The $Γ$-limit for singularly perturbed functionals of Perona-Malik type in arbitrary dimension

0.24 MB·English

by  Giovanni Bellettini

The Γ-limit for singularly perturbed functionals of
Perona-Malik type in arbitrary dimension
Giovanni Bellettini ∗ Antonin Chambolle † Michael Goldman ‡
3
1
0
2
Abstract
n
a
In this paper we generalize to arbitrary dimensions a one-dimensional equicoercive-
J
ness and Γ-convergenceresult for a second derivative perturbation of Perona-Maliktype
2
functionals. Our proof relies on a new density result in the space of special functions
2
of bounded variation with vanishing diffuse gradient part. This provides a direction of

] investigationtoderiveapproximationforfunctionalswithdiscontinuitiespenalizedwitha
P
“cohesive”energy,thatis,whosecostdepends ontheactualopeningofthe discontinuity.
A
.
h
1 Introduction
t
a
m
We investigate in this paper a singular pertubation problem whose limit is defined on piece-
[
wise constant functions, and corresponds essentially to a subadditive penalization of the

1 discontinuity.
v
More precisely, we consider for Ω a bounded open subset of Rn with Lipschitz boundary, and
0
7 ν ∈ (0,1], the functional F :L2(Ω) → [0,+∞] defined by
ν
1
5 1 1
. ν3|∇2u|2+ φ(|∇u|) dx if u ∈ H2(Ω),
1 2 νφ(1/ν)
0 Fν(u) :=  ZΩ(cid:20) (cid:21) (1.1)
3 
1  +∞ elsewhere in L2(Ω).
:
v 

i Here ∇2u is thehessian of u and, for a symmetric real (n × n)-matrix M, we set |M| :=
X
max{hMξ,ξi : ξ ∈ Rn,|ξ| ≤ 1}. Moreover the function φ : R → [0,+∞) is continuous, even,
r
a nondecreasing, typically nonconvex and nonconcave with φ−1(0) = {0}, and has sublinear
growth at infinity, in the sense that
φ(λp)
∃ a ∈ [0,1) such that lim = λa, λ > 0. (1.2)
p→+∞ φ(p)
∗DipartimentodiMatematica,Universit`adiRomaTorVergata,viadellaRicercaScientifica1,00133Roma,
Italy,and LNF-INFN,via E. Fermi 40, 00044 Frascati, Italy. E-mail: [email protected]
†CMAP, Ecole Polytechnique, CNRS,Palaiseau, France.
E–mail: [email protected]
‡Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany. E-mail: [email protected],
partially funded by a Von Humboldt PostDoc fellowship
1
In this paper we prove that the sequence (F ) is equicoercive and Γ-converges [25, 16] as
ν
ν → 0+ to the functional F :L2(Ω)→ [0,+∞] defined as follows:
σa |u+−u−|42−+aa dHn−1 if u ∈X(Ω)∩L2(Ω),
F(u) :=  ZJu (1.3)


 +∞ elsewhere in L2(Ω),

where X(Ω) consists ofspecial functions of bounded variation in Ω having gradient with no
absolutely continuous part (see Section 2) and σ > 0 is a constant depending only on a.
a
Our results generalize an equicoerciveness and Γ-convergence result obtained in [9] in the
one-dimensional case.
The interest in these results are twofold. A first important application of such variational
problems is in the numerical analysis of fracture mechanics, and in particular the variational
approach to fracture growth popularized in the 90’s by G. Francfort and J. J. Marigo, built
upon the Mumford-Shah functional of image processing [31, 40]. The original model allows
to approach the so-called “Griffith” model, wherethe cost for openinga crack is proportional
to its length or area surface. Its success for predicting realistic fractures is impressive [14],
however its physical relevance is still a matter of discussion. More physical models (known as
“cohesive” and initially introduced by Barenblatt), consider that for a fracture with a small
opening, the cost should rather be proportional (through some non-linear correspondence)
to the size of the discontinuity. Mathematically, the study of such models is more tricky.
Also, finding reasonable approximations of such energies is a difficult problem (still only
partially solved, if one really wants to consider linearized or non-linear elasticity energies).
Some approaches need to consider a small non-cohesive term [7, 26]. A more physical term
(with slope one when the opening goes to zero) is obtained in [3], which can approach quite
general cohesive energies in the scalar setting (and is, to our knowledge, the best and most
useful result in this direction so far). It is built as a variant of Ambrosio and Tortorelli’s
approximation of the Mumford-Shah functional [8]. The variant in [36], built upon finite-
differences approximations, lies in between the previous results since it allows to approach a
partially cohesive term with an infinite slope for infinitesimal openings.
This latter result is close to the model that we present here, which is however built upon
different ideas. Instead of introducing a phase-field [3] or considering finite differences, we
penalize thevariations of the gradient by a higher order term. Thisis similar to the two-wells
problem which has been studied in the celebrated paper [21] of Conti, Fonseca, Leoni. Our
energy could be considered as a special case, where one well is at 0 and the other at infinity,
just as [13, 41] are to the standard Modica-Mortola energy1.
The second motivation is the relation of our result with the long-time behaviour of solutions
to the Perona-Malik equation [42], obtained as the formal L2-gradient flow of the functional
1
PM(u) := log(1+|∇u|2) dx,
2
ZΩ
and corresponding to the choice
φ(x) = log(1+|x|2), x ∈ Rn, (1.4)
1 Seealso [1, 30, 22, 15] for related problems.
2
and a = 0 in (1.2). The Perona-Malik equation therefore reads as
∇u
u = div , (1.5)
t 1+|∇u|2
(cid:18) (cid:19)
anditisill-posed2,duetothenonconvexityofφ. Inordertoovercomethebackwardparabolic
character of (1.4), various regularizations have been suggested in the literature (see [9] and
references therein); in particular [27] one can consider, for ε ∈ (0,1], the functionals
1
PM (u) := ε2|∇2u|2+log 1+|∇u|2 dx,
ε
2
ZΩ
(cid:0) (cid:1)
and take the corresponding gradient flow equations, which amounts to add to the equation
(1.5) a fourth order term multiplied by ε2; see also [44]. On the basis of the numerical
experiments [9] performed when n = 1, one observes various distinct time scales for the
regularized equation; in particular, in a slow time scale, the regularized solutions seem to
converge to a piecewise constant function with the plateaus suitably evolving in the vertical
direction. The functionals F are related to this slow time scale3 setting ε2 := ν4log(1+ 1 )
ν ν2
and taking PMε . Therefore, one expects that the asymptotic limit as ν → 0+ of the
νlog(1+ 1 )
ν2
gradient flow of F should shade some light on the behaviour of solutions to (1.5) for large
ν
times. This problem has been addressed in [9] when n = 1. The n-dimensional case seems
much more difficult, and requires a preliminar study of the asymptotic limit of F and of its
ν
variational properties, and this is the content of the present paper. The most original part
of this paper concerns the proof of the Γ-limsup inequality (Theorem 5.5), which relies on a
newdensity resultof simpleplateaus functionsin thespace of specialfunctions with bounded
variation with vanishingdiffuse gradient part, see Lemma 4.1. Differently with respectto the
density theorem in [24], our result is valid without assumptions on the measure of the jump
set of the limit functions. This allows us to get a full Γ-convergence result, differently from
other related works [4, 39], see Remark 5.6.
The plan of the paper is the following. In Section 2 we introduce the notation and recall
some definitions about SBV functions and the slicing method. In Section 3 we remind the
results of [9] regarding the one-dimensional problem. In Section 4 we state and prove our
density result and eventually in Section 5 we prove the equicoerciveness and Γ-convergence
theorems.
2 Notation
In what follows n ≥ 1, Ω ⊂ Rn is a bounded open Lipschitz set, and A(Ω) is the class
of all open subsets of Ω. We denote by (e ,...,e ) a fixed orthonormal basis of Rn, by
1 n
| · | (respectively h·,·i) the euclidean norm (respectively the euclidean scalar product) in
Rn. We denote by Hn−1 and by dx, the (n − 1)-dimensional Hausdorff measure and the
Lebesgue measure in Rn. The open ball of radius ρ > 0 centered at x ∈ Ω is denoted by
B (x). Throughout the paper, with a small abuse of language, we call sequence a family
ρ
(u ) of functions labelled by a continuous parameter ν ∈ (0,1]. A subsequence of (u ) is any
ν ν
sequence (u ) such that ν → 0 as k → +∞.
νk k
2 Seefor instance [38, 37, 33, 35, 11, 10, 12, 28].
3We refer also to [20] for the studyof theslow time behaviourin a related semi-discrete approximation.
3
2.1 BV(Ω) and SBV(Ω) functions
BV(Ω) is the space of functions u∈ L1(Ω) having as distributional derivative Du a measure
with finite total variation. For u∈ BV(Ω), we denote by S the complement of the Lebesgue
u
set of u. That is, x ∈/ S if and only if lim |u(y)−z| dy = 0 for some z ∈ R. We
u ρ→0+
ZBρ(x)
say that x is an approximate jump point of u if there exist ξ ∈ Sn−1 and distinct a,b ∈ R
such that
1 1
lim |u(y)−a| dy = 0 and lim |u(y)−b| dy = 0,
ρ→0|Bρ+(x,ξ)| ZBρ+(x,ξ) ρ→0|Bρ−(x,ξ)| ZBρ−(x,ξ)
where B±(x,ξ) := {y ∈ B (x) : ±hy − x,ξi > 0}. Up to a permutation of a and b and a
ρ ρ
change of sign of ξ, this characterize thetriplet (a,b,ξ) whichis thendenoted by (u+,u−,ν ).
u
The set of approximated jump points is denoted by J . The following theorem holds [6].
u
Thxeorem 2.1. The set Su ixs countably Hn−1-rectifiable and Hn−1(Su\Ju) = 0. Moreover
Du J = (u+−u−)ν Hn−1 J .
u u u
We indicatexby Du = ∇u dx + Dsu the Radon-Nikodym decomposition of Du. Setting
Dcu:= Dsu (Ω\S ) we get the decomposition
u
x
Du= ∇u dx + (u+ −u−)ν Hn−1 J + Dcu,
u u
x
where denotes the restriction. When n = 1 we use the symbol u′ in place of ∇u, and
u(x±) to indicate the right and left limits at x. We let
SBV(Ω):= {u ∈ BV(Ω): Dcu = 0},
GSBV(Ω) := {u ∈L1(Ω): max(−T,min(T,u)) ∈ SBV(Ω) ∀T ∈R},
and
X(Ω) := {u∈ SBV(Ω): ∇u= 0}.
2.2 Slicing
Inthissectionwerecalltheslicingmethodforfunctionswithboundedvariation. Letξ ∈ Sn−1
and let
Π := {y ∈ Rn :hy,ξi = 0}.
ξ
If y ∈ Π and E ⊂ Rn, we define the one-dimensional slice
ξ
E := {t ∈ R :y+tξ ∈ E}.
ξy
For u: Ω → R, we define u : Ω → R as
ξy ξy
u (t) := u(y+tξ), t ∈ Ω . (2.1)
ξy ξy
Functions in GSBV(Ω) can be characterized by one-dimensional slices (see [15, Thm. 4.1]).
4
Theorem 2.2. Let u∈ GSBV(Ω). Then for all ξ ∈ Sn−1 we have
u ∈GSBV(Ω ) for Hn−1−a.e. y ∈ Π .
ξy ξy ξ
Moreover for such y, we have
u′ (t) = h∇u(y+tξ),ξi for a.e. t ∈ Ω , (2.2)
ξy ξy
J = {t ∈ R : y+tξ ∈J }, (2.3)
uξy u
and
u (t±)= u±(y+tξ) or u (t±)= u∓(y+tξ), (2.4)
ξy ξy
according to whether hν ,ξi > 0 or hν ,ξi < 0. Finally, for every Borel function g : Ω → R,
u u
g (t) dHn−1(y) = g |hν ,ξi| dHn−1. (2.5)
ξy u
ZΠξt∈XJuξy ZJu
Conversely if u ∈ L1(Ω) and if for all ξ ∈ {e ,...,e } and almost every y ∈ Π we have
1 n ξ
u ∈ SBV(Ω ) and
ξy ξy
|Du |(Ω ) dHn−1(y) < +∞,
ξy ξy
ZΠξ
then u∈ SBV(Ω).
3 The one-dimensional case
In this section we briefly record themain results of [9], obtained in dimensionn = 1, that will
be necessary in order to analyze the problem in arbitrary dimension. For I ⊂ R a bounded
open interval we consider the functional
1 1
ν3(u′′)2+ φ(u′) dx if u∈ H2(I),
2 νφ(1/ν)
F (u,I) :=  ZI(cid:20) (cid:21)
ν




+∞ if u∈ L1(I)\H2(I)


and 
σa |u(x+)−u(x−)|24+−aa if u∈ X(I),
F(u,I) :=
 xX∈Ju
+∞ if u∈ L1(I)\X(I),
where X(I) is defined, accordingly to the n-dimensional case, as X(I) := {u ∈ SBV(I) :

u′ = 0}.
Then the following results hold [9, Lemma 3.2 and Section 4].
Lemma 3.1. There exist ν > 0, a decreasing function ω :(0,ν¯) → (0,+∞) with lim ω(s) =
ν→0+
0, and a constant C > 0 such that for any u∈ H2(I) and ν ∈ (0,ν),
|u′| dx ≤ |I|ω(ν)+CF (u,I). (3.1)
ν
ZI
5
Theorem 3.2 (Equicoerciveness). Let (u ) ⊂ H2(I) be a sequence satisfying
ν
sup F (u ,I) < +∞
ν ν
ν∈(0,1]
and such that u dx = 0 for any ν ∈ (0,1]. Then there exist a function u ∈ X(I) and a
ν
ZI
subsequence of (u ) weakly* converging to u in BV(I).
ν
Theorem 3.3 (Γ-convergence). We have
Γ(L1(I))− lim F (·,I) = F(·,I),
ν
ν→0+
and the same result holds true for the Γ(L2(I))-limit.
We also remind the construction of the recovery sequence in the proof of the Γ − limsup
inequality in Theorem 3.3. For any η > 0 and s ≥0, let
Y (s) := ψ ∈ H2((0,η)) :ψ(0) = 0,ψ(η) = s,ψ′(0) = ψ′(η) = 0 .
η
(cid:8) (cid:9)
Define
1
e (s) := inf (ψ′′)2+|ψ′|a dx :η > 0,ψ ∈ Y (s) .
a η
2
(Z(0,η)h i )
Then it turns out that
2+a
ea(s)= σas4−a,
for a suitable constant σ > 0.
a
Fix now b >0 and s > 0. Let (η,ψ) ∈ (0,+∞)×Y (s) be such that
η
η
(ψ′′)2+|ψ′|a dx ≤ σas 42−+aa +b.
Z0 h i
For a function of X(I) of the form u = s 1 , a recovery sequence (u ) ⊂ H2(I) is given
[0,+∞) ν
by
0 x < 0,
u (x) := ψ(x) x ∈[0,ην],
ν  ν
s x > ην,

in the sense that for every δ > 0 we have that (uν) tends to u in L1((−δ,δ)) and
2+a
limsupFν(uν,(−δ,δ)) = σas 4−a +b = F(u,(−δ,δ))+b,
ν→0+
so that in particular lim limsupF (u ,(−δ,δ)) = F(u,(−δ,δ)). A similar construction can
ν ν
b→0+ ν→0+
be made for s < 0.
6
4 The n-dimensional case: a density result
Let θ :R → R+ be an even lower semicontinuous function, subadditive and nondecreasing on
θ(t)
(0,+∞), and continuous at t = 0, such that θ(0) = 0, lim = +∞. Let us notice that
t→0 t
2+a
the functions s → |s|4−a satisfy these properties.
Define
P(Ω) := {u∈ GSBV(Ω) :∇u =0 in Ω}
and for any A∈ A(Ω) set
θ u+−u− dHn−1 if u∈ P(A),
E(u,A) :=  ZJu∩A (4.1)
(cid:0) (cid:1)


 +∞ elsewhere in L1(A).


We recall that E(·,A) isL1(A)-lower semicontinuous [6, Ex. 5.23].
In this section we want to prove the following result, which is the main technical tool of this
paper, and will be used in the proof of Theorem 5.5.
Lemma 4.1 (Density). Let u ∈ P(Ω) be such that E(u,Ω) < +∞. Then there exists a
sequence (u )⊂ P(Ω) with the following properties:
h
- lim u = u in L1(Ω);
h
h→0+
- lim E(u ,Ω) = E(u,Ω);
h
h→0+
- J is contained in a finite union of facets of polytopes4 for any h ∈ N. In particular
uh
for any h∈ N,
Hn−1(Ω∩J )= Hn−1(Ω∩J ) and Hn−1(J ) < +∞.
uh uh uh
Proof. Let Ω′ ⊃⊃ Ω be an open set and let us denote still by u ∈ P(Ω′) an extension of u
such that Hn−1(J ∩∂Ω)= 0 (see [32]). We also fix a representative of u defined everywhere.
u
We will need an auxiliary functional E defined as follows: for any open set A ⊆ Ω′
0
kν k θ(v+−v−) dHn−1 if v ∈P(A),
v 1
E (v,A) :=  ZJv∩A (4.2)
0


 +∞ elsewhere in L1(A),

where, for ξ = (ξ ,...,ξ )∈ Rn, we let
1 n
kξk := |ξ |+···+|ξ |.
1 1 n
Again by [6, Ex. 5.23], the functional E (·,A) is L1(A)-lower semicontinuous.
0
4A polytope is a set whose boundary consists of a finitenumberof pieces of hyperplanes.
7
Observe that ν ∈ Sn−1 implies kνk ≥ 1, with equality only if ν coincides with one of the
1
vectors of the orthonormal basis (e ,...,e ) of Rn. In particular
1 n
E(u,Ω) ≤ E (u,Ω).
0
From our assumptions on θ, we have that E(u,Ω) decreases under truncation. By the lower
semicontinuity of E(u,Ω), this implies that setting u := max(−T,min(T,u)) for any T > 0,
T
then E(u ,Ω) converges to E(u,Ω) as T ↑ +∞. Hence, with no loss of generality, we can
T
assume that u∈ L∞(Ω).
We divide the proof into five steps. In the first step we construct a suitable discrete approx-
y
imation u of u, defined on points of a lattice. For any δ > 0 set
ε
Ωδ := {x ∈ Rn :dist(x,Ω) < δ}.
Step 1. Let ε > 0 be such that Ω := Ω2n1/2ε ⊂⊂Ω′ and, for any y ∈[0,1)n, set
ε
uy(x) := u(x+εy), x ∈ εZn∩Ω′.
ε
y y
Define the discrete energy D of u as
ε ε
n
Dy := εn−1 θ uy(x+εe )−uy(x) . (4.3)
ε ε i ε
Xi=1 x,x+Xεei∈Ωε/2 (cid:0) (cid:1)
where Ω := Ωn1/2ε. We claim that
ε/2
Dy dy ≤ E (u,Ω ). (4.4)
ε 0 ε
Z(0,1)n
To show the claim, we will follow some arguments in [34], [17] [18] and [19]. Let us first
establish an inequality in one dimension: given a bounded open interval I ⊂ R, ε > 0
sufficiently small and v ∈ P(I), we have
ε−1 θ(v(t+ε)−v(t)) dt ≤ θ(v+(t)−v−(t)). (4.5)
ZI∩(I−ε) tX∈Jv
Indeed, for almost any t ∈ I ∩(I −ε), using the subadditivity of θ it follows
θ(v(t+ε)−v(t)) ≤ θ(v+(s)−v−(s)).
s∈JvX∩(t,t+ε)
Therefore, using Fubini’s Theorem,
θ(v(t+ε)−v(t)) dt ≤ 1 (s)θ v+(s)−v−(s) dt
(t,t+ε)
ZI∩(I−ε) ZI∩(I−ε)sX∈Jv (cid:0) (cid:1)
= θ v+(s)−v−(s) 1 (s) dt,
(t,t+ε)
sX∈Jv (cid:0) (cid:1)ZI∩(I−ε)
8
which implies (4.5).
We can now prove claim (4.4). Since
n
Dy dy = εn−1 θ u(x+εy+εe ))−u(x+εy) dy,
ε i
Z(0,1)n Xi=1 x,x+Xεei∈Ωε/2 Z(0,1)n (cid:16) (cid:17)
making the variable change x′ := εy+x (for fixed x) gives
n
Dy dy ≤ε−1 θ u(x′+εe )−u(x′) dx′
ε i
Z(0,1)n i=1ZΩε/2
X (cid:0) (cid:1)
n
=ε−1 θ u(ζ +(t+ε)e )−u(ζ +te ) dt dHn−1(ζ).
i i
Xi=1ZΠei Z(Ωε/2)eiζ (cid:16) (cid:17)
If, for any i = 1,...,n and Hn−1-almost every ζ ∈ Π we define as in Section 2.2, the one-
ei
dimensional slice u as u (t) := u(ζ+te ) for almost each t ∈R such that ζ+te ∈ Ω′, we
eiζ eiζ i i
get,
n
Dy dy ≤ ε−1 θ u (t+ε)−u (t) dt dHn−1(ζ).
ε eiζ eiζ
Z(0,1)n Xi=1ZΠei Z(Ωε/2)eiζ (cid:16) (cid:17)
Hence, using (4.5),
n
Dy dy ≤ θ u+ (s)−u− (s) dHn−1(ζ).
ε eiζ eiζ
Z(0,1)n Xi=1ZΠei s∈JueiXζ∩(Ωε)eiζ (cid:16) (cid:17)
Therefore, by (2.5),
n
Dy dy ≤ |hν ,e i| θ u+−u− dHn−1 = E (u,Ω ),
ε u i 0 ε
Z(0,1)n i=1ZJu∩Ωε
X (cid:0) (cid:1)
which is claim (4.4). The proof of Step 1 is concluded.
y y
In the next step we define a suitable piecewise constant interpolation u of u having the
ε ε
y y
property that E(u ,Ω) and E (u ,Ω) coincide.
ε 0 ε
Step 2. Let ε and y be as in Step 1. Define the function uy : Ω → R as
ε
uyε(z) := uyε(x) 1Cεy(x)(z), z ∈ Ω, (4.6)
x∈εZn
X
y
where C (x) denotes the open coordinate hypercube centered at x+εy of side ε, i.e.,
ε
Cy(x) := x+εy+(−ε/2,ε/2)n.
ε
Clearly uε ∈P(Ω) and its jump set is contained in the union of the facets of the hypercubes
y
y
C (x). Let us prove that
ε
(i) lim kuy −uk dy = 0;
ε L1(Ω)
ε→0Z(0,1)n
9
y y y
(ii) E(u ,Ω) = E (u ,Ω) ≤ D .
ε 0 ε ε
To show (i) and (ii), we assume without loss of generality that all functions are extended to
0 outside Ω′. Making the variables change
(x,y) ∈ εZn×[0,1)n 7→ y′ := εy+x
and then
y′ ∈ Rn → ξ = z−y′ (for fixed z),
we obtain
Z(0,1)nkuyε −ukL1(Ω) dy =Z(0,1)nZRn x∈εZnu(εy+x)1Cεy(x)(z)−u(z) dzdy
(cid:12) X (cid:12)
(cid:12) (cid:12)
=Z(0,1)nZRn|x∈εZn u(εy+x)−u(z) 1Cεy(x)(z)| dzdy
X (cid:0) (cid:1)
≤Z(0,1)nx∈εZn ZRn|u(εy+x)−u(z)| 1Cεy(x)(z) dzdy
X
=ε−n |u(y′)−u(z)| 1(−ε,ε)n(z−y′) dzdy′
Rn Rn 2 2
Z Z
=ε−n |u(z−ξ)−u(z)| 1(−ε,ε)n(ξ) dzdξ
Rn Rn 2 2
Z Z
=ε−n ku(·−ξ)−u(·)kL1(Rn) dξ
Z(−2ε,2ε)n
= ku(·−εξ)−u(·)kL1(Rn) dξ,
Z(−12,12)n
which tends to zero as ε → 0 by the dominated convergence theorem. This proves (i).
The equality in (ii) follows from the fact that
kνuyε(x)k1 = 1 for Hn−1−a.e. x ∈ Juyε,
y
since the jump of u is contained in facets of coordinate hypercubes.
ε
y
The inequality in (ii) follows from the definition of E (see (4.2)) and definitions (4.3) of D
0 ε
y
and (4.6) of u .
ε
From (i) we deduce that there exists a set A ⊂ (0,1)n with |A| = 1 and a subsequence
(ε ) ⊂ (0,1) converging to zero such that uy → u in L1(Ω) as k → ∞ for any y ∈ A.
k εk
From Step 1 and Fatou’s lemma we have
liminfDy dy ≤ E (u,Ω).
Z(0,1)n k→+∞ εk 0
Hence there exists a point y ∈ A and a further subsequence (still denoted by (ε )) such that
k
y
in addition there exists the lim D and
k→+∞ εk
lim Dy ≤E (u,Ω).
k→+∞ εk 0
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