Geometric aspects and asymptotic analysis in phase ... · Geometric aspects and asymptotic analysis in phase separation of coupled elliptic equations Nicola Soave Justus Liebig Universit

Geometric aspects and asymptotic analysis inphase separation of coupled elliptic equations

Nicola Soave

Justus Liebig Universitat, Giessen (Germany)

Workshop on Nonlinear PDEsBruxelles

September 7th–11th, 2015

Joint work with Alessandro Zilio (CAMS/EHESS Paris)

Nicola Soave Asymptotic and geometric aspects in phase separation

Introduction

We aim at describing the asymptotic behaviour of solutions to competing(β > 0) systems of type

−∆ui,β = fi,β(x , ui,β)− βui,β

∑j 6=i u2

j,β in Ω

ui,β > 0 in Ω,

i = 1, . . . , k , in the limit of strong competition: β → +∞.

The problem is variational, and its solutions are critical points of

Jβ(u) :=

∫Ω

∑i

[1

2|∇ui |2 − Fi,β(x , ui )

]+β

2

∑i<j

∫Ω

u2i u2

j .


Introduction

We aim at describing the asymptotic behaviour of solutions to competing(β > 0) systems of type


∑j 6=i u2

j,β in Ω

ui,β > 0 in Ω,

i = 1, . . . , k , in the limit of strong competition: β → +∞.

The problem is variational, and its solutions are critical points of

Jβ(u) :=

∫Ω

∑i

[1

2|∇ui |2 − Fi,β(x , ui )

]+β

2

∑i<j

∫Ω

u2i u2

j .


Typical existence results assert for every β > 0 there are critical pointsuβ of the functional Jβ for which the energy bounds hold uniformly in β:

Jβ(uβ) ≤ C ∀β > 0.

As a consequence one can show that there exists C > 0 independent of βsuch that

β

∫Ω

u2i,βu2

j,β ≤ C =⇒ ui,βuj,β → 0 a.e. in Ω.

Phase separation: in the limit of strong competition different densitiestend to have disjoint support.


Great efforts devoted to the description of the asymptotic behaviour ofsolutions of competing systems as the competition parameter β diverges;the main goals have been:

(a) to investigate if one can guarantee convergence of the solutions tosome limit profile;

(b) to study the regularity of the class of limit profiles, both in term ofthe densities and in term of the emerging free boundary problem;

(c) to describe qualitative properties and to give precise estimates ofsuch convergence.


Uniform bounds, convergence, free boundary

Let uβ : β > 0 be a family of positive solutions to


∑j 6=i

aiju2j,β in Ω,

uniformly bounded in L∞(Ω). The following optimal results areestablished:

uβ : β > 0 is uniformly bounded in Liploc(Ω) [S., Zilio, ARMA 2015]

(Holder bounds in [Noris, Tavares, Terracini, Verzini, CPAM 2010]);up to subsequences, uβ → u = (u1, . . . , uk) in H1

loc(Ω) ∩ C0,α(Ω);u = (u1, . . . , uk) is a vector of Lipschitz continuous functionssatisfying (assuming the convergence fi,β → fi )

−∆ui = fi (x , ui ) in ui > 0uiuj = 0 segregation of the densities

[Noris Tavares Terracini Verzini, CPAM 2010].the free boundary Γ = ui = 0, i = 1, . . . , k can be decomposed ina finite number of C1,α-surfaces of dimension N − 1 and an irregularset of dimension at most N − 2 [Tavares Terracini, Calc. Var. 2012] (seealso [Caffarelli, Lin, JAMS 2008]).





∑j 6=i

aiju2j,β in Ω,











∑j 6=i

aiju2j,β in Ω,









Asymptotic estimates and geometric aspects in dimensionN = 1

Let us consider a family (uβ , vβ), uniformly bounded in L∞(a, b),solution to the 1-dimensional problem

−u′′β = fβ(uβ)− βuβv 2β in (a, b) ⊂ R

−v ′′β = gβ(vβ)− βvβu2β in (a, b)

uβ , vβ > 0,∫ b

au2β =

∫ b

av 2β = 1

uβ , vβ ∈ H10 (a, b).

In [Berestycki, Lin, Wei, Zhao, ARMA 2013], the authors use the uniformLipschitz bound to deduce that there exists C > 0 such that

βu2βv 2β < C in [a, b].

Moreover, if we consider points

xβ ∈ (a, b) such that uβ(xβ) = vβ(xβ) =: mβ

then there exists c > 0 such that

βu2β(xβ)v 2

β(xβ) = βm4β > c .


Theorem (Berestycki et al., ARMA 2013)

On the interface uβ = vβ, solutions decay as β−1/4 as β → +∞.

Using this estimate, it is also possible to show that:

Theorem (Berestycki et al., ARMA 2013)

Let xβ ∈ uβ = vβ, xβ → x0 ∈ (a, b). Then, if mβ = uβ(xβ), the scaledfamily

uβ(x) :=1

mβuβ(mβx + xβ), vβ(x) :=

1

mβvβ(mβx + xβ)

converges, up to a subsequence, in C 2loc(R) to solutions of

u′′ = uv 2,

v ′′ = u2v

u, v > 0

in R.


The second result suggest that the geometry of positive solutions to

u′′ = uv 2 v ′′ = u2v in R (S1)

reflects the geometry of (uβ , vβ) near the interface uβ = vβ. In thisperspective, we have (see [Berestycki, Terracini, Wang, Wei Advances in Math.

2013]):

Theorem

In dimension N = 1, system (S1) has a unique positive solution, up toscaling, translation, and exchange of the components.


One dimensional profile

uv

x0

The solution shadows (x+, x−).


The purpose of our work is to extend the previous analysis in higherdimension. Let uβ : β > 0 be a family of solutions to


∑j 6=i u2

j,β in Ω

ui,β > 0 in Ω,

i = 1, . . . , k , uniformly bounded in L∞(Ω). Then, uβ is bounded inLiploc(Ω), and u.t.s. converges to u, solutions of a free boundaryproblem. We suppose that u 6≡ 0 and we recall that Γ = u = 0.Let us define the interface of uβ as

Γβ = ui1,β = ui2,β ≥ uj,β ∀j 6= i1, i2 .

If xβ ∈ Γβ forall β and xβ → x ∈ Ω, then x ∈ Γ.


The purpose of our work is to extend the previous analysis in higherdimension. Let uβ : β > 0 be a family of solutions to


∑j 6=i u2

j,β in Ω

ui,β > 0 in Ω,

i = 1, . . . , k , uniformly bounded in L∞(Ω). Then, uβ is bounded inLiploc(Ω), and u.t.s. converges to u, solutions of a free boundaryproblem. We suppose that u 6≡ 0 and we recall that Γ = u = 0.Let us define the interface of uβ as

Γβ = ui1,β = ui2,β ≥ uj,β ∀j 6= i1, i2 .

If xβ ∈ Γβ forall β and xβ → x ∈ Ω, then x ∈ Γ.


Natural questions


Natural questions

is there a decay rate for

uβ(xβ) = vβ(xβ)

independent of the particular choice of xβ , as in the 1-D case, ornot? In particular, is the decay rate the same if x is a regular pointor a singular one?

Let us consider the dashed region U in the previous picture. Therein,the component u3 vanishes identically in the limit, differently u1 andu2. Can we prove that in U the decay of u3,β is faster?

Can we characterize the geometry of the uβ near the interface interms of entire solutions to an elliptic system?


Theorem (S., Zilio, preprint 2015)

Upper estimate: If K b Ω, then

βu2i,βu2

j,β ≤ C in K , for all i 6= j .

Lower estimate near the regular part: if xβ → x and x stays on theregular part of Γ, then ∀ε > 0 there exists Cε > 0 such that

β1+εu2i,β(xβ)u2

j,β(xβ) ≥ Cε

for some i 6= j .

Lower estimate in general? If xβ → x and x stays in the singularpart of Γ, then

βu2i,β(xβ)u2

j,β(xβ)→ 0 for all i 6= j .


Theorem (S., Zilio, preprint 2015)

Upper estimate: If K b Ω, then

βu2i,βu2

j,β ≤ C in K , for all i 6= j .

Lower estimate near the regular part: if xβ → x and x stays on theregular part of Γ, then ∀ε > 0 there exists Cε > 0 such that

β1+εu2i,β(xβ)u2

j,β(xβ) ≥ Cε

for some i 6= j .

Lower estimate in general? If xβ → x and x stays in the singularpart of Γ, then

βu2i,β(xβ)u2

j,β(xβ)→ 0 for all i 6= j .


Theorem

Improved upper estimate: under reasonable additional assumptionson xβ , we have

β6/5u2i,β(xβ)u2

j,β(xβ) ≤ C for all i 6= j ;

Faster decay of vanishing components: under the previous notation,there exist C1 and C2 such that in the dashed region

u3,β ≤ C1e−C1βC2

Uniform regularity of the interfaces: Far away from the singular partof Γ, the interfaces Γβ enjoy a uniform-in-β vanishing Reifenbergflatness condition (in particular, they are uniform-in-β C0,α graphs).


Theorem

Improved upper estimate: under reasonable additional assumptionson xβ , we have

β6/5u2i,β(xβ)u2

j,β(xβ) ≤ C for all i 6= j ;

Faster decay of vanishing components: under the previous notation,there exist C1 and C2 such that in the dashed region

u3,β ≤ C1e−C1βC2

Uniform regularity of the interfaces: Far away from the singular partof Γ, the interfaces Γβ enjoy a uniform-in-β vanishing Reifenbergflatness condition (in particular, they are uniform-in-β C0,α graphs).


Naive idea of the proof

There exists rβ , ρβ → 0 such that, if xβ ∈ Γβ , the scaled sequence isconvergent in

vi,β(x) =1

ρβui,β(xβ + rβx)

is convergent in C2loc(RN) to a nonnegative solution of

∆Vi =∑j 6=i

ViV2j . (S)


Classification results for (S)

Theorem (S., Terracini, Adv. Math. 2015)

In any dimension N ≥ 1, let V = (V1, . . . ,Vk) be a nonnegative solutionof (S).

if V has at most linear growth, then it has only 2 nontrivialcomponents and is 1-dimensional (for k = 2 [K. Wang, CommPDE 2014

and Manuscripta Math. 2015]);

if V has at most polynomial growth and

limxN→±∞

(V1(x ′, xN)− V2(x ′, xN)) = ±∞,

the limit being uniform in x ′ ∈ RN−1, then it has only 2 nontrivialcomponents and is 1-dimensional (for k = 2 [S., Farina, ARMA 2014]);

if (V1, . . . ,Vk) is not 1-dimensional, then

V1(x) + · · ·+ Vk(x) ' |x |3/2



The assumptions on xβ lead to profiles V, solutions to (S), satisfyingdifferent conditions;

Using the previous classification results, we obtain preciseinformation on V;

The idea is to switch back and forth from the sequence uβ to theprofile V in order to exchange information; this can be done bymeans of suitable monotonicity formulae of Almgren andAlt-Caffarelli-Friedman type.

Example

Let xβ → x in the singular part of Γ satisfying reasonable additionalassumptions; then the limit profile V is not 1-dimensional, and hence

V1(x) + · · ·+ Vk(x) ' |x |3/2.

Then we have

β3/2

2+2·(3/2) uβ(xβ) ≤ C =⇒ β6/5u2i,β(xβ)u2

j,β(xβ) ≤ C .






Example

Let xβ → x in the regular part of Γ; then the limit profile V has at mostlinear growth:

V1(x) + · · ·+ Vk(x) ' |x |1.

Then we have

β1

2+2·1 uβ(xβ) ≤ C =⇒ βu2i,β(xβ)u2

j,β(xβ) ≤ C .






Example

Let xβ → x in the regular part of Γ; then the limit profile V has at mostlinear growth:

V1(x) + · · ·+ Vk(x) ≤ C (1 + |x |1).

Then we have

β1

2+2·1 uβ(xβ) ≤ C =⇒ βu2i,β(xβ)u2

j,β(xβ) ≤ C .






Example

Let xβ → x , and let x be on the regular part of Γ. Then the limit profileV has at most linear growth, and hence it is 1-dimensional.


Thank you for the attention!


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Geometric aspects and asymptotic analysis in phase ... · Geometric aspects and asymptotic analysis in phase separation of coupled elliptic equations Nicola Soave Justus Liebig Universit