A modest review on stability investigations of numerical methods for systems of Itô-interpreted stochastic differential equations (SDEs) driven by m-dimensional Wiener processes W=(W1,W2,...,Wm) is presented in Rd. Since the problem of relevance of 1D test equations for multidimensional numerical methods has not completely been solved so far, we suggest to use the Krein-Perron-Frobenius theory of positive operators on positive cones of Rd×d, instead of classic stability functions with values in C1, which is only relevant for the very restricted case of "simultaneously diagonalizable" SDEs. Our main focus is put on the concept of asymptotic mean square and almost sure (a.s.) stability for systems with state-dependent noise (multiplicative case), and the concept of exact preservation of asymptotic probabilistic quantities for systems with state-independent noise (additive case). The asymptotic exactness of midpoint methods with any equidistant step size h is worked out in order to underline their superiority within the class of all drift-implicit, classic theta methods for multidimensional, bilinear systems of SDEs. Balanced implicit methods with appropriate weights can also provide a.s. exact, asymptotically stable numerical methods for pure diffusions. The review on numerical stability is based on "major breakthroughs" of research for systems of SDEs over the last 35 years, with emphasis on applicability to all dimensions d≥1.
Citation: Henri Schurz. A brief review on stability investigations of numerical methods for systems of stochastic differential equations[J]. Networks and Heterogeneous Media, 2024, 19(1): 355-383. doi: 10.3934/nhm.2024016
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A modest review on stability investigations of numerical methods for systems of Itô-interpreted stochastic differential equations (SDEs) driven by m-dimensional Wiener processes W=(W1,W2,...,Wm) is presented in Rd. Since the problem of relevance of 1D test equations for multidimensional numerical methods has not completely been solved so far, we suggest to use the Krein-Perron-Frobenius theory of positive operators on positive cones of Rd×d, instead of classic stability functions with values in C1, which is only relevant for the very restricted case of "simultaneously diagonalizable" SDEs. Our main focus is put on the concept of asymptotic mean square and almost sure (a.s.) stability for systems with state-dependent noise (multiplicative case), and the concept of exact preservation of asymptotic probabilistic quantities for systems with state-independent noise (additive case). The asymptotic exactness of midpoint methods with any equidistant step size h is worked out in order to underline their superiority within the class of all drift-implicit, classic theta methods for multidimensional, bilinear systems of SDEs. Balanced implicit methods with appropriate weights can also provide a.s. exact, asymptotically stable numerical methods for pure diffusions. The review on numerical stability is based on "major breakthroughs" of research for systems of SDEs over the last 35 years, with emphasis on applicability to all dimensions d≥1.
In this paper, for a bounded domain
Fε(u):={∫Ω{A(xε)∇u⋅∇u+|u|2}dx,if u∈H10(Ω),∞,if u∈L2(Ω)∖H10(Ω). | (1) |
The conductivity
ess-infy∈Yd(min{A(y)ξ⋅ξ:ξ∈Rd,|ξ|=1})≥0. | (2) |
This condition holds true when the conductivity energy density has missing derivatives. This occurs, for example, when the quadratic form associated to
Aξ⋅ξ:=A′ξ′⋅ξ′forξ=(ξ′,ξd)∈Rd−1×R, |
where
ess-infy∈Yd(min{A(y)ξ⋅ξ:ξ∈Rd,|ξ|=1})>0, | (3) |
combined with the boundedness implies a compactness result of the conductivity functional
u∈H10(Ω)↦∫ΩA(xε)∇u⋅∇udx |
for the
∫ΩA∗∇u⋅∇udx, |
where the matrix-valued function
A∗λ⋅λ:=min{∫YdA(y)(λ+∇v(y))⋅(λ+∇v(y))dy:v∈H1per(Yd)}. | (4) |
The
Fε(u)=∫Ωf(xε,Du)dx,foru∈W1,p(Ω;Rm), | (5) |
where
In this paper, we investigate the
Fε(u)≥∫Ω|u|2dx. |
This estimate guarantees that
V:={∫YdA1/2(y)Φ(y)dy:Φ∈L2per(Yd;Rd)withdiv(A1/2(y)Φ(y))=0inD′(Rd)} |
agrees with the space
the
F0(u):={∫Ω{A∗∇u⋅∇u+|u|2}dx,if u∈H10(Ω),∞,if u∈L2(Ω)∖H10(Ω), | (6) |
where the homogenized matrix
A∗λ⋅λ:=inf{∫YdA(y)(λ+∇v(y))⋅(λ+∇v(y))dy:v∈H1per(Yd)}. | (7) |
We need to make assumption (H1) since for any sequence
In the
In the present scalar case, we enlighten assumptions (H1) and (H2) which are the key ingredients to obtain the general
The lack of assumption (H1) may induce a degenerate asymptotic behaviour of the functional
The paper is organized as follows. In Section 2, we prove a general
Notation.
● For
●
●
● Throughout, the variable
● We write
uε⇀⇀u0 |
with
●
F1(f)(λ):=∫Re−2πiλxf(x)dx. |
Definition 1.1. Let
i)
ii)
Such a sequence
Recall that the weak topology of
In this section, we will prove the main result of this paper. As previously announced, up to a subsequence, the sequence of functionals
Theorem 2.1. Let
V:={∫YdA1/2(y)Φ(y)dy:Φ∈L2per(Yd;Rd)withdiv(A1/2(y)Φ(y))=0inD′(Rd)} | (8) |
agrees with the space
Then,
FεΓ(L2)−w⇀F0, |
where
Proof. We split the proof into two steps which are an adaptation of [11,Theorem 3.3] using the sole assumptions (H1) and (H2) in the general setting of conductivity.
Step 1 -
Consider a sequence
lim infε→0Fε(uε)≥F0(u). | (9) |
If the lower limit is
Fε(uε)≤C. | (10) |
As
uε⇀⇀u0. | (11) |
Assumption (H1) ensures that
u0(x,y)=u(x)is independent ofy, | (12) |
where, according to the link between two-scale and weak
uε⇀uweakly inL2(Ω). |
Since all the components of the matrix
A(xε)∇uε⇀⇀σ0(x,y)withσ0∈L2(Ω×Yd;Rd), |
and also
A1/2(xε)∇uε⇀⇀Θ0(x,y)withΘ0∈L2(Ω×Yd;Rd). | (13) |
In particular
εA(xε)∇uε⇀⇀0. | (14) |
Consider
div(A1/2(y)Φ(y))=0inD′(Rd), | (15) |
or equivalently,
∫YdA1/2(y)Φ(y)⋅∇ψ(y)dy=0∀ψ∈H1per(Yd). |
Take also
∫ΩA1/2(xε)∇uε⋅Φ(xε)φ(x)dx=−∫ΩuεA1/2(xε)Φ(xε)⋅∇φ(x)dx. |
By using [1,Lemma 5.7],
∫Ω×YdΘ0(x,y)⋅Φ(y)φ(x)dxdy=−∫Ω×Ydu(x)A1/2(y)Φ(y)⋅∇φ(x)dxdy. | (16) |
We prove that the target function
N:=∫YdA1/2(y)Φ(y)dy, | (17) |
and varying
∫Ω×YdΘ0(x,y)⋅Φ(y)φ(x)dxdy=−∫Ωu(x)N⋅∇φ(x)dx |
Since the integral in the left-hand side is bounded by a constant times
∫Ωu(x)N⋅∇φ(x)dx=∫Ωg(x)φ(x)dx, |
which implies that
N⋅∇u∈L2(Ω). | (18) |
In view of assumption (H2),
u∈H1(Ω). | (19) |
This combined with equality 16 leads us to
∫Ω×YdΘ0(x,y)⋅Φ(y)φ(x)dxdy=∫Ω×YdA1/2(y)∇u(x)⋅Φ(y)φ(x)dxdy. | (20) |
By density, the last equality holds if the test functions
divy(A1/2(y)ψ(x,y))=0inD′(Rd), |
or equivalently,
∫Ω×Ydψ(x,y)⋅A1/2(y)∇yv(x,y)dxdy=0∀v∈L2(Ω;H1per(Yd)). |
The
K:={A1/2(y)∇yv(x,y):v∈L2(Ω;H1per(Yd))}. |
Thus, the equality 20 yields
Θ0(x,y)=A1/2(y)∇u(x)+S(x,y) |
for some
A1/2(y)∇yvn(x,y)→S(x,y)strongly inL2(Ω;L2per(Yd;Rd)). |
Due to the lower semi-continuity property of two-scale convergence (see [1,Proposition 1.6]), we get
lim infε→0‖A1/2(x/ε)∇uε‖2L2(Ω;Rd)≥‖Θ0‖2L2(Ω×Yd;Rd)=limn‖A1/2(y)(∇xu(x)+∇yvn)‖2L2(Ω×Yd;Rd). |
Then, by the weak
lim infε→0Fε(uε)≥limn∫Ω×YdA(y)(∇xu(x)+∇yvn(x,y))⋅(∇xu(x)+∇yvn(x,y))dxdy+∫Ω|u|2dx≥∫Ωinf{∫YdA(y)(∇xu(x)+∇yv(y))⋅(∇xu(x)+∇yv(y))dy:v∈H1per(Yd)}dx+∫Ω|u|2dx. |
Recalling the definition 7, we immediately conclude that
lim infε→0Fε(uε)≥∫Ω{A∗∇u⋅∇u+|u|2}dx, |
provided that
It remains to prove that the target function
∫Ω×YdΘ0(x,y)⋅Φ(y)φ(x)dxdy=∫ΩN⋅∇u(x)φ(x)dx−∫∂ΩN⋅ν(x)u(x)φ(x)dH, | (21) |
where
∫∂ΩN⋅ν(x)u(x)φ(x)dH=0, |
which leads to
N⋅ν(x0)u(x0)=0. | (22) |
In view of assumption (H2) and the arbitrariness of
u∈H10(Ω). |
This concludes the proof of the
Step 2 -
We use the same arguments of [12,Theorem 2.4] which can easily extend to the conductivity setting. We just give an idea of the proof, which is based on a perturbation argument. For
Aδ:=A+δId, |
where
Fδ(u):={∫Ω{A∗δ∇u⋅∇u+|u|2}dx,ifu∈H10(Ω),∞,ifu∈L2(Ω)∖H10(Ω), |
for the strong topology of
F0(u)≤Fδ(u)≤lim infεj→0∫Ω{Aδ(xεj)∇uεj⋅∇uεj+|uεj|2}dx≤lim infεj→0∫Ω{A(xεj)∇uεj⋅∇uεj+|uεj|2}dx+O(δ)=F0(u)+O(δ). |
It follows that
limδ→0A∗δ=A∗. | (23) |
Thanks to the Lebesgue dominated convergence theorem and in view of 23, we get that
Now, let us show that
limε→0Fε(¯uε)=F0(u), |
which proves the
The next proposition provides a characterization of Assumption
Proposition 1. Assumption
Ker(A∗)=V⊥. |
Proof. Consider
H1λ(Yd):={u∈H1loc(Rd):∇uisYd-periodic and∫Yd∇u(y)dy=λ}. |
Recall that
0=A∗λ⋅λ=inf{∫YdA(y)∇u(y)⋅∇u(y)dy:u∈H1λ(Yd)}. |
Then, there exists a sequence
limn→∞∫YdA(y)∇un(y)⋅∇un(y)dy=0, |
which implies that
A1/2∇un→0strongly inL2(Yd;Rd). | (24) |
Now, take
∫YdA1/2(y)∇un(y)⋅Φ(y)dy=∫Yd∇un(y)⋅A1/2(y)Φ(y)dy=λ⋅∫YdA1/2(y)Φ(y)dy+∫Yd∇vn(y)⋅A1/2(y)Φ(y)dy=λ⋅∫YdA1/2(y)Φ(y)dy, | (25) |
where the last equality is obtained by integrating by parts the second integral combined with the fact that
0=λ⋅(∫YdA1/2(y)Φ(y)dy), |
for any
Ker(A∗)⊆V⊥. |
Conversely, by 23 we already know that
limδ→0A∗δ=A∗, |
where
A∗δλ⋅λ=min{∫YdAδ(y)∇uδ(y)⋅∇uδ(y)dy:uδ∈H1λ(Yd)}. | (26) |
Let
A∗δλ⋅λ=∫YdAδ(y)∇¯uδ(y)⋅∇¯uδ(y)dy=∫Yd|A1/2δ(y)∇¯uδ(y)|2dy≤C, |
which implies that the sequence
Now, we show that
A(y)=R(y)D(y)RT(y)for a.e. y∈Yd, |
where
A1/2δ(y)=R(y)(D(y)+δId)1/2RT(y)for a.e. y∈Yd, |
which implies that
Now, passing to the limit as
div(A1/2δΦδ)=div(Aδ∇¯uδ)=0in D′(Rd), |
we have
div(A1/2Φ)=0in D′(Rd). |
This along with
A∗δλ=∫YdAδ(y)∇¯uδ(y)dy=∫YdA1/2δ(y)Φδ(y)dy. |
Hence, taking into account the strong convergence of
A∗λ=limδ→0A∗δλ=limδ→0∫YdA1/2δ(y)Φδ(y)dy=∫YdA1/2(y)Φ(y)dy, |
which implies that
A∗λ⋅λ=0, |
so that, since
V⊥⊆Ker(A∗), |
which concludes the proof.
In this section we provide a geometric setting for which assumptions (H1) and (H2) are fulfilled. We focus on a
Z1=(0,θ)×(0,1)d−1andZ2=(θ,1)×(0,1)d−1, |
where
Z#i:=Int(⋃k∈Zd(¯Zi+k))fori=1,2. |
Let
X1:=(0,θ)×Rd−1andX2:=(θ,1)×Rd−1, |
and we denote by
The anisotropic phases are described by two constant, symmetric and non-negative matrices
A(y1):=χ(y1)A1+(1−χ(y1))A2fory1∈R, | (27) |
where
We are interested in two-phase mixtures in
A1=ξ⊗ξandA2is positive definite, | (28) |
for some
Proposition 2. Let
From Theorem 2.1, we easily deduce that the energy
Proof. Firstly, let us prove assumption (H1). We adapt the proof of Step 1 of [11,Theorem 3.3] to two-dimensional laminates. In our context, the algebra involved is different due to the scalar setting.
Denote by
0=−limε→0ε∫ΩA(xε)∇uε⋅Φ(x,xε)dx=limε→0∫Ωuεdivy(A1Φ(x,y))(x,xε)dx=∫Ω×Z#1u10(x,y)divy(A1Φ(x,y))dxdy=−∫Ω×Z#1A1∇yu10(x,y)⋅Φ(x,y)dxdy, |
so that
A1∇yu10(x,y)≡0inΩ×Z#1. | (29) |
Similarly, taking
A2∇yu20(x,y)≡0inΩ×Z#2. | (30) |
Due to 29, in phase
∇yu10∈Ker(A1)=Span(ξ⊥), |
where
u10(x,y)=θ1(x,ξ⊥⋅y)a.e.(x,y)∈Ω×X1, | (31) |
for some function
u20(x,y)=θ2(x)a.e.(x,y)∈Ω×X2, | (32) |
for some function
(A1−A2)Φ⋅e1=0on∂Z#1. | (33) |
Note that condition 33 is necessary for
0=−limε→0ε∫ΩA(y)∇uε⋅Φφ(x,xε)dx=limε→0∫Ωuεdivy(A(y)Φφ(x,y))(x,xε)dx=∫Ω×Z1u10(x,y)divy(A1Φφ(x,y))dxdy+∫Ω×Z2θ2(x)divy(A2Φφ(x,y))dxdy. |
Take now
φ#(x,y):=∑k∈Z2φ(x,y+k) |
as new test function. Then, we obtain
0=∫Ω×Z1u10(x,y)divy(A1Φφ#(x,y))dxdy+∫Ω×Z2θ2(x)divy(A2Φφ#(x,y))dxdy=∑k∈Z2∫Ω×(Z1+k)u10(x,y)divy(A1Φφ(x,y))dxdy+∑k∈Z2∫Ω×(Z2+k)θ2(x)divy(A2Φφ(x,y))dxdy=∫Ω×Z#1u10(x,y)divy(A1Φφ(x,y))dxdy+∫Ω×Z#2θ2(x)divy(A2Φφ(x,y))dxdy. | (34) |
Recall that
Φ∈R2↦(A1e1⋅Φ,A2e1⋅Φ)∈R2 |
is one-to-one. Hence, for any
A1Φ⋅e1=A2Φ⋅e1=f. | (35) |
In view of the arbitrariness of
A1e1⋅Φ=A2e1⋅Φ=1on∂Z#1. | (36) |
Since
0=∫Ω×∂Zv0(x,y)φ(x,y)dxdHy, |
where we have set
v0(x,⋅)=0on∂Z. | (37) |
Recall that
θ1(x,ξ1y2)=θ2(x)on∂Z. |
Since
We prove assumption (H2). The proof is a variant of the Step 2 of [11,Theorem 3.4]. For arbitrary
A1/2(y)Φ(y):=χ(y1)αξ+(1−χ(y1))(αξ+βe2)for a.e.y∈R2. | (38) |
Such a vector field
N:=∫Y2A1/2(y)Φ(y)dy=αξ+(1−θ)βe2. |
Hence, due to
From Proposition 1, it immediately follows that the homogenized matrix
We are going to deal with three-dimensional laminates where both phases are degenerate. We assume that the symmetric and non-negative matrices
Ker(Ai)=Span(ηi)fori=1,2. | (39) |
The following proposition gives the algebraic conditions so that assumptions required by Theorem 2.1 are satisfied.
Proposition 3. Let
Invoking again Theorem 2.1, the energy
Proof. We first show assumption (H1). The proof is an adaptation of the first step of [11,Theorem 3.3]. Same arguments as in the proof of Proposition 2 show that
Ai∇yui0(x,y)≡0inΩ×Z#ifori=1,2. | (40) |
In view of 39 and 40, in phase
ui0(x,y)=θi(x,ηi⋅y)a.e.(x,y)∈Ω×Xi, | (41) |
for some function
0=−limε→0ε∫ΩA(y)∇uε⋅Φφ(x,xε)dx=∫Ω×Z1u10(x,y)divy(A1Φφ(x,y))dxdy+∫Ω×Z2u20(x,y)divy(A2Φφ(x,y))dxdy. | (42) |
Take
φ#(x,y):=∑k∈Z3φ(x,y+k) |
as test function in 42, we get
∫Ω×Z#1u10(x,y)divy(A1Φφ(x,y))dxdy+∫Ω×Z#2u20(x,y)divy(A2Φφ(x,y))dxdy=0. | (43) |
Since the vectors
Φ∈R3↦(A1e1⋅Φ,A2e1⋅Φ)∈R2 |
is surjective. In particular, for any
A1Φ⋅e1=A2Φ⋅e1=f. | (44) |
In view of the arbitrariness of
∫Ω×∂Z[u10(x,y)−u20(x,y)]φ(x,y)dxdHy=0, |
which implies that
u10(x,⋅)=u20(x,⋅)on∂Z. | (45) |
Fix
θ1(x,b1y2+c1y3)=θ2(x,b2y2+c2y3)on∂Z, | (46) |
with
z1:=b1y2+c1y3,z2:=b2y2+c2y3, |
is a change of variables so that 46 becomes
θ1(x,z1)=θ2(x,z2)a.e.z1,z2∈R. |
This implies that
It remains to prove assumption (H2). To this end, let
E:={(ξ1,ξ2)∈R3×R3:(ξ1−ξ2)⋅e1=0,ξ1⋅η1=0,ξ2⋅η2=0}. | (47) |
For
A1/2(y)Φ(y):=χ(y1)ξ1+(1−χ(y1))ξ2a.e.y∈R3. | (48) |
The existence of such a vector field
Note that
If we identify the pair
with
Now, let
Let us show that
(49) |
In view of the definition of
This combined with the linear independence of
Hence,
Thanks to Proposition 1, the homogenized matrix
In this section we are going to construct a counter-example of two-dimensional laminates with two degenerate phases, where the lack of assumption (H1) provides an anomalous asymptotic behaviour of the functional
Let
for some positive constant
(50) |
with
(51) |
Thus, the energy
(52) |
We denote by
Throughout this section,
(53) |
where
Proposition 4. Let
where
(54) |
The
(55) |
where
(56) |
where
(57) |
Moreover, any two-scale limit
Remark 1. From 57, we can deduce that
for any
Proof. We divide the proof into three steps.
Step 1 -
Consider a sequence
(58) |
If the lower limit is
(59) |
It follows that the sequence
(60) |
In view of 51, we know that
(61) |
In particular,
(62) |
Take
Passing to the limit in both terms with the help of 60 and 62 leads to
which implies that
(63) |
Due to the link between two-scale and weak
(64) |
Now consider
(65) |
Since
In view of the convergences 60 and 61 together with 63, we can pass to the two-scale limit in the previous expression and we obtain
(66) |
Varying
which yields
(67) |
Then, an integration by parts with respect to
Since for any
which implies that
This combined with 67 yields
Finally, an integration by parts with respect to
Since the orthogonal of divergence-free functions is the gradients, from the previous equality we deduce that there exists
(68) |
Now, we show that
(69) |
To this end, set
Since
(70) |
hence, by periodicity, we also have
(71) |
for some positive constant
(72) |
From the inequality
we get
(73) |
In view of 71, the first integral on the right-hand side of 73 can be estimated as
Hence, passing to the limit as
Thanks to 70, we take the limit as
so that in view of 72, passing to the limit as
This combined with 68 proves 69.
By 63, we already know that
This combined with 69 implies that
(74) |
Now, we extend the functions in
(75) |
We minimize the right-hand side with respect to
for any
(76) |
for a.e.
(77) |
Note that 77 proves in particular that the two-scale limit
In light of the definition 54 of
(78) |
By using Plancherel's identity with respect to the variable
which proves the
Step 2-
For the proof of the
Lemma 4.1. Let
(79) |
where
(80) |
with
Let
(81) |
to the problem 80. Taking the Fourier transform
(82) |
where
(83) |
Let
Recall that rapidly oscillating
Due to 81, we can apply [1,Lemma 5.5] so that
(84) |
where the function
This together with 84 implies that, for
which proves the
Now, we extend the previous result to any
The associated metric
(85) |
Recall that
(86) |
A direct computation of
which implies that
(87) |
where
(88) |
This combined with the Plancherel identity yields
(89) |
where
Now, take
(90) |
In particular, due to 85, we also have that
which proves the
Step 3 - Alternative expression of
The proof of the equality between the two expressions of the
Lemma 4.2. Let
(91) |
By applying Plancherel's identity with respect to
(92) |
Recall that the Fourier transform of
(93) |
On the other hand, by applying Plancherel's identity with respect to
In view of the expansion of
which concludes the proof.
Proof of Lemma 4.1. In view of 87, the equality 79 becomes
(94) |
Since
the right-hand side of 94 belongs to
Applying the Plancherel identity, we obtain that
(95) |
We show that
Thanks to Plancherel's identity, we infer from 79 that
In view of 88 and thanks to the Plancherel identity, we obtain
By the Lebesgue dominated convergence theorem and since
(96) |
To conclude the proof, it remains to show the regularity of
On the other hand, the solution
(97) |
where
This combined with 96 and 97 proves that
which concludes the proof.
We prove now the Lemma 4.2 that we used in Step
Proof of Lemma 4.2. By the convolution property of the Fourier transform on
(98) |
where
Since
which yields 91. This concludes the proof.
Remark 2. Thanks to the Beurling-Deny theory of Dirichlet forms [3], Mosco [15, Theorem 4.1.2] has proved that the
(99) |
where
we have
In the present context, we do not know if the
We are going to give an explicit expression of the homogenized matrix
Set
(100) |
with
Proposition 5. Let
(101) |
If
(102) |
Furthermore, if one of the following conditions is satisfied:
i) in two dimensions,
ii) in three dimensions,
then
Remark 3. The condition
Proof. Assume that
(103) |
where, for
with
(104) |
If
We prove that
(105) |
This combined with the definition 101 of
(106) |
In view of definition 100 of
Plugging this equality in 106, we deduce that
(107) |
which proves that
Now, assume
which implies that the lamination formula 104 becomes
This combined with the convergence 103 yields to the expression 102 for
To conclude the proof, it remains to prove the positive definiteness of
Case (i).
Assume
(108) |
Recall that the Cauchy-Schwarz inequality is an equality if and only if one of vectors is a scalar multiple of the other. This combined with 108 leads to
(109) |
From the definition 101 of
(110) |
Recall that
Case (ii).
Assume that
(111) |
(112) |
Let
In view of 111, the discriminant of
(113) |
Recall that
(114) |
Similarly, recalling that
(115) |
Since the vectors
In light of definition 101 of
which implies that
Note that when
Then, it is easy to check that
This problem was pointed out to me by Marc Briane during my stay at Institut National des Sciences Appliquées de Rennes. I thank him for the countless fruitful discussions. My thanks also extend to Valeria Chiadò Piat for useful remarks. The author is also a member of the INdAM-GNAMPA project "Analisi variazionale di modelli non-locali nelle scienze applicate".
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