Entire solutions for competition-diffusion systems and optimal partitions

Susanna Terracini

Dipartimento di Matematica e Applicazioni

Università di Milano Bicocca

GNAMPA School

“DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS”

June 11-15 2012, Gaeta

1 Competition diffusion systems with Lotka-Volterra interactions: symmetric competition rates

With symmetric interspecific competition rates βi,j = βj,i large:

−∆ui = fi(ui)− ui h∑ j=1 j 6=i

βi,juj in Ω,

2 Competition diffusion systems with Lotka-Volterra interactions : asymmetric competition rates

With asymmetric interspecific competition rates βi,j 6= βj,i large:

3 Energy minimizing configurations of Bose–Einstein condensates in multiple spin–states with repulsive interaction potentials

E(ψ1, · · · , ψh) = ∫ ∑h

i 1 2|∇ψi|

2 + Fi(|ψi|2) + ∑h

j 6=i βi,j|ψi|2|ψj|2 in Ω,∫ |ψ|2i = mi , i = 1, . . . , h

Defocusing: S.M. Chang, C.S. Lin, T.C. Lin, and W.W. Lin, Phys. D 196, 341–361 (2004)

Focusing: Conti M., Terracini S., Verzini G., J. Functional Analysis, 198 (2003) 160-196

4 Optimal partition problems for Dirichlet eigenvalues

min

{ h∑ i=1

λp1(ωi) : (ω1, · · · , ωk) ∈ Bh(Ω))

}

where Bh = {(ω1, . . . , ωh) : ωi open, |ωi ∩ ωj| = 0 for i 6= j and ∪i ωi ⊆ Ω} .

B. Bourdin, D. Bucur, and . Oudet, Optimal Partitions for Eigenvalues, SIAM J. Sci. Comput. 31, 2009/10 pp. 4100-4114

With more and more nodal components:

With higher eigenvalues:

5 Energy minimizing segregated configurations

Let Ω be a bounded open subset of RN (N ≥ 2) and let us call segregated state a k–uple U = (u1, . . . , uk) ∈ (H1(Ω))k where

ui(x) · uj(x) = 0 i 6= j, a.e. x ∈ Ω.

We define the internal energy of U as

J(U) = ∑

i=1,··· ,k

{∫ Ω

( 1

2 d 2i (x) |∇ui(x)|2 − Fi(x, ui(x))

) dx

} ,

Our first goal is to minimize J among a class of segregated states subject to some boundary and positivity conditions.

6 Assumptions

Let N ≥ 2; let Ω ⊂ RN be a connected, open bounded domain with regular boundary . On the boundary data φi’s: φi ∈ H1/2(∂Ω), φi ≥ 0, and

φi · φj = 0, ∀i 6= j, a.e. on∂Ω

On the diffusions di’s and the fi’s: di ∈ W 2,∞(Ω), di > 0 on Ω (A1) fi(x, s) is Lipschitz in s, uniformly in x,

fi(x, 0) ≡ 0

(A2) there exists bi ∈ L∞(Ω) such that both

|fi(x, s)| ≤ bi(x)s ∀x ∈ Ω; s ≥ s̄ >> 1,∫ Ω

( d2i (x)|∇w(x)|2 − bi(x)w2(x)

) dx > 0 ∀w ∈ H10(Ω).

The last assumption ensures the coercivity of the functional.

7 Questions

Existence and continuous dependence on data

Uniqueness vs multiplicity

Extremality conditions

Regularity

of the minimizers of the interfaces

The multiplicity of a point x ∈ Ω is

m(x) = ] {i : meas ({ui > 0} ∩B(x, r)) > 0 ∀ r > 0} .

Denote by Zh(U) = {x ∈ Ω : m(x) ≥ h} the set of points of multiplicity greater or equal than h.

Structure of Z3(U) and expansion at multiple intersection points

8 Uniqueness of minimizers

Surprisingly enough, the minimizer of J turns out to be unique in the globally convex case:

Theorem 1 Assume moreover that

di ≡ dj ,∀i, j ∂2Fi ∂s2

(x, s) < 0,∀x ∈ Ω

Then, for each fixed boundary data, there is an unique minimizer.

Proof: Let U = (u1, · · · , uk) and V = (v1, · · · , vk) be two minimizers; consider

ûi = ui − ∑ h6=i

uh , v̂i = vi − ∑ h6=i

vh

Key: the following convexity–like property (for every λ ∈ (0, 1))

w (λ) i = [λûi + (1− λ)v̂i]

+ =⇒ J(w(λ)i ) < λJ(ui) + (1− λ)J(vi) .

At first, we note that the wλi have disjoint supports. Now, let us denote

Γ (λ) i = {x : w

λ i (x) > 0}.

Of course we have

Using the convexity of the quadratic part of the functional and keeping in mind that both the ui’s and vi’s have disjoint supports, we obtain that, for every λ ∈ (0, 1),

9 Extremality Conditions

Again denote

ûi = ui − ∑ h6=i

uh

and similarly

f̂ (x, ûi) = ∑ j

fj(x, ûi)χsupp (uj) =

 fi(x, ui) if x ∈ supp (ui)−fj(x, uj) if x ∈ supp (uj), j 6= i.

Theorem 2 Let U be a minimizer. Then, for every i, we have, in distributional sense,

−∆ûi ≥ f̂ (x, ûi)

10 Examples

Case k = 2 (two densities). Let

gi(x, s) =

{ fi(x, s) if s ≥ 0 −fj(x,−s) if s ≤ 0, j 6= i.

Note that gj(x, s) = −gi(x,−s). Then

−∆(ui − uj) = gi(x, ui − uj) = aij(x)(ui − uj) ,

where aij(x) = gi(x, ui − uj)/(ui − uj). The Lipschitz regularity of U follows. Also, the local structure of the interface is that of the nodal set of a solution to an elliptic divergence–type equation of the form

−div(b(x)∇v) = 0 .

Structure of the nodal set ⇒ Hartman–Winter, Alessandrini (N = 2), Caffarelli, Fang-Hua Lin, Hardt-Hoffmann-Ostenhof, Nadirashvili.

11 The class S

S = {

(u1, · · · , uk) ∈ (H1(ω))h : ui ≥ 0, ui · uj = 0 if i 6= j −∆ûi ≥ f̂ (x, ûi),∀i = 1, . . . , k

}

Basic properties in S : Proposition 1 Let x ∈ Ω: (a) If m(x) = 0, then there is r > 0 such that ui ≡ 0 on B(x, r), for every i. (b) If m(x) = 1, then there are i and r > 0 such that ui > 0 and −∆ui = fi(x, ui) on B(x, r). (c) If m(x) = 2, then are i, j and r > 0 such that uk ≡ 0 for k 6= i, j and −∆(ui − uj) =

gij(x, ui − uj) on B(x, r), where gi,j(x, s) = gi(x, s+)− gj(x, s−). Consequence: if m(x) = 2

lim y→x

y∈supp (ui)

∇ui(y) = − limy→x y∈supp (uj)

∇uj(y) .

12 The class S∗: interior Lipschitz continuity.....

S∗M,h(ω) =

(u1, · · · , uh) ∈ (H1(ω))h : ui ≥ 0, ui · uj = 0 if i 6= j

−∆ui ≤M, −∆ûi ≥ −M



Theorem 3 Let M > 0 and k be a fixed integer. Let U ∈ S∗M,‖(⊗): then U is Lipschitz continuous in the interior of Ω.

.... and global Lipschitz continuity

Theorem 4 Let ∂Ω be of class C2, U ∈ S with ui|∂Ω = φi and φi ∈ W 1,∞(∂Ω) for every i. Then U ∈ W 1,∞(Ω).

13 The Monotonicity formula by Alt–Caffarelli–Friedman

A key tool in the regularity theory is:

Lemma 1 (Alt-Caffarelli-Friedman) Let w1, w2 ∈ H1 ∩ L∞ such that −∆wi ≤ 0, w1(x) · w2(x) = 0 a.e. and x0 ∈ ∂(supp (wi)), i = 1, 2. Then the function

Φ(r) =

2∏ i=1

1

r2

∫ B(x0,r)

|∇wi(x)|2

|x− x0|N−2 dx

is non decreasing.

As a consequence, if the gradient of one density is not bounded at one point of multiplicity, the other must vanish. A generalization to k densities:

Lemma 2 Let wi ∈ H1 ∩L∞, i = 1, . . . , k such that −∆wi ≤ 0, wi(x) ·wj(x) = 0 a.e. if i 6= j and x0 ∈ ∂(supp (wi)), i = 1, 2. Then, for every k ≥ 3 and N ≥ 2, there exists β(k,N) > 2 such that the function

Φ(r) = k∏ i=1

1

rβ(k,N)

∫ B(x0,r)

|∇wi(x)|2

|x− x0|N−2 dx

is non decreasing.

14 An optimal partition problem for the ACF monotonicity formula

The proof of the ACF monotonicity lemma relies upon the fact that the partition into two equal half–spheres minimizes, over all possible partitions in two disjoint ωi (i = 1, 2) of the unit sphere SN−1, the sum of the characteristic values

β(2, N) := inf P(k,N)

2∑ i=1

(√(N − 2 2

)2 + λ1(ωi)−

N − 2 2

) .

Given the first eigenfunction with Dirichlet boundary conditions on ω ⊂ ΣN−1, the characteristic value

β =

(√(N − 2 2

)2 + λ1(ω)−

N − 2 2

) ,

is the exponent needed to have

∆(rβϕ(θ)) = 0.

on the solid angle spanned by ω.

15 The value β(N, k)

Our next objective consists in developing a variant of this formula for the case of many subharmonic densities having mutually disjoint supports. Recall the optimal partition value involved in the generalizations of the monotonicity lemma

β(k,N) := inf P(k,N)

2

k

k∑ i=1

(√(N − 2 2

)2 + λ1(ωi)−

N − 2 2

) ,

where the minimization is taken over all possible partitions in k disjoint parts of the unit sphere SN−1.

In dimension N = 2 the harmonic weight is not involved: thus one can easily compute:

β(k, 2) = k.

Moreover, in any dimension, when there are only two parts, the optimal partition is achieved by the equator–cut sphere, in other words: λ1(ωi) = N − 1. In this case

β(2, N) = 2.

Finally we can prove that β(k,N) > 2 for k ≥ 3.

16 A perturbed monotonicity lemma

Consider the reaction-diffusion system with Lotka-Volterra interactions:

 −∆ui(x)