

(⇢�"⌫)(u"tt, v)dt =

⇢�"⌫

2|u"

t(T )�u1|2 + ⇢�"⌫

2

Z T

0

|u"t�u1|2dt.

Finally, we have thatZ T

0

⌫(u"t , v)dt = ⌫

Z T

0

|u"t |2dt+ ⌫

Z T

0

Z t

0

|u"t |2ds dt� ⌫

Z T

0

(u"t , u

1)dt

� ⌫

Z T

0

Z t

0

(u"t , u

1)ds dt, (4.3)

Z T

0

hDE(u"), vi = E(u"(T ))� (1 + T )E(u0) +

Z T

0

E(u")dt�Z T

0

hDE(u"), u1idt

�Z T

0

Z t

0

hDE(u"), u1ids dt. (4.4)

Therefore, by summing up equations (4.2)–(4.4) and using the final conditions in (3.7) wecome to the following

(1+T )"2⇢

2|u"

tt(0)|2 +✓⇢�"⌫

2� "⇢

◆|u"

t(T )�u1|2 + ⌫

Z T

0

|u"t |2dt+ 2"⇢

Z T

0

Z t

0

|u"tt|2ds dt

+ E(u"(T )) +

Z T

0

E(u")dt+ ⇢

✓2"�3"2

2

◆Z T

0

|u"tt|2dt+ ⌫

Z T

0

Z t

0

|u"t |2ds dt+

⇢�"⌫

2

Z T

0

|u"t�u1|2dt

= (1+T )E(u0) +

Z T

0

hDE(u"), u1idt+Z T

0

Z t

0

hDE(u"), u1ids dt

+ ⌫

Z T

0

(u"t , u

1)dt+ ⌫

Z T

0

Z t

0

(u"t , u

1)ds dt. (4.5)


Hence, by using the growth conditions (2.1) and Young’s inequality we have shown that forsmall " estimate (4.1) holds.

4.2. Proof of Theorem 2.1. Let us now come to the proof of our main result. Let u" be theunique minimizer of W". Owing to Lemma 4.1 we can extract a not relabeled subsequenceu" such that u" * u in U . In order to check that u solves (2.4) let w 2 C1

0 ([0, T );V \X) begiven and define v"(t) := et/"w(t)� t(wt(0)+w(0)/")�w(0) such that v" 2 K(0, 0). We havethat

v"t (t) = et/"wt(t)+1

"et/"w(t)�wt(0)�

1

"w(0) and v"tt(t) = et/"wtt(t)+

2

"et/"wt(t)+

1

"2et/"w(t).

Since v" 2 C10 ([0, T );V \X) ⇢ K(u0, u1), from the variational equality (3.4) one obtains

0 =

Z T

0

e�t/"�"2⇢(u"

tt, v"tt) + "⌫(u"

t , v"t ) + hDE(u"), v"i

�dt

=

Z T

0

⇣⇢(u"

tt, "2wtt+2"wt+w) + ⌫(u"

t , "wt+w)� "⌫(u"t , h

")

+ hDE(u"), w�th"�e�t/"w(0)i⌘dt,

where we have used the short-hand notation h"(t) = e�t/"(wt(0)+w(0)/"). Hence, by inte-gration by parts we obtain

Z T

0

��⇢(u"

t , wt) + ⌫(u"t , w) + hDE(u"), wi

�dt+ ⇢(u"

t(0), w(0))

=

Z T

0

�u"t , "

2⇢wttt+2"⇢wtt+"⌫w+"⌫h"

�dt+

Z T

0

hDE(u"), th"+e�t/"w(0))idt

� ⇢�u"t(0), "

2wtt(0)+2"wt(0)�.

We easily check that t 7! th"(t) + e�t/"w(0) converges strongly in Lq(0, T ;V \ X) to 0 forevery q 2 [1,1). In particular, letting " ! 0 we exploit the weak continuity (2.3) andu"t(0) = u1 to obtain

Z T

0

��⇢(ut, wt) + ⌫(ut, w) + hDE(u), wi

�dt+ ⇢(u1, w(0)) = 0.

Namely, u solves the equation in (2.4), where utt makes sense in L2(0, T ;V 0) + Lp(0, T ;X 0).The initial condition u(0) = u0 follows from the precompactness of u" in U while ut(0) = u1

follows from the weak formulation of the limit equation.


4.3. �-convergence. As already mentioned, a remarkable trait of the WIDE approach isits independence of the character of the equation (1.1) as long as ⇢ + ⌫ > 0. In particular,the WIDE formalism is well-suited in order to describe limiting behaviors in the parameters.First of all, by inspecting the proof of Theorem 2.1 it is apparent that minimizers of theWIDE functional pass to limits ⇢ ! 0 and ⌫ ! 0 as well as to joint limits (⇢, ") ! (0, 0)and (⌫, ") ! (0, 0). On the other hand, by keeping " fixed we can argue from a variationalviewpoint by addressing the limits ⇢ ! 0 and ⌫ ! 0 within the frame of �-convergence [7].

Let us momentarily modify the notation for the WIDE functionals W", the function spaceV , and the set K by highlighting the dependence on the parameters ⇢ and ⌫ as W ⇢⌫

" , V⇢, andK⇢(u0, u1), respectively. Moreover, for the sake of notational simplicity we incorporate theconstraint u 2 K⇢(u0, u1) directly in the functional by letting

W⇢⌫

" = W⇢⌫" on K⇢(u0, u1) and W

⇢⌫

" = 1 elsewhere.

We have the following.

Lemma 4.2 (�-limit ⌫ ! 0). W⇢⌫

"�! W

⇢0

" w.r.t. both the strong and weak topology of V⇢.

Proof. The existence of a recovery sequence is immediate by pointwise convergence. The�-lim inf inequality follows from the fact that W

⇢⌫

" � W⇢0

" pointwise and W⇢0

" is lower semi-continuous with respect to the weak topology of V⇢. ⇤

As for the purely viscous limit we have the following.

Lemma 4.3 (�-limit ⇢ ! 0). W⇢⌫

"�! W

0⌫

" w.r.t. both the strong and weak topology of U .

Proof. The �-lim inf inequality is immediate as W⇢⌫

" � W0⌫

" pointwise and the latter is lowersemicontinuous with respect to the weak topology of U . As for the recovery sequence, weshall resort here to some singular perturbation technique (in time). In particular, for anygiven u 2 K(u0, u1) and almost every x 2 ⌦ we can find t 7! v⇢(x, t) 2 H1

0(0, T ) solvingweakly

v⇢(x, ·)�p⇢v⇢tt(x, ·) = ut(x, ·)� u1(x).

Then, it is a standard matter to prove that u⇢(x, t) := u0(x) + tu1(x) +R t

0v⇢(x, s) ds 2

K(u0, u1) is such that u⇢ ! u strongly in U andp⇢u⇢

tt ! 0 strongly in L2(0, T ;H). We

hence have that W⇢⌫

" (u⇢) ! W0⌫

" (u). ⇤Note that the above results entail �-convergence with respect to both strong and weak

topology. This circumstance is usually referred to as Mosco convergence [18] and plays aprominent role in classical convex analysis [4].

Before closing this subsection let us stress that the above �-limits are taken for " fixed andrecord that combined �-convergence analyses for both parameters and " ! 0 are presentlynot available. Additional material on �-convergence for WIDE functionals in the paraboliccase is however to be found in [?, 15, 16].


5. Time-discretization

The above proof of Theorem 2.1 rests upon the possibility of proving the key estimate(4.1). We achieve this by investigating a time-discrete version of the WIDE principle and,in particular, of the argument of Subsection 4.1. We replace the functional W" by a time-discrete WIDE functional W"⌧ . From here on we directly focus on the situation ⇢ > 0, thecase ⇢ = 0 being covered in [17]. We start by recalling the notation for the constant time-step⌧ := T/n (n 2 N) and introduce the space

V⌧ :=�(u0, . . . , un) 2 Hn+1 : (u2, . . . , un�2) 2 (V \X)n�3

.

Moreover, we define the functional W"⌧ : V⌧ ! R by

W"⌧ (u0, . . . , un) ="2⇢

2

nX

j=2

⌧e"⌧,j|�2uj|2 +"⌫

2

n�1X

j=2

⌧e"⌧,j+1|�uj|2 +n�2X

j=2

⌧e"⌧,j+2E(uj).

Given the vector (w0, . . . , wn), in the latter we have used the notation �w for its discrete

derivative �wj := (wj � wj�1)/⌧ for j = 1, . . . , n and �2w = �(�w), �3w = �(�2w) and so on.Moreover we have used the weights e"⌧,1, . . . , e"⌧,n given by

e"⌧,i =

✓"

"+ ⌧

◆i

for i = 1, . . . , n. (5.1)

These weights are nothing but the discrete version of the exponentially decaying weightt 7! exp(�t/") for we have that �e"⌧,i + e"⌧,i/" = 0. Namely, e"⌧,i is the solution of theconstant time-step implicit Euler discretization of the problem e0 + e/" = 0, with the initialcondition e(0) = 1. Finally, we denote the discrete counterpart of K(u0, u1) by K⌧ (u0, u1),i.e.,

K⌧ (u0, u1) = {(u0, . . . , un) 2 V⌧ : u0 = u0, ⇢�u1 = ⇢u1}.

The discrete WIDE functional W"⌧ represents a discrete version of the original time-continuous WIDE functional W". We shall drop the subscript "⌧ from e"⌧,j in the remainderof this section for the sake of notational simplicity.

5.1. Well-posedness of the discrete minimum problem. Exactly as in the time-continu-ous situation, in case E is �-convex the functional W"⌧ turns out to be uniformly convex forall su�ciently small ". Note that for all (u0, . . . , un) 2 K⌧ (u0, u1) we have that

nX

k=2

⌧ |uk|2 C

|u0|2 + |u1|2 +

nX

k=2

⌧ |�2uk|2!

(5.2)

where C depends on T . Hence, the functional W"⌧ is coercive on K⌧ (u0, u1). Indeed, thecoercivity of W"⌧ in V n�3 with respect to (u2, . . . , un�2) is immediate. As for the coercivityin H we see that, due to (5.2), the discrete WIDE functional W"⌧ controls the norm in H(up to constants depending on T, ⇢, ⌫, ", and ⌧).


Lemma 5.1 (Well-posedness of the discrete minimum problem). For " and ⌧ small and all

u0, u1 2 H, the discrete WIDE functional W"⌧ admits a unique minimizer in K⌧ (u0, u1).

Proof. This argument is the discrete analogue of the proof of Theorem 3.1. In particular, westart by decomposing W"⌧ into a quadratic part Q"⌧ and a convex remainder R"⌧ as

W"⌧ (u0, . . . , un)

=

"2⇢

nX

j=2

⌧ej2|�2uj|2 + "⌫

n�1X

j=2

⌧ej+1

2|�uj|2 � ��

n�2X

j=2

⌧ej+2

2|uj|2 +

n�2X

j=2

⌧ej+2

2kujk2V

!

+n�2X

j=2

⌧ej+2

Z

⌦

G(uj) dx =: Q"⌧ (u0, . . . , un) +R"⌧ (u0, . . . , un)

with G(u) = F (u) + ��|u|2/2. The result follows by checking that, for small " and ⌧ , thefunctional Q"⌧ is uniformly convex. To this end, for (u0, . . . , un) 2 K⌧ (u0, u1) let (v0, . . . , vn)be defined as vi =

peiui. Then, we compute thatpei�ui = `"⌧�vi + �"⌧vi

pei�

2ui = `"⌧�2vi + `"⌧�"⌧�vi�1 + �"⌧�vi + �2

"⌧vi�1

where `"⌧ =p

"/("+⌧) and �"⌧ = (1�`"⌧ )/⌧ such that `"⌧ ! 1 and �"⌧ ! 1/(2") for ⌧ ! 0.Moreover, we used that

�1

pei

= �"⌧1

pei.

Hence, we can rewrite Q"⌧ (u0, . . . , un) as

Q"⌧ (u0, . . . , un) = "2⇢nX

j=2

⌧

2

�`2"⌧ |�2vj|2 + �2

"⌧ |�vj|2 + `2"⌧�2"⌧ |�vj�1|2 + �4

"⌧ |vj�1|2�

+ "⌫n�1X

j=2

⌧`2"⌧2

�`2"⌧ |�vj|2 + �2

"⌧ |vj|2��

n�2X

j=2

⌧`4"⌧2

|vj|2

+ "2⇢

nX

j=2

⌧⇣`2"⌧�"⌧ (�

2vj, �vj�1) + `"⌧�"⌧ (�2vj, �vj) + `"⌧�

2"⌧ (�

2vj, vj�1)

+ `"⌧�2"⌧ (�vj, �vj�1) + �3

"⌧ (�vj, vj�1) + `"⌧�3"⌧ (�vj�1, vj�1)

⌘

+ "⌫

n�1X

j=2

⌧`3"⌧�"⌧ (�vj, vj) +n�2X

j=2

⌧`4"⌧2

kvjk2V .


Next, we collect all terms involving |vj| and obtain

"2⇢

nX

j=2

⌧

2�4"⌧ |vj�1|2 + "⌫

n�1X

j=2

⌧

2`2"⌧�

2"⌧ |vj|2 � ��

n�2X

j=2

⌧`4"⌧2

|vj|2

=⇥"2⇢�4

"⌧ + "⌫`2"⌧�2"⌧ � ��`4"⌧

⇤ n�2X

j=2

⌧

2|vj|2

+⌧"2⇢�4

"⌧

2(|v1|2 + |vn�1|2) +

⌧"⌫`2"⌧�2"⌧

2|vn�1|2.

We check that the coe�cient in square brackets in front of the sum in the above right-handside is positive for su�ciently small " and ⌧ . For instance, one can choose ⌧ = 3" andcompute `"⌧ = 1/2 and (1�`"⌧ )/⌧ = 1/(6"). In particular, the coe�cient reads

⇥"2⇢�4

"⌧ + "⌫`2"⌧�2"⌧ � ��`4"⌧

⇤=

⇢

64"2+

⌫

4(62)"� ��

16

which basically corresponds to the coe�cient multiplying |v|2 in the first line of (3.3). Next,we look at the mixed terms. Using summing by parts, we obtain

nX

j=2

⌧(�2vj, vj�1) = (�vn, vn)� (�v1, v1)�nX

j=2

⌧ |�vj|2.

Hence, we have that

"2⇢nX

j=2

⌧

✓1

2�2"⌧ |�vj|2 + `"⌧�

2"⌧ (�

2vj, vj�1)

◆

= "2⇢nX

j=2

⌧�2"⌧

✓1

2� `"⌧

◆|�vj|2 + "2⇢`"⌧�

2"⌧

�(�vn, vn)� (�v1, v1)

�.

Finally, using again summation by parts for the remaining mixed terms, we check as in thetime-continuous case that the quadratic form Q" is uniformly convex. Hence, the existenceof unique minimizers follows by the Direct Method. ⇤

5.2. Discrete Euler–Lagrange system. As in Section 3.1 the unique minimizer (u0, . . . , un)of the time-discrete functional W"⌧ solves the corresponding Euler–Lagrange system and wedirectly compute that

0 = "2⇢nX

j=2

⌧ej(�2uj, �

2vj) + "⌫n�1X

j=2

⌧ej+1(�uj, �vj) +n�2X

j=2

⌧ej+2hDE(uj), vji

8(v0, . . . , vn) 2 K⌧ (0, 0).

(5.3)


Let us now proceed as in the continuous case. First of all, we sum-by-parts and obtain that

⇢nX

j=2

⌧ej(�2uj, �

2vj) = ⇢nX

j=2

ej(�2uj, �vj)� ⇢

nX

j=2

ej(�2uj, �vj�1)

= ⇢en(�2un, �vn)� ⇢e2(�

2u2, �v1)� ⇢n�1X

j=2

⌧(�(e�2u)j+1, �vj)

⇢�v1=0= ⇢en(�

2un, �vn)� ⇢n�1X

j=2

(�(e�2u)j+1, vj) + ⇢n�1X

j=2

(�(e�2u)j+1, vj�1)

= ⇢en(�2un, �vn)� ⇢(�(e�2u)n, vn�1) + ⇢(�(e�2u)3, v1) + ⇢

n�2X

j=2

⌧(�2(e�2u)j+2, vj)

v0=0= ⇢en(�

2un, �vn)� ⇢(�(e�2u)n, vn�1) + ⇢n�2X

j=2

⌧(�2(e�2u)j+2, vj),

where we have used that ⇢�v1 = 0 and v0 = 0. Similarly, we obtain

⌫n�1X

j=2

⌧ej+1(�uj, �vj) = ⌫en(�un�1, vn�1)� ⌫n�2X

j=2

⌧(�(ej+2�uj+1), vj).

Next, by means of the definition of ej in (5.1) and some tedious computations we check that

⇢�2(e�2u)j+2 = ⇢ej+2

⇣�4uj+2 �

2

"�3uj+1 +

1

"2�2uj

⌘,

⇢�(ej+2�uj+1) = ⇢ej+2

⇣�2uj+1 �

1

"�uj

⌘,

and rewrite relation (5.3) in the equivalent form

0 =N�2X

j=2

⌧ej+2

⇣⇢"2�4uj+2 � 2⇢"�3uj+1 + ⇢�2uj � "⌫�2uj+1 + ⌫�uj, vj

⌘+ hDE(uj), vji

+ "2⇢⇣en(�

2un, �vn)� (�(e�2u)n, vn�1)⌘+ "⌫en(�un�1, vn�1).

This holds for arbitrary v 2 K⌧ (0, 0) and we have therefore proved the following.

Lemma 5.2 (Discrete Euler-Lagrange system). Let (u0, . . . , un) 2 K⌧ (u0, u1) be the unique

minimizer of the discrete WIDE functional W"⌧ . Then, (u0, . . . , un) solves

"2⇢�4uj+2 � 2"⇢�3uj+1 + ⇢�2uj � "⌫�2uj+1 + ⌫�uj +DE(uj) = 0, (5.4)


subject to the initial and final conditions

u0 = u0, ⇢�u1 = ⇢u1, and (5.5)

"2⇢�2un = 0, "2⇢�3un = "⇢�2un�1 + "⌫�un�1. (5.6)

Equations (5.4)–(5.6) are the discrete analogue of equations (3.5)–(3.7).

5.3. Discrete estimate. The argument of Subsection 4.1 can be made rigorous at the time-discrete level. We present here a time-discrete version of estimate (4.5) by using the time-discrete Euler-Lagrange system (5.4). Namely, we aim at proving the following.

Lemma 5.3 (Discrete estimate). Let (u0, . . . , un) minimize the discrete WIDE functional

W"⌧ over K⌧ (u0, u1). Then, for all " and ⌧ su�ciently small we have

(⇢+⌫)n�2X

j=2

⌧ |�uj|2 +n�2X

j=2

⌧E(uj) C. (5.7)

Proof. Let us assume from the very beginning that ⇢ > 0 throughout this proof. Indeed, thecase ⇢ = 0 (and correspondingly ⌫ > 0) is already detailed in [17]. We argue by mimickingat the discrete level the estimate of Subsection 4.1. Namely, we shall perform the following:

n�2X

k=2

⌧(5.4)��v=�uk�u1 +

n�2X

k=2

⌧

kX

i=1

⌧(5.4)��v=�ui�u1

!. (5.8)

At first, let us test the time-discrete Euler–Lagrange equation in (5.4) on v = ⌧(�uj�u1)and sum for j = 2, . . . , k n�2 in order to get that

"2⇢

kX

j=2

⌧(�4uj+2, �uj�u1)� 2"⇢kX

j=2

⌧(�3uj+1, �uj�u1)� "⌫

kX

j=2

⌧(�2uj+1, �uj�u1)

+ ⇢

kX

j=2

⌧(�2uj, �uj�u1) + ⌫

kX

j=2

⌧(�uj, �uj�u1) +kX

j=2

⌧hDE(uj), �uj�u1i = 0. (5.9)


We now treat separately all terms in the above left-hand side. The fourth-order-in-time termcan be handled as follows.

"2⇢kX

j=2

⌧(�4uj+2, �uj�u1) = "2⇢kX

j=2

(�3uj+2��3uj+1, �uj�u1)

= "2⇢(�3uk+2, �uk�u1)� "2⇢(�3u3, �u2 � u1)� "2⇢kX

j=3

⌧(�3uj+1, �2uj)

= "2⇢(�3uk+2, �uk�u1)� "2⇢kX

j=2

⌧(�3uj+1, �2uj)

= "2⇢(�3uk+2, �uk�u1)� "2⇢

2|�2uk+1|2 +

"2⇢

2|�2u2|2 +

"2⇢

2

kX

j=2

|�2uj+1��2uj|2, (5.10)

where we also used that �u1 = u1. Next, we treat the third-order-in-time term of (5.9)somehow similarly as

� 2"⇢kX

j=2

⌧(�3uj+1, �uj�u1) = �2"⇢(�2uk+1, �uk�u1) + 2"⇢kX

j=2

⌧ |�2uj|2. (5.11)

As for the remaining discrete time derivatives in (5.9) we compute

⇢kX

j=2

⌧(�2uj, �uj�u1) =⇢

2

kX

j=2

|�uj��uj�1|2 +⇢

2|�uk�u1|2, (5.12)

"⌫kX

j=2

⌧(�2uj+1, �uj�u1) ="⌫

2|�uk+1�u1|2 � "⌫

2|�u2�u1|2 � "⌫

2

kX

j=2

|�uj+1��uj|2, (5.13)

⌫kX

j=2

⌧(�uj, �uj�u1) = ⌫kX

j=2

⌧ |�uj|2 � ⌫kX

j=2

⌧(�uj, u1). (5.14)

By using the �-convexity of F we obtain for the last term in (5.9)

kX

j=2

⌧hDE(uj), �uj�u1i �u1=u1

=kX

j=1

⌧hDE(uj), �uj�u1i

� E(uk)� E(u0)�kX

j=1

⌧hDE(uj), u1i+ �

2

kX

j=1

|uj�uj�1|2. (5.15)


We now recollect computations (5.10)–(5.15) into equation (5.9) in order to deduce that

"2⇢(�3uk+2, �uk�u1)� "2⇢

2|�2uk+1|2 � 2"⇢(�2uk+1, �uk�u1) + 2"⇢

kX

j=2

⌧ |�2uj|2

+⇢

2|�uk�u1|2 � "⌫

2|�uk+1�u1|2 + ⌫

kX

j=2

⌧ |�uj|2 + E(uk) +�⌧

2

kX

j=2

⌧ |�uj|2

C +kX

j=1

⌧hDE(uj), u1i+ ⌫

kX

j=2

⌧(�uj, u1). (5.16)

Let us now move to the consideration of the second term in (5.8). We multiply (5.16) by⌧ and take the sum for k = 2, . . . , n� 2 getting

"2⇢n�2X

k=2

⌧(�3uk+2, �uk�u1)� "2⇢

2

n�2X

k=2

⌧ |�2uk+1|2 � 2"⇢n�2X

k=2

⌧(�2uk+1, �uk�u1)

+ 2"⇢n�2X

k=2

⌧kX

j=2

⌧ |�2uj|2 +⇢

2

n�2X

j=2

⌧ |�uk�u1|2 � "⌫

2

n�2X

k=2

⌧ |�uk+1�u1|2

+ ⌫n�2X

k=2

⌧kX

j=2

⌧ |�uj|2 +n�2X

k=2

⌧E(uk) +�⌧

2

n�2X

k=2

⌧kX

j=2

⌧ |�uj|2

C +n�2X

k=2

⌧kX

j=1


n�2X

k=2

⌧kX

j=2

⌧(�uj, u1). (5.17)

By summing by parts, we can write the first term in (5.17) in the following way:

"2⇢n�2X

k=2

⌧(�3uk+2, �uk�u1) = "2⇢n�2X

k=2

(�2uk+2��2uk+1, �uk � u1)

= "2⇢(�2un, �un�2�u1)� "2⇢(�2u3, �u2 � u1)� "2⇢n�2X

k=3

⌧(�2uk+1, �2uk)

= �"2⇢

n�2X

k=2

⌧(�2uk+1, �2uk) (5.18)


where we have used the initial condition ⇢�u1 = ⇢u1 and the final condition ⇢�2un = 0.Moreover, we can also compute that

� 2"⇢n�2X

k=2

⌧(�2uk+1, �uk�u1) = 2"⇢n�2X

k=2

(�uk��uk+1, �uk�u1)

= "⇢|�u2 � u1|2 + "⇢

n�3X

k=2

|�uk+1��uk|2 � "⇢|�un�2�u1|2 + 2"⇢(�un�2��un�1, �un�2�u1)

= "⇢|�u2 � u1|2 + "⇢n�3X

k=2

|�uk+1��uk|2 � "⇢|�un�2�u1|2 � 2"⇢⌧(�2un�1, �un�2�u1)

� �3

2"⇢|�un�2�u1|2 � 2"⇢⌧ 2|�2un�1|2. (5.19)

Furthermore, we observe that

2"⇢n�2X

k=2

⌧ |�2uk|2 � "2⇢n�2X

k=2

⌧(�2uk+1, �2uk)�

"2⇢

2

n�2X

k=2

⌧ |�2uk+1|2

� 2"⇢n�2X

k=2

⌧ |�2uk|2 �"2⇢

2

n�2X

k=2

⌧ |�2uk+1|2 �"2⇢

2

n�2X

k=2

⌧ |�2uk|2 �"2⇢

2

n�2X

k=2

⌧ |�2uk+1|2

� ⇢

✓2"�3

2"2◆ n�2X

k=2

⌧ |�2uk|2 � "2⇢⌧ |�2un�1|2. (5.20)

Let us now write estimate (5.16) by choosing k = n�2 and taking advantage of the finalboundary conditions in (5.6)

"⌫(�un�1, �un�2�u1)� "⇢(�2un�1, �un�2�u1)� "2⇢

2|�2un�1|2

+ 2"⇢n�2X

j=2

⌧ |�2uj|2 +⇢

2|�un�2�u1|2 � "⌫

2|�un�1�u1|2 + ⌫

n�2X

j=2

⌧ |�uj|2 + E(un�2)

C +n�2X

j=1


n�2X

j=2

⌧(�uj, u1). (5.21)

Note that due to the final conditions in (5.6) the following identity holds:

�⇢�2un�1 =⌧⌫

⌧+"�un�1.


Hence, we have that

"⌫(�un�1, �un�2�u1)� "⇢(�2un�1, �un�2�u1)

= "⌫|�un�1|2 � "⌫(�un�1, �un�1��un�2)� "⌫(�un�1, u1) +

⌧"⌫

⌧+"(�un�1, �un�2�u1)

= "⌫|�un�1|2 � "⌫⌧(�un�1, �2un�1)� "⌫

✓1 +

⌧

⌧+"

◆(�un�1, u

1) + "⌫

✓⌧

⌧+"

◆|�un�1|2

� "⌫⌧

⌧+"(�un�1, �un�1��un�2)

= "⌫

✓1 +

⌧

⌧+"+

⌫⌧ 2

⇢(⌧+")+

⌫⌧ 3

⇢(⌧+")2

◆|�un�1|2 � "⌫

✓1 +

⌧

⌧+"

◆(�un�1, u

1)

� ⌫

✓"�"2

2

◆|�un�1|2 � 2⌫|u1|2.

Therefore, by recalling that ⇢ > 0 we have from (5.21) that

✓⌫"

4�⌫"2

2�"2⌧

⇢�2"⌧ 2⌫2

⇢

◆|�un�1|2 + 2"⇢

n�2X

j=2

⌧ |�2uj|2

+⇢

2|�un�2�u1|2 + ⌫

n�2X

j=2

⌧ |�uj|2 + E(un�2) C +n�2X

j=1


n�2X

j=2

⌧(�uj, u1)

(5.22)

where we have used that

"⌫

2|�un�1�u1|2 3

4"⌫|�un�1|2 + C,

�"2⇢

2|�2un�1|2 = �"2⌫2

2⇢

✓⌧

⌧+"

◆2

|�un�1|2 � �"2⌫2

2⇢|�un�1|2.


By taking the sum of (5.22) and (5.17), using the equalities (5.18), (5.19), (5.20), andrecalling the fact that ⇢ > 0 we obtain that

⇢

✓1

2�3

2"

◆|�un�2�u1|2 +

✓⌫"

4�⌫"2

2�"2⌧

⇢�2"⌧ 2⌫2

⇢�3"⌫⌧

4

◆|�un�1|2

+ ⌫ (1�")n�2X

j=2

⌧ |�uj|2 + E(un�2) + 2"⇢n�2X

k=2

⌧kX

j=2

⌧ |�2uj|2 + ⇢

✓2"� 3"2

2

◆ n�2X

k=2

⌧ |�2uk|2

+⇢

2

n�2X

k=2

⌧ |�uk�u1|2 +✓⌫��⌧

2

◆ n�2X

k=2

⌧

kX

j=2

⌧ |�uj|2 +n�2X

k=2

⌧E(uk)

C +n�2X

j=1


n�2X

j=2

⌧(�uj, u1) +

n�2X

k=2

⌧kX

j=1


n�2X

k=2

⌧kX

j=2

⌧(�uj, u1).

As " and ⌧ are assumed to be small, by using the growth conditions in (2.1) and Young’sinequality we readily get the estimate. ⇤

5.4. Proof of Lemma 4.1. In order to conclude the proof of Lemma 4.1 we need to showthat the time-discrete energy estimate in Lemma 5.3 passes to the limit as ⌧ ! 0 (for

fixed " > 0). To this aim, we check the discrete-to-continuous �-convergence W"⌧�! W"

with respect to the weak topology on U (see [5, 7] for relevant definitions and results on�-convergence).

For all vectors (w0, . . . , wn), we indicate by w⌧ and w⌧ its backward constant and piecewisea�ne interpolants on the partition, respectively. Namely,

w⌧ (0) = w⌧ (0) := w0, w⌧ (t) := wi, w⌧ (t) := ↵i(t)wi + (1�↵i(t))wi�1

for t 2 ((i� 1)⌧, i⌧ ], i = 1, . . . , n

where we have used the auxiliary functions

↵i(t) := (t� (i�1)⌧)/⌧ for t 2 ((i�1)⌧, i⌧ ], i = 1, . . . , n.

With these definitions we can reformulate the estimate in Lemma 5.3 as

(⇢+⌫)

Z T�2⌧

⌧

|@tu"⌧ |2dt+

Z T�2⌧

⌧

E(u"⌧ )dt C (5.23)

where u"⌧ and u"

⌧ denote the interpolants associated with the minimizer (u0, . . . , un) of thediscrete WIDE functional W"⌧ .

Lemma 5.4 (Discrete-to-continuous �-convergence). Let

bV⌧ := {u : [0, T ] ! V \X : u is piecewise a�ne on the time partition}


and define the functionals W ", W "⌧ : V ! [0,1] as

W "(u) :=

(W"(u) if u 2 K(u0, u1),

1 elsewhere,

W "⌧ (u) :=

(W"⌧ (u(0), u(⌧), . . . , u(T )) if u 2 bV⌧ \K(u0, u1),

1 elsewhere.

Then, W "⌧ �-converges to W " with respect to both the strong and the weak topology of V.

Before going on, let us remark that

e⌧ , e⌧ , e⌧ (·+ ⌧), e⌧ (·+ 2⌧) !⇣t 7! e�t/"

⌘strongly in L1(0, T ), (5.24)

the convergence of e⌧ being actually strong in W 1,1(0, T ).

Proof. The proof is classically divided into (i) proving the �-liminf inequality and (ii) checkingthe existence of a recovery sequence (see [7, 5]).

Ad (i). Assume to be given a sequence u⌧ 2 bV⌧ such that u⌧ * u with respect to the weaktopology on V and lim inf⌧!0 W "⌧ (u⌧ ) < 1. Let us denote by u⌧ 2 C1([0, T ];V \ X) thepiecewise-quadratic-in-time interpolant of ui := u⌧ (i⌧), i = 0, . . . , n, defined by the relations

u⌧ (t) := u⌧ (t) for t 2 [0, ⌧ ]

and @tu⌧ = ↵⌧ (t)@tu⌧ (t) + (1�↵⌧ (t))@tu⌧ (t�⌧) for t 2 (⌧, T ]

where we have used the notation ↵⌧ (t) := ↵i(t) for t 2 ((i�1)⌧, i⌧ ], i = 1, . . . , n. Wepreliminarily observe that

@tu⌧ (t) = @tu⌧ (t� ⌧) + ⌧↵⌧ (t)@ttu⌧ (t) 8t 2 (⌧, T ]. (5.25)

Since lim inf⌧!0 W "⌧ (u⌧ ) < 1 we can extract a not relabeled subsequence such that we haveu⌧ (0) = u0 and

lim sup⌧!0

✓"2⇢

2

Z T

⌧

e⌧ |@ttu⌧ |2dt+"⌫

2

Z T�⌧

⌧

e⌧ (·+ ⌧)|@tu⌧ |2dt+Z T�2⌧

⌧

e⌧ (·+ 2⌧)E(u⌧ )dt

◆< 1.

Then, owing to convergences 5.24 we have that, for small ⌧ ,

⇢

Z T

⌧

|@ttu⌧ |2dt+ ⌫

Z T�⌧

⌧

|@tu⌧ |2dt+Z T�2⌧

⌧

E(u⌧ )dt C.


Hence, by using the growth conditions (2.1) and by possibly further extracting a not relabeledsubsequence (and considering standard projections for t > T � 2⌧) we have that

u⌧ * u weakly in Lp(0, T ;X), (5.26)

u⌧ * u weakly in L2(0, T ;V ), (5.27)

u⌧ * u weakly in H1(0, T ;H), (5.28)

u⌧ * v weakly in H2(0, T ;H), (5.29)

⇢@tu⌧ * ⇢vt strongly in C0(0, T ;H). (5.30)

Indeed, we have that v = u. In order to check this fix w 2 L2(0, T ;H) and compute that

⇢

Z T

0

(@tu⌧�ut, w)dt = ⇢

Z ⌧

0

(@tu⌧�ut, w)dt+ ⇢

Z T

⌧

(@tu⌧ (·� ⌧) + ⌧↵⌧@ttu⌧�ut, w)dt

= ⇢

Z T

0

(@tu⌧�ut, w)dt+ ⇢

Z T

⌧

(@tu⌧ (·� ⌧)� @tu⌧ + ⌧↵⌧@ttu⌧ , w)dt

= ⇢

Z T

0

(@tu⌧�ut, w)dt� ⇢⌧

Z T

⌧

(1�↵⌧ )(@ttu⌧ , w)dt ! 0

where we have used (5.25), (5.28), |↵⌧ | 1, and the boundedness ofp⇢@ttu in L2(0, T ;H).

Namely, ⇢@tu⌧ ! ⇢ut weakly in L2(0, T ;H) and v = u. In particular, owing to convergence(5.30) we have proved that ⇢u1 = ⇢@tu⌧ (0) = ⇢ut(0) and u 2 K(u0, u1).

Eventually, we exploit the convergences in (5.24) and (5.26)-(5.29) in order to get thatZ T

0

e�t/" "2⇢

2|utt|2dt lim inf

⌧!0

Z T

⌧

e⌧"2⇢

2|@ttu⌧ |2dt = lim inf

⌧!0

nX

j=2

⌧ej"2⇢

2|�2uj|2,

Z T

0

e�t/" "⌫

2|ut|2dt lim inf

⌧!0

Z T�⌧

⌧

e⌧ (·+ ⌧)"⌫

2|@tu⌧ |2dt = lim inf

⌧!0

n�1X

j=2

⌧ej+1"⌫

2|�uj|2,

Z T

0

e�t/"E(u)dt lim inf⌧!0

Z T�2⌧

⌧

e⌧ (·+ 2⌧)E(u⌧ )dt = lim inf⌧!0

n�2X

j=2

⌧ei+2E(ui).

In particular, these three inequalities ensure that

W "(u) lim inf⌧!0

nX

j=2

⌧ej"2⇢

2|�2uj|2 +

n�1X

j=2

⌧ej+1"⌫

2|�uj|2 +

n�2X

j=2

⌧ej+2E(uj)

!

= lim inf⌧!0

W"⌧ (u0, . . . , un) = lim inf⌧!0

W "⌧ (u⌧ ),

which is the desired �-lim inf inequality.


Ad (ii). Let us define the backward floating mean operator M⌧ on L1(0, T ;H) by setting

M⌧ (u)(t) :=

8<

:

u0 for t 2 [0, ⌧)1

⌧

Z t

t�⌧

u(s) ds for t 2 [⌧, T ].

Then, let u 2 K(u0, u1) be fixed and define (u0, . . . , un) by

u0 = u0, ⇢u1 = ⇢u0 + ⌧⇢u1, ui = M⌧ (u)(i⌧) for i = 2, . . . , n.

We denote by u⌧ and u⌧ the piecewise a�ne and constant interpolants, respectively, associatedwith (u0, . . . , un).

We aim to show that u⌧ is a recovery sequence for u. Indeed, we clearly have that u⌧

converges strongly to u in L2(0, T ;V ) \ Lp(0, T ;X), while u⌧ converges at least weakly to uin L2(0, T ;V ) \ Lp(0, T ;X). Moreover, one can check that

Z T

0

|@tu⌧�ut|2dt =Z ⌧

0

|u1�ut|2dt+nX

i=2

Z i⌧

(i�1)⌧

��1

⌧ 2

Z i⌧

(i�1)⌧

�u(s)� u(s�⌧)

�ds� ut

��2

dt

=

Z ⌧

0

|u1�ut|2dt+nX

i=2

Z i⌧

(i�1)⌧

��Z i⌧

(i�1)⌧

M⌧ (ut)ds� ut

��2

dt (5.31)

Hence, as one has that M⌧ (ut) ! ut strongly in L2(0, T ;H), we conclude that u⌧ ! ustrongly in H1(0, T ;H). In particular, we have verified that u⌧ * u weakly in U

Next, we exploit the �-convexity of F and compute that

n�2X

i=2

⌧ei+2E(ui) =n�2X

i=2

Z i⌧

(i�1)⌧

ei+2

�E(u⌧ )�E(u)

�+ ei+2E(u)dt

Z T�2⌧

⌧

e⌧ (·+ 2⌧)⇣hAu⌧ , u⌧�ui+ hF (u⌧ ), u⌧�ui � �

2|u⌧�u|2 + E(u)

⌘dt

In particular, by taking the lim sup as ⌧ ! 0 and recalling that u⌧ ! u strongly inL2(0, T ;V ) \ Lp(0, T ;X) and the convergences (5.24), we have that

lim sup⌧!0

n�2X

i=2

⌧ej+2E(ui)

!Z T

0

e�t/"E(u)dt. (5.32)


Next, we deal with the second-order derivatives in time like we did in (5.31). We compute

⇢

Z T

⌧

|�2u⌧�utt|2dt = ⇢

nX

i=2

Z i⌧

(i�1)⌧

��ui�2ui�1 + ui�2

⌧ 2� utt

��2

dt

= ⇢

nX

i=2

Z i⌧

(i�1)⌧

��1

⌧ 3

Z i⌧

(i�1)⌧

�u�u(·� ⌧)

�ds� 1

⌧ 3

Z (i�1)⌧

(i�2)⌧

�u�u(·� ⌧)

�ds� utt

��

2

dt

= ⇢nX

i=2

Z i⌧

(i�1)⌧

��1

⌧ 3

Z i⌧

(i�1)⌧

✓Z s

s�⌧

�ut�ut(·� ⌧)

�dr

◆ds� utt

��2

dt

= ⇢nX

i=2

Z i⌧

(i�1)⌧

��Z i⌧

(i�1)⌧

M⌧

�M⌧ (utt)

�ds� utt

��2

dt ! 0 (5.33)

where the convergence to 0 is ensured by the fact that M⌧ (M⌧ (utt)) ! utt strongly inL2(0, T ;H). In particular, the convergence in (5.33) shows that (see the proof of (i) forthe definition of u⌧ )

@ttu⌧ ! utt strongly in L2(0, T ;H).

Finally, combining (5.31)–(5.33) we have proved that

W "(u) =

Z T

0

e�t/"

✓"2⇢

2|utt|2 +

"⌫

2|ut|2 + E(u)

◆dt

� lim sup⌧!0

✓Z T

⌧

e⌧"2⇢

2|@ttu⌧ |2dt+

Z T�⌧

⌧

e⌧ (·+ ⌧)"⌫

2|@tu⌧ |2dt+

Z T�2⌧

⌧

e⌧ (·+ 2⌧)E(u⌧ )dt

◆

= lim sup⌧!0

nX

i=2

⌧ei"2⇢

2|�2ui|2 +

n�1X

i=2

⌧ei+1"⌫

2|�ui|2 +

n�2X

i=2

⌧ej+2E(ui)

!

= lim sup⌧!0

W"⌧ (u0, . . . , un)

= lim sup⌧!0

W "⌧ (u⌧ ).

Namely, u⌧ is a recovery sequence for u. ⇤

Proof of Lemma 4.1. The minimizers u"⌧ of the discrete functional W "⌧ fulfill estimate

(5.23) and are hence weakly precompact in U . As W "⌧ �-converges to W " with respect tothe same topology by Lemma 5.4, we have, by the Fundamental Theorem of �-convergence(see [7, Ch. 7] and [5, Sect. 1.5]), that u"

⌧ * u" weakly in U , where u" is the unique minimizerof W ". Finally, estimate (5.23) passes to the limit and we have proved Lemma 4.1.


Acknowledgments

U. S. and M.L. are partly supported by FP7-IDEAS-ERC-StG Grant # 200947 BioSMA.U. S. acknowledges the partial support of CNR-AVCR Grant SmartMath, the CNR-JSPSGrant VarEvol, and the Alexander von Humboldt Foundation. Furthermore, M. L. thanksthe Istituto di Matematica Applicata e Tecnologie Informatiche Enrico Magenes in Pavia,where part of the work was conducted, for its kind hospitality.

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Weierstraß-Institut fur Angewandte Analysis und Stochastik, Mohrenstr. 39, D-10117Berlin, Germany.

Istituto di Matematica Applicata e Tecnologie Informatiche “Enrico Magenes” - CNR, v.Ferrata 1, I-27100 Pavia, Italy.

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