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Stability results for the nonlocal Mullins-Sekerkaand for the Hell-Shaw flow
Nicola Fusco
A Mathematical Tribute to Ennio De GiorgiPisa, September 19-23, 2016
Copolymer = a polymer derived from two or more monomericstructures
Diblock Copolymer = structure given by two differentchemical blocks of polymers
.....A-A-A-A-A-A-B-B-B-B-A-A-A-A-A-A-B-B-B-B....
Microphase separationFormation of nanostructures
The relative lenghts of each block =⇒ different morphologies
Copolymer = a polymer derived from two or more monomericstructures
Diblock Copolymer = structure given by two differentchemical blocks of polymers
.....A-A-A-A-A-A-B-B-B-B-A-A-A-A-A-A-B-B-B-B....
Microphase separationFormation of nanostructures
The relative lenghts of each block =⇒ different morphologies
Lamellae Spheres
Lamellae Spheres
Denote u : Ω→ [−1, 1] the function describing the density
u(x) =
1 on phase A
−1 on phase B
Ohta-Kawasaki = Cahn-Hilliard (Modica-Mortola) + Non local term
Eε(u) = ε
∫
Ω|∇u|2 dx +
1ε
∫
Ω(1− u2)2 dx
︸ ︷︷ ︸attractive short range interaction
+ γ0
∫
Ω|∇(∆−1u)|2 dx
︸ ︷︷ ︸repulsive long range interaction
Letting ε→ 0, the functionals Eε Γ-converge (Ren-Wei, 2003) to
E(u) =12|∇u|(Ω) +
3γ0
16
∫
Ω|∇(∆−1u)|2 dx
where
u ∈ BV (−1, 1), u = uE := χE −χΩ\E , |∇u|(Ω) = 2P(E ; Ω)
Denote u : Ω→ [−1, 1] the function describing the density
u(x) =
1 on phase A
−1 on phase B
Ohta-Kawasaki = Cahn-Hilliard (Modica-Mortola) + Non local term
Eε(u) = ε
∫
Ω|∇u|2 dx +
1ε
∫
Ω(1− u2)2 dx
︸ ︷︷ ︸attractive short range interaction
+ γ0
∫
Ω|∇(∆−1u)|2 dx
︸ ︷︷ ︸repulsive long range interaction
Letting ε→ 0, the functionals Eε Γ-converge (Ren-Wei, 2003) to
E(u) =12|∇u|(Ω) +
3γ0
16
∫
Ω|∇(∆−1u)|2 dx
where
u ∈ BV (−1, 1), u = uE := χE −χΩ\E , |∇u|(Ω) = 2P(E ; Ω)
Denote u : Ω→ [−1, 1] the function describing the density
u(x) =
1 on phase A
−1 on phase B
Ohta-Kawasaki = Cahn-Hilliard (Modica-Mortola) + Non local term
Eε(u) = ε
∫
Ω|∇u|2 dx +
1ε
∫
Ω(1− u2)2 dx
︸ ︷︷ ︸attractive short range interaction
+ γ0
∫
Ω|∇(∆−1u)|2 dx
︸ ︷︷ ︸repulsive long range interaction
Letting ε→ 0, the functionals Eε Γ-converge (Ren-Wei, 2003) to
E(u) =12|∇u|(Ω) +
3γ0
16
∫
Ω|∇(∆−1u)|2 dx
where
u ∈ BV (−1, 1), u = uE := χE −χΩ\E , |∇u|(Ω) = 2P(E ; Ω)
Open problem: are minimizers of
J(E ) = P(E ; Ω) + γ
∫
Ω|∇(∆−1uE )|2 dx (almost) periodic?
Known in one dimension (Müller, 1993); if n ≥ 2 partial answer:Alberti-Choksi-Otto 2009, Spadaro 2009
Assume periodicity by taking Ω = Tn = flat torus
J(E ) = PTn(E ) + γ
∫
Tn
∫
Tn
G (x , y)uE (x)uE (y) dxdy
−∆yG (x , y) = δx − 1 in Tn
∫Tn G (x , y) dy = 0
u(x) =
1 if x ∈ E
−1 if x ∈ Tn \ E
Choksi-Sternberg, 2007: calculated J ′′ at critical points
Ren-Wei, 2002–2008: critical spheres, cylinders and lamellae withJ ′′ > 0 minimize the energy with respect to some special variations
Open problem: are minimizers of
J(E ) = P(E ; Ω) + γ
∫
Ω|∇(∆−1uE )|2 dx (almost) periodic?
Known in one dimension (Müller, 1993); if n ≥ 2 partial answer:Alberti-Choksi-Otto 2009, Spadaro 2009
Assume periodicity by taking Ω = Tn = flat torus
J(E ) = PTn(E ) + γ
∫
Tn
∫
Tn
G (x , y)uE (x)uE (y) dxdy
−∆yG (x , y) = δx − 1 in Tn
∫Tn G (x , y) dy = 0
u(x) =
1 if x ∈ E
−1 if x ∈ Tn \ E
Choksi-Sternberg, 2007: calculated J ′′ at critical points
Ren-Wei, 2002–2008: critical spheres, cylinders and lamellae withJ ′′ > 0 minimize the energy with respect to some special variations
Open problem: are minimizers of
J(E ) = P(E ; Ω) + γ
∫
Ω|∇(∆−1uE )|2 dx (almost) periodic?
Known in one dimension (Müller, 1993); if n ≥ 2 partial answer:Alberti-Choksi-Otto 2009, Spadaro 2009
Assume periodicity by taking Ω = Tn = flat torus
J(E ) = PTn(E ) + γ
∫
Tn
∫
Tn
G (x , y)uE (x)uE (y) dxdy
−∆yG (x , y) = δx − 1 in Tn
∫Tn G (x , y) dy = 0
u(x) =
1 if x ∈ E
−1 if x ∈ Tn \ E
Choksi-Sternberg, 2007: calculated J ′′ at critical points
Ren-Wei, 2002–2008: critical spheres, cylinders and lamellae withJ ′′ > 0 minimize the energy with respect to some special variations
Open problem: are minimizers of
J(E ) = P(E ; Ω) + γ
∫
Ω|∇(∆−1uE )|2 dx (almost) periodic?
Known in one dimension (Müller, 1993); if n ≥ 2 partial answer:Alberti-Choksi-Otto 2009, Spadaro 2009
Assume periodicity by taking Ω = Tn = flat torus
J(E ) = PTn(E ) + γ
∫
Tn
∫
Tn
G (x , y)uE (x)uE (y) dxdy
−∆yG (x , y) = δx − 1 in Tn
∫Tn G (x , y) dy = 0
u(x) =
1 if x ∈ E
−1 if x ∈ Tn \ E
Choksi-Sternberg, 2007: calculated J ′′ at critical points
Ren-Wei, 2002–2008: critical spheres, cylinders and lamellae withJ ′′ > 0 minimize the energy with respect to some special variations
J(E ) = PTn(E ) + γ
∫
Tn
∫
Tn
G (x , y)uE (x)uE (y) dxdy
can be also written as
J(E ) = PTn(E ) + γ
∫
Tn
|∇vE |2 dx (Pattern Formation)
where
−∆vE = uE −m in Tn
∫Tn vE = 0
m = |E | − |Tn \ E |
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
1122 XIAOFENG REN AND JUNCHENG WEI
Fig. 3. Spots on Gold Nugget Plecostomus and stripes on Distichodus Sexfasciatus.
Nishiura and Ohnish identified (1.1) as a formal singular limit of the Euler–Lagrangeequation of (1.3) [12]. Ren and Wei noted that (1.1) is the Γ-limit of (1.3) as ϵ tendsto 0 [15] and hence put the convergence of I to its singular limit J under a rigorousmathematical framework. The lamellar phase of diblock copolymers has been studiedin [5, 16, 2], the cylindrical phase in [18, 17], and the spherical phase in [19].
Another place where one finds (1.2) is the Gierer–Meinhardt theory for morpho-genesis in cell development. It is a minimal model that provides a theoretical bridgebetween observations on the one hand and the deduction of the underlying molecular-genetic mechanisms on the other hand. Mathematically it is an activator-inhibitortype reaction-diffusion system with two unknowns of space variable x ∈ D ⊂ R2 andtime variable t > 0. The first unknown, denoted by u, describes the short-range au-tocatalytic substance, i.e., the activator, and the second unknown, denoted by v, isits long-range antagonist, i.e., the inhibitor. They satisfy the equations
(1.4) ut = ϵ2∆u − u +up
(1 + κup)vq, ιvt = d∆v − v +
ur
vs.
Here u and v satisfy the Neumann condition on the boundary of D, i.e.,
(1.5) ∂νu(x, t) = ∂νv(x, t) = 0 ∀x ∈ ∂D, ∀t > 0,
where ∂ν is the outward normal derivative operator on the boundary of D.Activator-inhibitor systems were studied by Turing [21]. They may be used to
model animal coats and skin pigmentation; see Figure 3. In Appendix A we give aformal justification for the convergence of steady states of (1.4) to solutions of (1.2).In this paper we study the first stage of the saturation process depicted in the secondimage of Figure 1. We show that when a is sufficiently small and γ is in a particularrange, on a generic domain there exist two solutions to (1.2), each of which has theshape of a small oval set. The location and direction of each oval droplet solutionare determined via the regular part R of the Green’s function of the domain D.The precise definition of R is given in (2.1). Note that the regular part R(x, y) =R(x1, x2, y1, y2) is a function of two sets of variables x ∈ D and y ∈ D, each of whichhas two components. The diagonal of R, given by R(z) = R(z, z), is a function definedon D. If z → ∂D, R(z) → ∞. Hence R has at least one global minimum in D.
It is often convenient to use another parameter ρ in place of a. We set
(1.6) ρ =
!a|D|π
.
It is the average radius of a set E whose measure is fixed at a|D|. In other wordsif E were a round disc of the same measure a|D|, ρ would be the radius of E. The
Hypostomus Plecostomus Distichodus Sexfasciatus
J(E ) = PTn(E ) + γ
∫
Tn
∫
Tn
G (x , y)uE (x)uE (y) dxdy
can be also written as
J(E ) = PTn(E ) + γ
∫
Tn
|∇vE |2 dx (Pattern Formation)
where
−∆vE = uE −m in Tn
∫Tn vE = 0
m = |E | − |Tn \ E |
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
1122 XIAOFENG REN AND JUNCHENG WEI
Fig. 3. Spots on Gold Nugget Plecostomus and stripes on Distichodus Sexfasciatus.
Nishiura and Ohnish identified (1.1) as a formal singular limit of the Euler–Lagrangeequation of (1.3) [12]. Ren and Wei noted that (1.1) is the Γ-limit of (1.3) as ϵ tendsto 0 [15] and hence put the convergence of I to its singular limit J under a rigorousmathematical framework. The lamellar phase of diblock copolymers has been studiedin [5, 16, 2], the cylindrical phase in [18, 17], and the spherical phase in [19].
Another place where one finds (1.2) is the Gierer–Meinhardt theory for morpho-genesis in cell development. It is a minimal model that provides a theoretical bridgebetween observations on the one hand and the deduction of the underlying molecular-genetic mechanisms on the other hand. Mathematically it is an activator-inhibitortype reaction-diffusion system with two unknowns of space variable x ∈ D ⊂ R2 andtime variable t > 0. The first unknown, denoted by u, describes the short-range au-tocatalytic substance, i.e., the activator, and the second unknown, denoted by v, isits long-range antagonist, i.e., the inhibitor. They satisfy the equations
(1.4) ut = ϵ2∆u − u +up
(1 + κup)vq, ιvt = d∆v − v +
ur
vs.
Here u and v satisfy the Neumann condition on the boundary of D, i.e.,
(1.5) ∂νu(x, t) = ∂νv(x, t) = 0 ∀x ∈ ∂D, ∀t > 0,
where ∂ν is the outward normal derivative operator on the boundary of D.Activator-inhibitor systems were studied by Turing [21]. They may be used to
model animal coats and skin pigmentation; see Figure 3. In Appendix A we give aformal justification for the convergence of steady states of (1.4) to solutions of (1.2).In this paper we study the first stage of the saturation process depicted in the secondimage of Figure 1. We show that when a is sufficiently small and γ is in a particularrange, on a generic domain there exist two solutions to (1.2), each of which has theshape of a small oval set. The location and direction of each oval droplet solutionare determined via the regular part R of the Green’s function of the domain D.The precise definition of R is given in (2.1). Note that the regular part R(x, y) =R(x1, x2, y1, y2) is a function of two sets of variables x ∈ D and y ∈ D, each of whichhas two components. The diagonal of R, given by R(z) = R(z, z), is a function definedon D. If z → ∂D, R(z) → ∞. Hence R has at least one global minimum in D.
It is often convenient to use another parameter ρ in place of a. We set
(1.6) ρ =
!a|D|π
.
It is the average radius of a set E whose measure is fixed at a|D|. In other wordsif E were a round disc of the same measure a|D|, ρ would be the radius of E. The
Hypostomus Plecostomus Distichodus Sexfasciatus
Distance between (equivalence classes) of sets:
d(E ,F ) = minτ|E4(F + τ)|
E ⊂ Tn is a (strict) local minimizers if ∃δ > 0 s.t.
J(F ) > J(E )
whenever F ⊂ Tn with 0 < d(E ,F ) < δ, and |F | = |E |
Minimizers of J(E ) under a volume constraint satisfy
(∗) H∂E (x) + 4γvE (x) = λ on ∂E
where H∂E = sum of principal curvatures
while a solution E ∈ C 2 of equation (∗) is a critical configuration
TheoremIf E ⊂ Tn is a local minimizer of J, then ∂E \ Σ is C 3,α, for anyα<1, and Σ is a closed set such that dimH(Σ) ≤ n − 8.In fact, ∂E \ Σ is C∞ (Julin-Pisante, 2014).
E ⊂ Tn is a (strict) local minimizers if ∃δ > 0 s.t.
J(F ) > J(E )
whenever F ⊂ Tn with 0 < d(E ,F ) < δ, and |F | = |E |
Minimizers of J(E ) under a volume constraint satisfy
(∗) H∂E (x) + 4γvE (x) = λ on ∂E
where H∂E = sum of principal curvatures
while a solution E ∈ C 2 of equation (∗) is a critical configuration
TheoremIf E ⊂ Tn is a local minimizer of J, then ∂E \ Σ is C 3,α, for anyα<1, and Σ is a closed set such that dimH(Σ) ≤ n − 8.In fact, ∂E \ Σ is C∞ (Julin-Pisante, 2014).
E ⊂ Tn is a (strict) local minimizers if ∃δ > 0 s.t.
J(F ) > J(E )
whenever F ⊂ Tn with 0 < d(E ,F ) < δ, and |F | = |E |
Minimizers of J(E ) under a volume constraint satisfy
(∗) H∂E (x) + 4γvE (x) = λ on ∂E
where H∂E = sum of principal curvatures
while a solution E ∈ C 2 of equation (∗) is a critical configuration
TheoremIf E ⊂ Tn is a local minimizer of J, then ∂E \ Σ is C 3,α, for anyα<1, and Σ is a closed set such that dimH(Σ) ≤ n − 8.In fact, ∂E \ Σ is C∞ (Julin-Pisante, 2014).
E ⊂ Tn is a (strict) local minimizers if ∃δ > 0 s.t.
J(F ) > J(E )
whenever F ⊂ Tn with 0 < d(E ,F ) < δ, and |F | = |E |
Minimizers of J(E ) under a volume constraint satisfy
(∗) H∂E (x) + 4γvE (x) = λ on ∂E
where H∂E = sum of principal curvatures
while a solution E ∈ C 2 of equation (∗) is a critical configuration
TheoremIf E ⊂ Tn is a local minimizer of J, then ∂E \ Σ is C 3,α, for anyα<1, and Σ is a closed set such that dimH(Σ) ≤ n − 8.In fact, ∂E \ Σ is C∞ (Julin-Pisante, 2014).
Our Problem:Under which conditions regular critical points arelocal minimizers?
Second variation?
Let E ∈ C 2, and fix a C 2 vector field X : Tn 7→ Tn. Then, let usconsider
Φ : Tn × (−1, 1) 7→ Tn the associated flow
∂Φ
∂t= X (Φ), Φ(x , 0) = x
and set Et := Φ(·, t)(E )
We also require∫
∂EX · ν = 0 =⇒ d |Et |
dt∣∣t=0
= 0
Then we set J ′′(E )[X ] :=d2
dt2J(Et)∣∣
t=0
Our Problem:Under which conditions regular critical points arelocal minimizers? Second variation?
Let E ∈ C 2, and fix a C 2 vector field X : Tn 7→ Tn. Then, let usconsider
Φ : Tn × (−1, 1) 7→ Tn the associated flow
∂Φ
∂t= X (Φ), Φ(x , 0) = x
and set Et := Φ(·, t)(E )
We also require∫
∂EX · ν = 0 =⇒ d |Et |
dt∣∣t=0
= 0
Then we set J ′′(E )[X ] :=d2
dt2J(Et)∣∣
t=0
Our Problem:Under which conditions regular critical points arelocal minimizers? Second variation?
Let E ∈ C 2, and fix a C 2 vector field X : Tn 7→ Tn. Then, let usconsider
Φ : Tn × (−1, 1) 7→ Tn the associated flow
∂Φ
∂t= X (Φ), Φ(x , 0) = x
and set Et := Φ(·, t)(E )
We also require∫
∂EX · ν = 0 =⇒ d |Et |
dt∣∣t=0
= 0
Then we set J ′′(E )[X ] :=d2
dt2J(Et)∣∣
t=0
Our Problem:Under which conditions regular critical points arelocal minimizers? Second variation?
Let E ∈ C 2, and fix a C 2 vector field X : Tn 7→ Tn. Then, let usconsider
Φ : Tn × (−1, 1) 7→ Tn the associated flow
∂Φ
∂t= X (Φ), Φ(x , 0) = x
and set Et := Φ(·, t)(E )
We also require∫
∂EX · ν = 0 =⇒ d |Et |
dt∣∣t=0
= 0
Then we set J ′′(E )[X ] :=d2
dt2J(Et)∣∣
t=0
Our Problem:Under which conditions regular critical points arelocal minimizers? Second variation?
Let E ∈ C 2, and fix a C 2 vector field X : Tn 7→ Tn. Then, let usconsider
Φ : Tn × (−1, 1) 7→ Tn the associated flow
∂Φ
∂t= X (Φ), Φ(x , 0) = x
and set Et := Φ(·, t)(E )
We also require∫
∂EX · ν = 0 =⇒ d |Et |
dt∣∣t=0
= 0
Then we set J ′′(E )[X ] :=d2
dt2J(Et)∣∣
t=0
Theorem (Choksi-Sternberg 2007)If E is a critical point and X is as above, then
J ′′(E )[X ] =
∫
∂E
(|Dτ (X · ν)|2 − |B∂E |2(X · ν)2
)dHn−1
+ 8γ∫
∂E
∫
∂EG (x , y)
(X · ν
)(x)(X · ν
)(y)dσx dσy
+ 4γ∫
∂E∂νvE (X · ν)2 dσ
|B∂E |2 = sum of the squares of principal curvatures
Since the second variation depends only on X · ν,
we define for a C 2 critical point E and for ϕ ∈ H1(∂E )
∂2J(E )[ϕ] =
∫
∂E
(|Dτϕ|2 − |B∂E |2ϕ2
)dHn−1
+ 8γ∫
∂E
∫
∂EG (x , y)ϕ(x)ϕ(y)dHn−1(x) dHn−1(y)
+ 4γ∫
∂E∂νvE ϕ
2 dHn−1
Remark:∫∂E X · ν dHn−1 = 0 =⇒ d |Et |
dt∣∣t=0
= 0
Space of admissible variations:
H1(∂E ) :=
ϕ ∈ H1(∂E ) :
∫
∂Eϕ = 0
︸ ︷︷ ︸volume pres.
,
∫
∂EϕνE = 0
︸ ︷︷ ︸translation inv.
Since the second variation depends only on X · ν,
we define for a C 2 critical point E and for ϕ ∈ H1(∂E )
∂2J(E )[ϕ] =
∫
∂E
(|Dτϕ|2 − |B∂E |2ϕ2
)dHn−1
+ 8γ∫
∂E
∫
∂EG (x , y)ϕ(x)ϕ(y)dHn−1(x) dHn−1(y)
+ 4γ∫
∂E∂νvE ϕ
2 dHn−1
Remark:∫∂E X · ν dHn−1 = 0 =⇒ d |Et |
dt∣∣t=0
= 0
Space of admissible variations:
H1(∂E ) :=
ϕ ∈ H1(∂E ) :
∫
∂Eϕ = 0
︸ ︷︷ ︸volume pres.
,
∫
∂EϕνE = 0
︸ ︷︷ ︸translation inv.
Since the second variation depends only on X · ν,
we define for a C 2 critical point E and for ϕ ∈ H1(∂E )
∂2J(E )[ϕ] =
∫
∂E
(|Dτϕ|2 − |B∂E |2ϕ2
)dHn−1
+ 8γ∫
∂E
∫
∂EG (x , y)ϕ(x)ϕ(y)dHn−1(x) dHn−1(y)
+ 4γ∫
∂E∂νvE ϕ
2 dHn−1
Remark:∫∂E X · ν dHn−1 = 0 =⇒ d |Et |
dt∣∣t=0
= 0
Space of admissible variations:
H1(∂E ) :=
ϕ ∈ H1(∂E ) :
∫
∂Eϕ = 0
︸ ︷︷ ︸volume pres.
,
∫
∂EϕνE = 0
︸ ︷︷ ︸translation inv.
Theorem (Acerbi-F.-Morini 2013)Let E be a C 2 critical configuration such that
∂2J(E )[ϕ] > 0 ∀ϕ ∈ H1(∂E ).
Then, E is a strict local minimizer. Precisely, there exists δ > 0,s.t. for every set of finite perimeter F ⊂ Tn, with d(E ,F ) < δ
(∗∗) J(F ) ≥ J(E ) + C0d(E ,F )2
Consequences: γ = 0 =⇒ quantitative isop. ineq.
CorollaryLet E ⊂ Tn be smooth open set with constant mean curvature. If∫
∂E
(|Dτϕ|2 − |B∂E |2ϕ2) dHn−1 > 0 ∀ ϕ ∈ T⊥(∂E ) \ 0 ,
there exist δ, C > 0 s.t. for F ⊂ Tn, with |F | = |E | andd(E ,F ) < δ
PTn(F ) ≥ PTn(E ) + C [d(E ,F )]2 .
The local minimality w.r.t. L∞ perturbations (B.White, 1994)or w.r.t. L1 perturbations (⇒ n ≤ 7, Morgan-Ros, 2010)
Consequences: γ = 0 =⇒ quantitative isop. ineq.
CorollaryLet E ⊂ Tn be smooth open set with constant mean curvature. If∫
∂E
(|Dτϕ|2 − |B∂E |2ϕ2) dHn−1 > 0 ∀ ϕ ∈ T⊥(∂E ) \ 0 ,
there exist δ, C > 0 s.t. for F ⊂ Tn, with |F | = |E | andd(E ,F ) < δ
PTn(F ) ≥ PTn(E ) + C [d(E ,F )]2 .
The local minimality w.r.t. L∞ perturbations (B.White, 1994)or w.r.t. L1 perturbations (⇒ n ≤ 7, Morgan-Ros, 2010)
Application: Global and local minimality of lamellae
(P) MinJ(E ) = PTn(E ) + γ
∫
Tn
|∇vE |2 dx , |E | = d
−∆vE = uE −m in Tn
∫Tn vE = 0
uE = χE − χTn\E , m = 2|E | − 1
For 0 < d < 1, k ≥ 1, set
Lk = Tn−1 × ∪ki=1[i−1k , i−1
k + dk
]
Theorem (Acerbi-F.-Morini, 2013)If L1 is the unique global minimizer of the periodic isoperimetricproblem in Tn, then it is also the unique global minimizer of (P),provided γ is sufficiently small.Moreover, let d and γ > 0. Then there exists k0 such that ifk ≥ k0 the set Lk is a strict local minimizer for (P).
Application: Global and local minimality of lamellae
(P) MinJ(E ) = PTn(E ) + γ
∫
Tn
|∇vE |2 dx , |E | = d
−∆vE = uE −m in Tn
∫Tn vE = 0
uE = χE − χTn\E , m = 2|E | − 1
For 0 < d < 1, k ≥ 1, set
Lk = Tn−1 × ∪ki=1[i−1k , i−1
k + dk
]
Theorem (Acerbi-F.-Morini, 2013)If L1 is the unique global minimizer of the periodic isoperimetricproblem in Tn, then it is also the unique global minimizer of (P),provided γ is sufficiently small.Moreover, let d and γ > 0. Then there exists k0 such that ifk ≥ k0 the set Lk is a strict local minimizer for (P).
Application: Global and local minimality of lamellae
(P) MinJ(E ) = PTn(E ) + γ
∫
Tn
|∇vE |2 dx , |E | = d
−∆vE = uE −m in Tn
∫Tn vE = 0
uE = χE − χTn\E , m = 2|E | − 1
For 0 < d < 1, k ≥ 1, set
Lk = Tn−1 × ∪ki=1[i−1k , i−1
k + dk
]
Theorem (Acerbi-F.-Morini, 2013)If L1 is the unique global minimizer of the periodic isoperimetricproblem in Tn, then it is also the unique global minimizer of (P),provided γ is sufficiently small.Moreover, let d and γ > 0. Then there exists k0 such that ifk ≥ k0 the set Lk is a strict local minimizer for (P).
Application: Global and local minimality of lamellae
(P) MinJ(E ) = PTn(E ) + γ
∫
Tn
|∇vE |2 dx , |E | = d
−∆vE = uE −m in Tn
∫Tn vE = 0
uE = χE − χTn\E , m = 2|E | − 1
For 0 < d < 1, k ≥ 1, set
Lk = Tn−1 × ∪ki=1[i−1k , i−1
k + dk
]
Theorem (Acerbi-F.-Morini, 2013)If L1 is the unique global minimizer of the periodic isoperimetricproblem in Tn, then it is also the unique global minimizer of (P),provided γ is sufficiently small.Moreover, let d and γ > 0. Then there exists k0 such that ifk ≥ k0 the set Lk is a strict local minimizer for (P).
Critical 2d k-lamellar patterns
o ay1
y2
...
yk
1
Ωa := (0, a)× (0, 1)We consider only the casem = 0
Theorem (Morini-Sternberg, 2014)For any positive integer k , if
a < π√
k2γ ,
then the k-lamellar critical configuration uk is an isolated L1-localminimizer in Ωa := (0, a)× (0, 1).
Critical 2d k-lamellar patterns
o ay1
y2
...
yk
1
Ωa := (0, a)× (0, 1)We consider only the casem = 0
Theorem (Morini-Sternberg, 2014)For any positive integer k , if
a < π√
k2γ ,
then the k-lamellar critical configuration uk is an isolated L1-localminimizer in Ωa := (0, a)× (0, 1).
Application: Periodic local minimizers
Cristoferi (2015): for every critical set E that has positive secondvariation for the perimeter, there exists a set F locally minimizingfor J, which closely resemble a rescaled version of E .
Application: Periodic local minimizers
Cristoferi (2015): for every critical set E that has positive secondvariation for the perimeter, there exists a set F locally minimizingfor J, which closely resemble a rescaled version of E .
Extension: Global minimality of a single droplet
Theorem (Cicalese-Spadaro, 2013)If Ω is C 2 and bounded and
γr3| log r |<<1 (n = 2), γr3<<1 (n ≥ 3),
then the unique global minimizer is a convex set E such that
∂E = x + (r + ϕ(ω))ω : ω ∈ Sn−1, ‖ϕ‖C1(Sn−1) ≤ c(n)γrn+3
Moreover E is a ball iff Ω is a ball.
J(E ) = P(E ; Ω) + γ
∫
Ω|∇vE |2 dx , |E | = d = ωnr
n < |Ω|
Julin-Pisante, 2014: Local minimality (with a quantitativeestimate) for critical points of in a smooth open set underNeumann boundary condition
Extension: Global minimality of a single droplet
Theorem (Cicalese-Spadaro, 2013)If Ω is C 2 and bounded and
γr3| log r |<<1 (n = 2), γr3<<1 (n ≥ 3),
then the unique global minimizer is a convex set E such that
∂E = x + (r + ϕ(ω))ω : ω ∈ Sn−1, ‖ϕ‖C1(Sn−1) ≤ c(n)γrn+3
Moreover E is a ball iff Ω is a ball.
J(E ) = P(E ; Ω) + γ
∫
Ω|∇vE |2 dx , |E | = d = ωnr
n < |Ω|
Julin-Pisante, 2014: Local minimality (with a quantitativeestimate) for critical points of in a smooth open set underNeumann boundary condition
Evolutionary counterpart: the nonlocal Mullins-Sekerka flowA smooth flow of sets (Et)t ⊂⊂ Ω is a solution to the nonlocalMullins-Sekerka flow if
Vt = [∂νwt ] on ∂Et ,∆wt = 0 in Ω \ ∂Et ,wt = H∂Et + 4γvEt on ∂Et ,−∆vEt = uEt −−
∫Ω uEt , in Ω,
We have set Vt = normal velocity at time t
[∂νwt ] = jump of the normal derivative of wt
Note:d
dt|Et | =
∫
∂Et
Vt dH2 =
∫
∂Et
[∂νtwt ] dH2 = 0
⇓
The flow is volume preserving.
Evolutionary counterpart: the nonlocal Mullins-Sekerka flowA smooth flow of sets (Et)t ⊂⊂ Ω is a solution to the nonlocalMullins-Sekerka flow if
Vt = [∂νwt ] on ∂Et ,∆wt = 0 in Ω \ ∂Et ,wt = H∂Et + 4γvEt on ∂Et ,−∆vEt = uEt −−
∫Ω uEt , in Ω,
We have set Vt = normal velocity at time t
[∂νwt ] = jump of the normal derivative of wt
Note:d
dt|Et | =
∫
∂Et
Vt dH2 =
∫
∂Et
[∂νtwt ] dH2 = 0
⇓
The flow is volume preserving.
Evolutionary counterpart: the nonlocal Mullins-Sekerka flowA smooth flow of sets (Et)t ⊂⊂ Ω is a solution to the nonlocalMullins-Sekerka flow if
Vt = [∂νwt ] on ∂Et ,∆wt = 0 in Ω \ ∂Et ,wt = H∂Et + 4γvEt on ∂Et ,−∆vEt = uEt −−
∫Ω uEt , in Ω,
We have set Vt = normal velocity at time t
[∂νwt ] = jump of the normal derivative of wt
Note:d
dt|Et | =
∫
∂Et
Vt dH2 =
∫
∂Et
[∂νtwt ] dH2 = 0
⇓
The flow is volume preserving.
Evolutionary counterpart: the nonlocal Mullins-Sekerka flowA smooth flow of sets (Et)t ⊂⊂ Ω is a solution to the nonlocalMullins-Sekerka flow if
Vt = [∂νwt ] on ∂Et ,∆wt = 0 in Ω \ ∂Et ,wt = H∂Et + 4γvEt on ∂Et ,−∆vEt = uEt −−
∫Ω uEt , in Ω,
We have set Vt = normal velocity at time t
[∂νwt ] = jump of the normal derivative of wt
Note:d
dt|Et | =
∫
∂Et
Vt dH2 =
∫
∂Et
[∂νtwt ] dH2 = 0
⇓
The flow is volume preserving.
The nonlocal Mullins-Sekerka flow
I It arises as the sharp interface limit of the Ohta-Kawaskiequation
∂tu = −∆(ε∆u − W ′(u)
ε− γv) in Ω× (0,T ),
−∆v = u −−∫
Ω u in Ω× (0,T ),
Proved by Alikakos, Bates, Chen (1994, for γ = 0), Le (2010in the general case)
I It can be seen as the H−12 -gradient flow of the sharp-interface
Ohta-Kawaski energy
I When γ = 0 we recover the Hele-Shaw flow with surfacetension. In this case the normal velocity [∂νwt ] is related to∆
12H
The nonlocal Mullins-Sekerka flow
I It arises as the sharp interface limit of the Ohta-Kawaskiequation
∂tu = −∆(ε∆u − W ′(u)
ε− γv) in Ω× (0,T ),
−∆v = u −−∫
Ω u in Ω× (0,T ),
Proved by Alikakos, Bates, Chen (1994, for γ = 0), Le (2010in the general case)
I It can be seen as the H−12 -gradient flow of the sharp-interface
Ohta-Kawaski energy
I When γ = 0 we recover the Hele-Shaw flow with surfacetension.
In this case the normal velocity [∂νwt ] is related to∆
12H
The nonlocal Mullins-Sekerka flow
I It arises as the sharp interface limit of the Ohta-Kawaskiequation
∂tu = −∆(ε∆u − W ′(u)
ε− γv) in Ω× (0,T ),
−∆v = u −−∫
Ω u in Ω× (0,T ),
Proved by Alikakos, Bates, Chen (1994, for γ = 0), Le (2010in the general case)
I It can be seen as the H−12 -gradient flow of the sharp-interface
Ohta-Kawaski energy
I When γ = 0 we recover the Hele-Shaw flow with surfacetension. In this case the normal velocity [∂νwt ] is related to∆
12H
Features of the nonlocal Mullins-Sekerka flow
I Singularities may appear
I Lack of a comparison principle
I Convexity is not preserved along the flow
I Local-in-time existence theory by Escher-Nishiura, 2002
Features of the nonlocal Mullins-Sekerka flow
I Singularities may appear
I Lack of a comparison principle
I Convexity is not preserved along the flow
I Local-in-time existence theory by Escher-Nishiura, 2002
Features of the nonlocal Mullins-Sekerka flow
I Singularities may appear
I Lack of a comparison principle
I Convexity is not preserved along the flow
I Local-in-time existence theory by Escher-Nishiura, 2002
Features of the nonlocal Mullins-Sekerka flow
I Singularities may appear
I Lack of a comparison principle
I Convexity is not preserved along the flow
I Local-in-time existence theory by Escher-Nishiura, 2002
Nonlinear stability
Vt = [∂νwt ] on ∂Et ,∆wt = 0 in Ω \ ∂Et ,wt = H∂Et + 4γvEt on ∂Et ,−∆vEt = uEt −−
∫Ω uEt , in Ω,
Theorem (Acerbi-F.-Julin-Morini, 2016)Let F ⊂ T3 be a strictly stable set.
There exists δ0 > 0 such thatif the initial datum E0 satisfies
|E0| = |F | , distC1(E0,F ) ≤ δ0 , and∫
T3|DwE0 |2 dx ≤ δ0 ,
then the flow (Et)t starting from E0 is defined for all t > 0.Moreover, Et → F + σ in H5/2 exponentially fast as t → +∞, forsome σ ∈ R3.
Previous related results: exponential stability of spheres for theHele-Shaw flow (Chen 1993, Escher-Simonett 1998)
Nonlinear stability
Vt = [∂νwt ] on ∂Et ,∆wt = 0 in Ω \ ∂Et ,wt = H∂Et + 4γvEt on ∂Et ,−∆vEt = uEt −−
∫Ω uEt , in Ω,
Theorem (Acerbi-F.-Julin-Morini, 2016)Let F ⊂ T3 be a strictly stable set. There exists δ0 > 0 such thatif the initial datum E0 satisfies
|E0| = |F | , distC1(E0,F ) ≤ δ0 , and∫
T3|DwE0 |2 dx ≤ δ0 ,
then the flow (Et)t starting from E0 is defined for all t > 0.
Moreover, Et → F + σ in H5/2 exponentially fast as t → +∞, forsome σ ∈ R3.
Previous related results: exponential stability of spheres for theHele-Shaw flow (Chen 1993, Escher-Simonett 1998)
Nonlinear stability
Vt = [∂νwt ] on ∂Et ,∆wt = 0 in Ω \ ∂Et ,wt = H∂Et + 4γvEt on ∂Et ,−∆vEt = uEt −−
∫Ω uEt , in Ω,
Theorem (Acerbi-F.-Julin-Morini, 2016)Let F ⊂ T3 be a strictly stable set. There exists δ0 > 0 such thatif the initial datum E0 satisfies
|E0| = |F | , distC1(E0,F ) ≤ δ0 , and∫
T3|DwE0 |2 dx ≤ δ0 ,
then the flow (Et)t starting from E0 is defined for all t > 0.Moreover, Et → F + σ in H5/2 exponentially fast as t → +∞, forsome σ ∈ R3.
Previous related results: exponential stability of spheres for theHele-Shaw flow (Chen 1993, Escher-Simonett 1998)
Nonlinear stability
Vt = [∂νwt ] on ∂Et ,∆wt = 0 in Ω \ ∂Et ,wt = H∂Et + 4γvEt on ∂Et ,−∆vEt = uEt −−
∫Ω uEt , in Ω,
Theorem (Acerbi-F.-Julin-Morini, 2016)Let F ⊂ T3 be a strictly stable set. There exists δ0 > 0 such thatif the initial datum E0 satisfies
|E0| = |F | , distC1(E0,F ) ≤ δ0 , and∫
T3|DwE0 |2 dx ≤ δ0 ,
then the flow (Et)t starting from E0 is defined for all t > 0.Moreover, Et → F + σ in H5/2 exponentially fast as t → +∞, forsome σ ∈ R3.
Previous related results: exponential stability of spheres for theHele-Shaw flow (Chen 1993, Escher-Simonett 1998)
Nonlinear stability
Theorem (Acerbi-F.-Julin-Morini, 2016)Let F ⊂ T3 be a strictly stable set. There exists δ0 > 0 such that ifthe initial datum E0 satisfies
|E0| = |F | , distC1(E0,F ) ≤ δ0 , and∫
T3|DwE0 |2 dx ≤ δ0 ,
then the flow (Et)t starting from E0 is defined for all t > 0.Moreover, Et → F + σ in H5/2 exponentially fast as t → +∞, forsome σ ∈ R3.
Previous related results: exponential stability of spheres for theHele-Shaw flow (Chen 1993, Escher-Simonett 1998)
Nonlinear stability
Theorem (Acerbi-F.-Julin-Morini, 2016)Let F ⊂ T3 be a strictly stable set. There exists δ0 > 0 such that ifthe initial datum E0 satisfies
|E0| = |F | , distC1(E0,F ) ≤ δ0 , and∫
T3|DwE0 |2 dx ≤ δ0 ,
then the flow (Et)t starting from E0 is defined for all t > 0.Moreover, Et → F + σ in H5/2 exponentially fast as t → +∞, forsome σ ∈ R3.
Previous related results: exponential stability of spheres for theHele-Shaw flow (Chen 1993, Escher-Simonett 1998)
Nonlinear stability: ingredients of the proof
Step 1 (Energy identity)
d
dt
∫
T3|Dwt |2 dx = −2∂2J(Et) [Vt ] + R(Et)
whereR(Et) =
∫
∂Et
(∂νtw+t + ∂νtw
−t )[∂νtwt ]
2 dH2 ,
Step 2 (Stopping time) Let
t := supt > 0 : distC1(Et ,F ) < 2δ0 and
∫
T3|Dwt |2 dx < 2δ0 for all t ∈ (0, t),
Nonlinear stability: ingredients of the proofStep 3 (t =maximal time of existence T ∗)Assume by contradiction that
∫
T3|Dwt |2 dx = 2δ0 for some t < T ∗
By stability (+ estimate on the translational part of the flow)
∂2J(Et)[Vt ] ≥ σ‖Vt‖2H1(∂Et)(1)
Moreover,
|R(Et)| ≤ ε‖Vt‖2H1(∂Et)and
∫
T3|Dwt |2 dx ≤ C‖Vt‖2H1(∂Et)
(2)
The last inequalities follow from delicate boundary estimates for
−∆w = fH2 ∂E
Nonlinear stability: ingredients of the proofStep 3 (t =maximal time of existence T ∗)Assume by contradiction that
∫
T3|Dwt |2 dx = 2δ0 for some t < T ∗
By stability (+ estimate on the translational part of the flow)
∂2J(Et)[Vt ] ≥ σ‖Vt‖2H1(∂Et)(1)
Moreover,
|R(Et)| ≤ ε‖Vt‖2H1(∂Et)and
∫
T3|Dwt |2 dx ≤ C‖Vt‖2H1(∂Et)
(2)
The last inequalities follow from delicate boundary estimates for
−∆w = fH2 ∂E
Nonlinear stability: ingredients of the proofStep 3 (t =maximal time of existence T ∗)Assume by contradiction that
∫
T3|Dwt |2 dx = 2δ0 for some t < T ∗
By stability (+ estimate on the translational part of the flow)
∂2J(Et)[Vt ] ≥ σ‖Vt‖2H1(∂Et)(1)
Moreover,
|R(Et)| ≤ ε‖Vt‖2H1(∂Et)and
∫
T3|Dwt |2 dx ≤ C‖Vt‖2H1(∂Et)
(2)
The last inequalities follow from delicate boundary estimates for
−∆w = fH2 ∂E
Nonlinear stability: ingredients of the proofStep 3 (t =maximal time of existence T ∗)Assume by contradiction that
∫
T3|Dwt |2 dx = 2δ0 for some t < T ∗
By stability (+ estimate on the translational part of the flow)
∂2J(Et)[Vt ] ≥ σ‖Vt‖2H1(∂Et)(1)
Moreover,
|R(Et)| ≤ ε‖Vt‖2H1(∂Et)and
∫
T3|Dwt |2 dx ≤ C‖Vt‖2H1(∂Et)
(2)
The last inequalities follow from delicate boundary estimates for
−∆w = fH2 ∂E
Boundary estimates
Proposition (Boundary estimates for harmonic functions)Let E ⊂ T3 be of class C 1,α, f ∈ Cα(∂E ) (with zero average) andu ∈ H1(T3) be the solution of
−∆u = fH2 ∂E
with zero average in T3. Then, for every 1 < p <∞ there exists aconstant C , which depends on the C 1,α bounds on ∂E and on p,such that:(i) ‖∂νE u+‖L2(∂E) + ‖∂νE u−‖L2(∂E) ≤ C‖u‖H1(∂E);(ii) ‖∂νE u+‖Lp(∂E) + ‖∂νE u−‖Lp(∂E) ≤ C‖f ‖Lp(∂E).(iii) Moreover, if f ∈ H1(∂E ), then for every 1 ≤ p < +∞
‖f ‖Lp(∂E) ≤ C‖f ‖p−1p
H1(∂E)‖u‖
1p
L2(∂E).
Nonlinear stability: ingredients of the proof
Combining (1) and (2) with energy identity
d
dt
∫
T3|Dwt |2 dx = −2∂2J(Et) [Vt ] + R(Et)
≤ −c0‖Vt‖2H1(∂Et)≤ −c1
∫
T3|Dwt |2 dx
Thus, ∫
T3|Dwt |2 dx ≤
∫
T3|Dw0|2 dxe−c0t ≤ δ0e−c0t
which contradicts the assumption∫
T3|Dwt |2 dx = 2δ0
Step 4 T ∗ = +∞ and conclusion
Nonlinear stability: ingredients of the proof
Combining (1) and (2) with energy identity
d
dt
∫
T3|Dwt |2 dx = −2∂2J(Et) [Vt ] + R(Et)
≤ −c0‖Vt‖2H1(∂Et)≤ −c1
∫
T3|Dwt |2 dx
Thus, ∫
T3|Dwt |2 dx ≤
∫
T3|Dw0|2 dxe−c0t ≤ δ0e−c0t
which contradicts the assumption∫
T3|Dwt |2 dx = 2δ0
Step 4 T ∗ = +∞ and conclusion
Nonlinear stability: ingredients of the proof
Combining (1) and (2) with energy identity
d
dt
∫
T3|Dwt |2 dx = −2∂2J(Et) [Vt ] + R(Et)
≤ −c0‖Vt‖2H1(∂Et)≤ −c1
∫
T3|Dwt |2 dx
Thus, ∫
T3|Dwt |2 dx ≤
∫
T3|Dw0|2 dxe−c0t ≤ δ0e−c0t
which contradicts the assumption∫
T3|Dwt |2 dx = 2δ0
Step 4 T ∗ = +∞ and conclusion
Nonlinear stability: ingredients of the proof
Combining (1) and (2) with energy identity
d
dt
∫
T3|Dwt |2 dx = −2∂2J(Et) [Vt ] + R(Et)
≤ −c0‖Vt‖2H1(∂Et)≤ −c1
∫
T3|Dwt |2 dx
Thus, ∫
T3|Dwt |2 dx ≤
∫
T3|Dw0|2 dxe−c0t ≤ δ0e−c0t
which contradicts the assumption∫
T3|Dwt |2 dx = 2δ0
Step 4 T ∗ = +∞ and conclusion
Thank you for your attention!