Entire solutions for competition-diffusion systems and optimalpartitions

Susanna Terracini

Dipartimento di Matematica e Applicazioni

Universita di Milano Bicocca

GNAMPA School

“DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS”

June 11-15 2012, Gaeta

1 Competition diffusion systems with Lotka-Volterra interactions: symmetriccompetition rates

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

−∆ui = fi(ui)− uih∑j=1j 6=i

βi,juj in Ω,

2 Competition diffusion systems with Lotka-Volterra interactions : asymmetriccompetition rates

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

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

E(ψ1, · · · , ψh) =∫ ∑h

i12|∇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))

whereBh = (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

2d 2i (x) |∇ui(x)|2 − Fi(x, ui(x))

)dx

,

Our first goal is to minimize J among a class of segregated states subject tosome 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 ≥ s >> 1,∫Ω

(d2i (x)|∇w(x)|2 − bi(x)w2(x)

)dx > 0 ∀w ∈ H1

0(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 minimizersof 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

ui = ui −∑h6=i

uh , vi = vi −∑h6=i

vh

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

w(λ)i = [λui + (1− λ)vi]

+ =⇒ 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 : wλ

i (x) > 0.

Of course we have

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

9 Extremality Conditions

Again denote

ui = ui −∑h6=i

uh

and similarly

f (x, ui) =∑j

fj(x, ui)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,

−∆ui ≥ f (x, ui)

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 localstructure of the interface is that of the nodal set of a solution to an elliptic divergence–type equationof 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

−∆ui ≥ f (x, ui),∀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

limy→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,−∆ui ≥ −M

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

.... and global Lipschitz continuity

Theorem 4Let ∂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−2dx

is non decreasing.

As a consequence, if the gradient of one density is not bounded at one point of multiplicity, theother 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= jand x0 ∈ ∂(supp (wi)), i = 1, 2. Then, for every k ≥ 3 and N ≥ 2, there exists β(k,N) > 2such that the function

Φ(r) =

k∏i=1

1

rβ(k,N)

∫B(x0,r)

|∇wi(x)|2

|x− x0|N−2dx

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 equalhalf–spheres minimizes, over all possible partitions in two disjoint ωi (i = 1, 2) of the unit sphereSN−1, the sum of the characteristic values

β(2, N) := infP(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 characteristicvalue

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 subharmonicdensities having mutually disjoint supports. Recall the optimal partition value involved in thegeneralizations of the monotonicity lemma

β(k,N) := infP(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 sphereSN−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 bythe equator–cut sphere, in other words: λ1(ωi) = N − 1. In this case

β(2, N) = 2.

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

16 A perturbed monotonicity lemma

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

−∆ui(x) = −ui(x)∑j 6=i

aijuj(x) x ∈ RN

ui(x) ≥ 0 x ∈ RN

Letf be a smooth function f (r) : [0,∞) → (0,∞) such that f (r) = 1/rN−2 when r > 1 and∆f (|x|) := 2m(|x|) is bounded and vanishes outside the ball of radius 1. Now we are ready toprove

Lemma 3 (Conti-T-Verzini ’05) Let N ≥ 2 and let (u1, . . . , uk) be a solution of the systemsuch that ui > 0 for all i. Let h ≤ k be any integer, let h′ < β(h,N) and define

Φ(r) =

h∏i=1

1

rh′Φi(r)

where

Φi(r) =

∫B(0,r)

[f (|x|)

(|∇ui(x)|2 + u2

i (x)∑

1≤j≤kj 6=i

aijuj(x))−m(|x|)u2

i (x)]dx .

Then there exists r′ = r(h′) > 1 such that Φi > 0 and Φ is an increasing function in [r′,∞).

17 A Liouville type theorem

As a consequence of the perturbed monotonicity formula we have a Liouville–type theorem forcompetition diffusion systems:

Theorem 5 (Conti-T-Verzini ’05) Let k ≥ 2 and let U = (u1, . . . , uk) be a solution of

−∆ui(x) = −ui(x)∑j 6=i

aijuj(x) x ∈ RN

ui(x) ≥ 0 x ∈ RN

for every i. Let α ∈ (0, β(k,N)/2) such that

maxi=1,...,k

supx∈RN

|ui(x)|1 + |x|α

<∞.

Then at least one component vanishes identically. If α < 1 then all components (but possiblyone) vanish.

18 Uniform bounds in Holder spaces

Consider strongly competing systems with either of the Lotka-Volterra or gradient type interactions

−∆ui(x) = f (ui)− βui(x)∑j 6=i

aijuj(x)

Theorem 6 (Conti-T-Verzini ’05) Let Uβ be a family of H1–bounded solutions. Thereexists Lα > 0 such that

supx,y∈Ω

ui,β(x)− ui,β(y)

|x− y|α< Lα

for all i = 1, . . . , h and for all β > 0. Moreover, the limiting profiles are Lipschitz continuous.

Ingredients of the proof. Assuming the contrary, after scaling and performing the blow–up analysis,after a lot of work we would arrive to the existence of an entire solution of

−∆ui(x) = −ui(x)∑j 6=i

aijuj(x)

satisfying a global bound in Holder norm, in contradiction with suitable perturbed monitonicityformulæ underlying the rules of the minimal spacial growth for competition systems.

19 A class of competition–diffusion systems

Consider a system of k competing densities:−∆ui(x) = −βui(x)

∑j 6=i

uj(x) + fi(x, ui) x ∈ Ω

ui(x) = ϕi(x) x ∈ ∂Ωui(x) > 0 x ∈ Ω

Question: asymptotics as β →∞.

Theorem 7 Let Uβ = (u1,β, ..., uk,β) be a solution of the system at fixed β. Let β →∞: then,there exists U such that, for all i = 1, . . . , k:

(1) up to subsequences, ui,β → ui strongly in H1

(2) if i 6= j then ui · uj = 0 a.e. in Ω

(3) −∆ui ≥ f (x, ui).

The class S is the limiting class of the solutions of competition–diffusion systems, as the inter-specific competition rates tends to infinity

20 Asymptotic estimates for the two–density problem−∆u1(x) = −βu1(x)u2(x) x ∈ Ω−∆u2(x) = −βu1(x)u2(x) x ∈ Ω

Subtracting the two equations, we obtain u1 − u2 = Φ, where Φ is an harmonic function on RN .With this notation, writing u = u1, the system becomes equivalent to

−∆u = −βu(u− Φ), u > Φ+ on Ω.

Theorem 8 (CTV ’05)For every positive β the elliptic equation has a unique solution uβwith

‖uβ − Φ+‖H1(Ω) = O(β−1/6) as β →∞.

Theorem 9 (CTV ’05)If the boundary data are lipschitz continuous, then uββ is equi–lipschitz.

Related results by Dancer–Du, Dancer–Hilhorst–Mimura & Peletier.

21 Asymmetric interspecific competition rates I−∆ui(x) = −βui(x)

∑j 6=i

aijuj(x) + fi(x, ui) x ∈ Ω

ui(x) = ϕi(x) x ∈ ∂Ωui(x) > 0 x ∈ Ω

Passing to the limit as β →∞ we find a new class S:

Define, for every i = 1, . . . , k,

ui := (ui −∑j 6=i

aijajiuj)

fi(x, s) = fi(x, s)χsupp (ui) −∑j 6=i

aijajifj(x,

ajiaijs)χsupp (uj)

then the limiting differential inequalities takes again the usual form

−∆ui ≥ fi(x, ui) in Ω.

22 Asymmetric interspecific competition rates II

What doesn’t changes:

• lipschitz continuity (local and global)

• equi–holderianity w.r. to β

• equi–lipschitzianity w.r. to β

• vanishing of the gradient at multiple points

What changes:

• local expansion at multiple points (in dimension N = 2):

Conjecture 1 Let N = 2. Let x0 ∈ Ω with m(x0) = h ≥ 3 that is isolated in Z3. Then thereexists α ∈ R and θ0 such that

U(r, θ) = rh2 exp (αθ)| cos(

h

2θ − α log r + θ0)))| + o(r

h2 )

as r → 0, where (r, θ) denotes a system of polar coordinates about x0 and U is a suitablyweighted sum of the components ui.

23 Some references

• H. W. Alt, L. Caffarelli and A. Friedman, Variational problem with two phases and theirfree boundaries, Trans. Amer. Math. Soc., vol.282(1984), pp. 431-461.

• Caffarelli, L. A.; Karakhanyan, A. L.; Lin, Fang-Hua, The geometry of solutions to a segre-gation problem for nondivergence systems. J. Fixed Point Theory Appl. 5 (2009), no. 2,319351.

• L. Caffarelli, F.-H- Lin, Singularly perturbed elliptic systems and multi-valued harmonic func-tions with free boundaries, J. Amer. Math. Soc. (2008)

• M. Conti, S. Terracini, G. Verzini, A variational problem for the spatial segregation ofreaction–diffusion systems, Indiana Univ. Math. J. 54, 3 (2005) 779–815

• M. Conti, S. Terracini, G. Verzini, Asymptotic estimates for the spatial segregation of com-petitive systems, Adv. Math. 195 (2005), no. 2, 524–560.

• M. Conti, S. Terracini, G. Verzini, On a class of optimal partition problems related to theFucık spectrum and to the monotonicity formulae, Calc. Var. Partial Differential Equations22, 1 (2005) 45–72

• M. Conti, S. Terracini, G. Verzini, Uniqueness and Least Energy Property for Solutions toStrongly Compeitng Systems, Interfaces and Free Boundaries, 8 (2006), 437–446

• E.N. Dancer and Z. Zhang, Dynamics of Lotka-Volterra competition systems with largeinteraction, Journal of Differential Equations, 182(2) (2002), 470- 489.

• E.N. Dancer, K. Wang and Z. Zhang, Uniform Holder estimate for singularly perturbedparabolic systems of Bose-Einstein Condensates and competing species, Journal of Differ-ential Equations, 251 (2011) 2737-2769.

• E.N. Dancer, K. Wang and Z. Zhang, Dynamics of Strongly Competing Systems with ManySpecies, Transactions of the American Mathematical Society, 364 (2012) 961-1005.

• E.N. Dancer, K. Wang and Z. Zhang, The limit equation for the Gross-Pitaevskii equationsand S. Terracini’s conjecture, Journal of Functional Analysis, 262 (2012), no. 2, 1087-1131

24 Uniform bounds in Holder spaces

Consider strongly competing systems with either of the Lotka-Volterra or gradient type interactions

−∆ui(x) = f (ui)− βui(x)∑j 6=i

u2j(x)

Theorem 10 (Conti-T-Verzini ’05, Noris-Tavares-T-Verzini ’10) Let Uβ be a familyof H1–bounded solutions. There exists Lα > 0 such that

supx,y∈Ω

ui,β(x)− ui,β(y)

|x− y|α< Lα

for all i = 1, . . . , h and for all β > 0. Moreover, the limiting profiles are Lipschitz continuous.

Ingredients of the proof. Assuming the contrary, after scaling and performing the blow–up analysis,after a lot of work we would arrive to the existence of an entire solution of

−∆ui(x) = −ui(x)∑j 6=i

u2j(x)

satisfying a global bound in Holder norm, in contradiction with suitable perturbed monitonicityformulæ underlying the rules of the minimal spacial growth for competition systems.

25 Liouville type theorems

The proof of Theorem 1 rests upon a blow–up argument and the following non existence theorem:

Theorem 11 (Conti-T-Verzini ’06, Noris-Tavares-T-Verzini ’10) Let k ≥ 2, aij > 0,and let U = (u1, . . . , uk) be a solution of

−∆ui(x) = −ui(x)∑j 6=i

aiju2j(x) x ∈ RN

ui(x) ≥ 0 x ∈ RN

for every i. Assume that, for some α ∈ (0, 1), there holds

maxi=1,...,k

supx∈RN

|ui(x)|1 + |x|α

<∞.

Then all components (but possibly one) vanish.

26 Classification of entire solutions

We understand that a key step of the theory is the classification of the solutions of the system, withrespect to their spatial growth. To this aim, we have at our disposal two major tools

The perturbed Alt-Caffarelli-Friedman monotonicity formula (for Lotka-Volterra and all othertypes of interaction): let

Φ(r) :=1

rα(k)−ε

∏i

∫Br(0)

K(x)

|∇ui|2 +∑j 6=i

aiju2iu

2j

,

where K(x) = |x|2−N for |x| ≥ 1. The exponent α(k) ≥ 2 depends on a spectral optimalpartition problem on the sphere. Then Φ is increasing.

The Almgren monotonicity formula (only for gradient systems): let

N(r) :=r∫Br(0)

∑i |∇ui|2 +

∑j 6=i u

2iu

2j∫

∂Br(0)

∑i u

2i

Then N is increasing. Moreover,

limr→+∞

N(r) ≥ 1

27 Two components in one space dimension

In order to understand the interplay of two neighboring components, we have to deal with thesolutions to the system

∆u = uv2

∆v = vu2 ,

u, v > 0 in RN

Of course, there are one variable solutions (depending on the energy h > 0):

u′′ = uv2

v′′ = vu2 ,

|u′|2 + |v′|2 − u2v2 = h

u(x) = v(−x) , u, v > 0 in R.

All these solution have the lowest possible growth

limr→+∞

N(r) = 1

28 Uniqueness of entire solutions in one space dimension

Theorem 12 (Berestycki-Lin-Wei-Zhao ’09, Berestycki-T-Wang-Wei ’12)) Up to trans-lations and reflections, there is only one one-parameter family of solutions to

u′′

= uv2,

v′′

= vu2,

u, v > 0 in R

For this family we have v(t∗ − t) = u(t∗ + t). In addition they are all stable.

Proof

Normalize the energy to one;

observe that u and v are positive convex functions;

assume, for instance, that u is increasing and v is decreasing. Moreover u decays superexponen-tially at −∞, v at +∞as x→ +∞, |u′(x)− 1| decays exponentially. This implies the existence of a positive constantA such that

|u(x)− x+| + |v(x)− x−| ≤ A.

29 The sliding method

In addition, the limits

limx→+∞

(u(x)− x+), limx→−∞

(v(x)− x−)

exist. So there are asymptotes at infinityfor both u and v

Now assume (u1, v1) and (u2, v2) are two entire solutions of the problem with the same unitaryenergy. For t > 0, denote

u1,t(x) := u1(x + t), v1,t(x) := v1(x + t).

Step 1. Thanks of the decay estimates, there exists t such that for all t ≥ t,

u1,t(x) ≥ u2(x), v1,t(x) ≤ v2(x) in R. (Sl)

Step 2. Decrease the t to an optimal value when (Sl) holds

t0 = inft′| such that (Sl) holds for all t ≥ t′.

Note that there holds −(u1,t0 − u2)

′′+ v2

1,t0(u1,t0 − u2) ≥ 0

−(v2 − v1,t0)′′

+ u21,t0

(v2 − v1,t0) ≤ 0

Hence, by the strong maximum principle,

either u1,t0(x) ≡ u2(x), v1,t0(x) ≡ v2(x) in Ror u1,t0(x) > u2(x), v1,t0(x) < v2(x) in R

Step 3. We show, using again some form of the maximum principle, that the second case cannot occur. This is done in two steps:

if the asymptotes do not coincide, then we work essentially on a bounded domain and we canuse the usual continuation argument to obtain a contradiction;

if some of the asymptotes coincides, the extremality of t0 implies that, for t0 − ε there is anegative part of either u1,t0−ε − u2 or −v1,t0−ε + v2. This part must be placed close to infinity.Using as a weight function g(x) = log(x2 + 1/2) we reach again a contradiction.

30 De Giorgi type conjecture. partial results in two dimensions

Theorem 13 (Berestycki-Lin-Wei-Zhao ’09(+BTWW’12)) Let (u, v) a solution to

∆u = uv2 , ∆v = vu2 , u, v > 0 in R2

such that

u(x) + v(x) ≤ C(1 + |x|).

and which is monotone in one direction. Then (u, v) is one dimensional , (i.e., there existsa ∈ R2, |a| = 1, b ∈ R such that (u, v) = (u0(a · x − b), v0(a · x − b)) where (u0, v0) is theone-dimensional solution).

A quite standard argument shows that monotone =⇒ stable. Recall that a stable solution (u, v)is such that the linearization is weakly positive definite. That is, it satisfies

∫Rn

[∇ϕ|2 + |∇ψ|2 + v2ϕ2 + u2ψ2 + 4uvϕψ] ≥ 0, ∀ϕ, ψ ∈ C∞0 (Rn).

31 Planar stable solutions and a theorem by Kelei Wang

Theorem 14 (Berestycki-T-Wang-Wei ’12) Let (u, v) be a stable solution to the systemin R2 of

∆u = uv2 , ∆v = vu2 , u, v > 0 in R2

such that

u(x) + v(x) ≤ C(1 + |x|).

Then (u, v) is one-dimensional, (i.e., there exists a ∈ R2, |a| = 1, b ∈ R such that (u, v) =(u0(a · x− b), v0(a · x− b)) where (u0, v0) is the one-dimensional solution)

Theorem 15 (Kelei Wang, preprint 2012) In any space dimensions, let (u, v) a solutionhaving at most linear growth and which is minimal (in the sense of Morse): the energy isminimized with respect to compact support variations. Then (u, v) is one dimensional .

32 Solutions with polynomial growth

A natural question is whether or not all solutions to the system should satisfy the linear growth.For every integer d, there are solutions to the system with polynomial growth |x|d. To describe thebehavior at infinity, let us consider the harmonic polynomial Φ of degree d as

Φ := Re(zd).

Note that Φ has some dihedral symmetry; indeed, let us take its d nodal lines L1, · · · , Ld anddenote the corresponding reflection with respect to these lines as T1, · · · , Td: then there holdsΦ(Tiz) = −Φ(z).

Theorem 16 (B-T-W-W’12) For each positive integer d ≥ 1, there exists a solution (u, v)to the system, satisfying

(1) u− v > 0 in Φ > 0 and u− v < 0 in Φ < 0;(2) u ≥ Φ+ and v ≥ Φ−;

(3) ∀i = 1, · · · , d, u(Tiz) = v(z);

(4) ∀r > 0, the Almgren frequency function satisfies

N(r) :=r∫Br(0) |∇u|

2 + |∇v|2 + u2v2∫∂Br(0) u

2 + v2≤ d = lim

r→+∞N(r);

33 Asymptotics at infinity

We consider the blow-down sequence

(uR(x), vR(x)) := (1

L(R)u(Rx)

1

L(R)v(Rx)),

where u(0) = v(0) and L(R) is chosen so that∫∂B1(0)

u2R + v2

R = 1.

We have the following

Theorem 17 Let (u, v) be a solution of the system such that d := limr→+∞

N(r) < +∞. As

R→∞, (uR, vR) defined above (up to a subsequence) converges to (Ψ+,Ψ−) uniformly on anycompact set of RN . Here Ψ is a homogeneous harmonic polynomial of degree d. If d = 1 then(u, v) is asymptotically linear at infinity.

In particular this applies to the solutions found by Theorem 6 to give

Corollary 1 Let (u, v) be a solution of the system given by Theorem 6: then

(uR(x), vR(x)) := (1

Rdu(Rx)

1

Rdv(Rx))

converges uniformly on compact subsets of R2 to a multiple of (Φ+,Φ−), where Φ := Re(zd).

34 Systems with many components

Theorem 18 (Berestycki-T-Wang-Wei ’12) There exists a positive solution to the system

∆ui = ui

k∑j 6=i,j=1

u2j , in C, i = 1, . . . , k

having the following symmetries (here z is the complex conjugate of z)

ui(z) = ui(Ghz), on C , i = 1, . . . , k

ui(z) = ui+1(Gz), on C , i = 1, . . . , k

uk+1(z) = u1(z), on Cuk+2−i(z) = ui(z), on C , i = 1, . . . , k

such that

limr→∞

1

r1+2d

∫∂Br(0)

k∑1

u2i = b ∈ (0,+∞) ;

and

limr→∞

r∫Br(0)

∑k1 |∇ui|2 +

∑i<j u

2iu

2j∫

∂Br(0)

∑k1 u

2i

= d .

35 Limiting profiles

As the interspecific competition rate β = minij βij tends to infinity we find a vectorU = (u1, · · · , uh)of functions

having mutually disjoint supports: ui · uj ≡ 0 in Ω for i 6= j,

satisfying

−∆ui = fi(x, ui) whenever ui 6= 0 , i = 1, . . . , h,

Questions:

Uniqueness vs multiplicity

Extremality conditions at the interface of the nodal components

Regularity

of the minimizersof the interfaces

36 Segregated critical configuration

Let Ω be an open bounded subset of RN , with N ≥ 2. Let U = (u1, . . . , uh) ∈ (H1(Ω))h be avector of (real, complex, vector-valued)

nontrivial Lipschitz functions in Ω,

having mutually disjoint supports: ui · uj ≡ 0 in Ω for i 6= j,

satisfying

−∆ui = fi(x, ui) whenever ui 6= 0 , i = 1, . . . , h,

where fi : Ω × R+ → R are C1 functions such that fi(x, s) = O(s) when s → 0, uniformly inx.

Our main interest is the study of the regularity of the nodal set of the segregated configurationsU = (u1, . . . , uh):

ΓU = x ∈ Ω : U(x) = 0

Obviously, without other conditions, there is no reason at all why the nodal set should be regular.We must add some information on the interaction between the components at the interface of theirsupports.

37 A weak reflection law

Theorem 19 (Tavares-T, 2010) Let us define, for every x0 ∈ Ω and r ∈ (0, dist(x0, ∂Ω))the energy

E(r) = E(x0, U, r) =1

rN−2

∫Br(x0)

|∇U |2 ,

then, E(x0, U, ·) is an absolutely continuous function of r, and we assume that it satisfies thefollowing differential equation

d

drE(x0, U, r) =

2

rN−2

∫∂Br(x0)

(∂νU)2 dσ+2

rN−1

∫Br(x0)

∑i

fi(x, ui)〈∇ui, x−x0〉.

Then, there exists a set ΣU ⊆ ΓU the regular part, relatively open in ΓU , such that

Hdim(ΓU \ ΣU) ≤ N − 2, and if N = 2 then actually ΓU \ ΣU is a locally finite set;

ΣU is a collection of hyper-surfaces of class C1,α (for every 0 < α < 1). Furthermore forevery x0 ∈ ΣU

limx→x+

0

|∇U(x)| = limx→x−0

|∇U(x)| 6= 0,

where the limits as x → x±0 are taken from the opposite sides of the hyper-surface. Further-more, if N = 2 then ΣU consists in a locally finite collection of curves meeting with equalangles at singular points.

38 Some remarks

d

dr

1

rN−2

∫Br(x0)

|∇U |2 =2

rN−2

∫∂Br(x0)

(∂νU)2 dσ+2

rN−1

∫Br(x0)

∑i

fi(x, ui)〈∇ui, x−x0〉. (WRL)

It is easy to check that equation (WRL) always holds for balls lying entirely inside one of thecomponent supports, as a consequence of the elliptic equation. Hence, for our class systems,it represents the only interaction between the different components ui through the commonboundary of their supports;

(WRL) is satisfied by the nodal components of solutions to a single semilinear elliptic equationof the form −∆u = f (u).

equation (WRL) can be seen as a weak form of a reflection property through the interfaces.Consider the following example: take two linear functions on complementary half-spaces:

u1(x) = a1x+1 u2(x) = a2x

−1 .

Then

(WRL) ⇐⇒ |a1| = |a2| .

More in general, when we have two components with a smooth interface between the supports,then

(WRL) ⇐⇒ limx→x+

0

|∇U(x)| = limx→x−0

|∇U(x)| .

(WRL) as an extremality condition

Although this hypothesis may look weird and may seem hard to check in applications, it occursnaturally in many situations where the vector U appears as a limit configuration in problems ofspatial segregation.

It has to be noted indeed that a form of (WRL) always holds for solutions of systems of interactingsemilinear equations and that it persists under strong H1 limits.

In addition, (WRL) holds for vector functions U minimizing Lagrangian functional associatedwith the system.

It is fullfilled also for strong limits to competition–diffusion systems, both those possessing avariational structure and those with Lotka-Volterra type interaction.

Our theorem extends also to sign changing, complex and vector valued functions ui. Lipschitzcontinuity can be weakened into Holder continuity for every α ∈ (0, 1)].

39 Domain variations and (WRL)

This was brought to my attention by Kelei Wang. Assume U minimizes a Lagrangian energy witha pointwise constraint of the type U(x) ∈ Σ, for almost every x ∈ Ω. Let Y ∈ C∞0 (Ω;RN). Then,differentiation of the energy with respect to ε with U(x) 7→ Uε(x) = U(x + εY (x)) yields the wellknown identity∫

Ω

dY (x)∇U(x) · ∇U(x)− divY (x)

[1

2|∇U(x)|2 − F (U(x))

]dx = 0 ,∀Y ∈ C∞0 (Ω;RN) .

By localizing to a regular bounded ω ⊂ Ω this implies that, for every smooth ω and ∀ Y ∈C∞(Ω;RN)∫

ω

dY (x)∇U(x) · ∇U(x)− divY (x)

[1

2|∇U(x)|2 − F (U(x))

]dx

=

∫∂ω

Y (x) · ∇U(x)ν(x) · ∇U(x)− ν(x) · Y (x)

[1

2|∇U(x)|2 − F (U(x))

]dσ . (*)

Next

(∗) +

(Y (x) = x− x0

ω = Br(x0)

)=⇒ (WRL)

On the other hand, once our regularity theorem is proved, it implies that

(WRL) =⇒ (∗)

40 Almgren’s monotonicity formula

E(r) =1

rN−2

∫Br(x0)

(|∇U |2 − 〈F (x, U), U〉

),

H(r) =1

rN−1

∫∂Br

|U |2,

We define the Almgren’s quotient as follows:

N(r) =E(r)

H(r).

Theorem 20 Assume (WRL). Then, there exist r, C > 0, such that for 0 < r < r we haveH(r) 6= 0 and also E(r) ≥ 0,

N ′(r) ≥ −CrN(r)

in particular the function N = eC2 r

2N(r) is non decreasing and has a limit as r → 0, and

moreoverd

drlog(H(r)) =

2

rN(r).

41 Ideas of the proof.

We follow the same general strategy used by L. Caffarelli and F. Lin for energy minimizing vectorvalued segregated harmonic maps (Singularly perturbed elliptic systems and multi-valued har-monic functions with free boundaries, J. Amer. Math. Soc. (2008)). But now, our configurationneeds not to be (even locally) energy minimizing. Define the Almgen’s quotient as

N(x0, r) =

1rN−2

∫Br(x0)

(|∇U |2 − 〈F (x, U), U〉

)1

rN−1

∫∂Br|U |2

Define the regular and singular parts of the boundary

ΣU = x ∈ ΓU ; : limr→0

N(x, r) = 1 S(U) = ΓU \ ΣU = x ∈ ΓU : limr→0

N(x, r) > 1

First we wish to apply Federer’s reduction principle:

• Almgren monotonicity formula at nodal points

• bounds in Holder spaces =⇒ convergence of blow-up sequences

• classification of conic solutions satisfying (WRL)

Next we analyze the regular part of the nodal set:

• flatness at regular points of the boundary

• separation lemma

Reflection principle:

• Pohozaev identity =⇒ equality of the gradient norms on the two sides.

42 Federer’s reduction principle

Let F ⊆ (L∞loc(RN))h, and define, for any given U ∈ F , x0 ∈ RN and t > 0, the rescaled andtranslated function

Ux0,t := U(x0 + t·).We say that Un → U in F iff Un → U uniformly on every compact set of RN . Assume that Fsatisfies the following conditions:

(A1) (Closure under rescaling, translation and normalization) Given any |x0| ≤ 1 − t, 0 < t < 1,ρ > 0 and U ∈ F , we have that also ρ · Ux0,t ∈ F .

(A2) (Convergence of normalized “blow–up” to a conic element) Given |x0| < 1, tk ↓ 0 and U ∈ F ,there exists a sequence ρk ∈ (0,+∞), a real number α ≥ 0 and a function U ∈ F homogeneousof degree α such that, if we define Uk(x) = U(x0 + tkx)/ρk, then

Uk → U in F , .

(A3) (Singular Set hypotheses) There exists a map S : F → C (where C := A ⊂ RN : A ∩B1(0) is relatively closed in B1(0)) such that

(i) Given |x0| ≤ 1− t, 0 < t < 1 and ρ > 0, it holds

S(ρ · Ux0,t) = (S(U))x0,t :=S(U)− x0

t.

(ii) Given |x0| < 1, tk ↓ 0 and U, U ∈ F such that there exists ρk > 0 satisfying Uk :=ρkUx0,tk → U in F , the following “continuity” property holds:

∀ε > 0 ∃k(ε) > 0 : k ≥ k(ε)⇒ S(Uk) ∩B1(0) ⊆ x ∈ RN : dist(x,S(U)) < ε.

Theorem 21 Define

d = max

dim L : L is a vector subspace of RN and there exist U ∈ F and α ≥ 0

such that S(U) 6= ∅ and Uy,t = tαU ∀y ∈ L, t > 0 ,

Then

either S(U) ∩B1(0) = ∅ for every U ∈ F ,

or else Hdim(S(U) ∩B1(0)) ≤ d for every U ∈ F .

Moreover in the latter case there exist a function V ∈ F , a d-dimensional subspace L ≤ RN

and a real number α ≥ 0 such that

Vy,t = tαV ∀y ∈ L, t > 0, and S(V )∩B1(0) = L∩B1(0).

If d = 0 then S(U) ∩Bρ(0) is a finite set for each U ∈ F and 0 < ρ < 1.

43 Conic functions

Lemma 4 Let N ≥ 2. Given U = rαG(θ) ∈ Liploc(RN) such that ∆U = 0 in U > 0, and(WRP) holds, then either α = 1 or α ≥ 1 + δN for some universal constant δN depending onlyon the dimension. Moreover if α = 1 then ΓU is an hyperplane.

U = rα(g1(θ), . . . , γk(θ)). Note that for every connected component A ⊆ gi > 0 ⊂ SN−1 itholds

−∆SN−1gi = λgi in A, with λ = α(α + N − 2) and λ = λ1(A).

Lemma 5 If G > 0 has at least three connected components then there exists an universalconstant δN > 0 such that α ≥ 1 + δN .

Proof: At least one of the connected components, say C, must have a measure less that one thirdof the measure of the sphere, and hence λ = λ1(C) ≥ λ1(E(π/2)). Moreover it is well know thatλ1(E(π/2)) = N − 1. This implies the existence of γ > 0 such that λ1(E(π/3)) = N − 1 + γ, and

thus α =

√(N−2

2

)2+ λ− N−2

2 ≥ 1 + δN for some δN > 0.

Hence we are left with the case of two nodal connected components.

Use an inductive argument on the dimension, starting with dimension N = 2. Use uppersemicontinuity of the limit N(x, 0+) with respect to x. The value at the vertex of the cone mustbe larger than that at the conic surface.

44 Flatness at the regular points

Let x be a regular point:

Lemma 6 For any given 0 < δ < 1 there exists R > 0 such that for every x ∈ Γ?∩ Ω = ΓU ∩ Ωand 0 < r < R there exists an hyper-plane H = Hx,r containing x such that

dH(ΓU ∩Br(x), H ∩Br(x)) ≤ δr.

Proposition 2 (Local Separation Property) Given x0 ∈ Γ? there exists a radius R0 > 0such that BR0(x0) ∩ Γ? = BR0(x0) ∩ ΓU and BR0(x0) \ ΓU = BR0(x0) ∩ U > 0 has exactlytwo connected components Ω1,Ω2. Moreover, for sufficiently small δ > 0, we have that giveny ∈ ΓU ∩ BR0(x0) and 0 < r < R − |y| there exist a hyper-plane Hy,r (passing through y) anda unitary vector νy,r (orthogonal to Hy,r) such that

x+ tνy,r ∈ Br(y) : x ∈ Hy,r, t ≥ δr ⊂ Ω1, x− tνy,r ∈ Br(y) : x ∈ Hy,r, t ≥ δr ⊂ Ω2.

45 The reflection principle in action

Lemma 7 (Reflection Principle) Let u, v ∈ Liploc(RN) be two non zero and non negativefunctions in RN such that u · v = 0 and

−∆u = f (x, u)− λ−∆v = g(x, v)− µ in RN

for some λ, µ ∈ Mloc(RN), locally non negative Radon measures supported on the commonzero set. Suppose moreover that (WRP ) holds. Then

λ = µ

and in particular −∆(u−v) = f (x, u)−g(x, v) in RN . Moreover ΓU is a regular hyper-surfaceof codimension 1 at regular points, and Then for every Borel set E ⊆ RN it holds

λ(E) =

∫E∩∂u>0

−∂νu dσ =

∫E∩∂v>0

−∂νv dσ = µ(E)

In the complex or vector valued case, we obtain that ‖λ‖ = ‖µ‖. With this and an iterativeargument by Caffarelli we deduce the C1 regularity of the regular part of the boundary.

46 Elliptic systems on Riemannian Manifolds

The main theorem extends to segregated configurations associated with systems of semilinear el-liptic equations on Riemannian manifolds, under an appropriate version of the weak reflection law.Following N. Garofalo and F.-H. Lin Monotonicity properties of variational integrals, Ap

weights and unique continuation. Indiana Univ. Math. J. 35 (1986), we consider with a systemof semilinear equations involving the Laplace-Beltrami operator on a Riemannian manifold M :

−∆Mui = f (x, ui) where ui > 0 .

We define the “energy” E as

E(r) = E(x0, U, r) =1

rN−2

∫Br(x0)

|∇MU |2dVM ,

where Br(x0) is the geodesic ball of radius r. Let us choose normal coordinates xi centered at x0.By Gauss Lemma we know that, denoting by ρ =

∑i(x

i)2 and θi the radial and angular coordinates,it holds

g = dρ2 + ρ2∑i,j

bij(ρ, θ)dθidθj.

Notice that the variation with respect to the euclidean metric is purely tangential; moreover, asρ→ 0, the tangential part of the metric converges to the standard metric on the sphere. Moreoverthe Christoffel symbols vanish at the origin.

47 The reflection law

In normal coordinates, denoting, as usual, gij = g(∂i, ∂j) the coefficients of the metric, we requirethat E satisfies the differential equation:

d

drE(x0, U, r) =

2

rN−2

∫∂Br(x0)

(∂ρU)2 dσM

+2

rN−1

∫Br(x0)

ρ∑i

fi(x, ui)∂ρui +1√g

∑k,j

∂ρ

(√g gkj

)∂kui∂jui

dVM .

Here g = | det(gkj)| and (gkj) is the inverse of the matrix (gkj). This identity is satisfied also inthe case of Lipschitz metrics , by any solution u of the semilinear equation

−∆Mu = f (x, u).

48 Some references:

L. Caffarelli, F.-H- Lin, Analysis on the junctions of domain walls, Discrete Contin. Dyn.Syst. no. 3, 915929,

L. A. Caffarelli, A. L. Karakhanyan, and F. Lin, The geometry of solutions to asegregation problem for non-divergence systems, Journal of Fixed Point Theory and Applications(2009)

L. Caffarelli, F.-H- Lin, Singularly perturbed elliptic systems and multi-valued harmonicfunctions with free boundaries, J. Amer. Math. Soc. (2008)

S.M. Chang, C.S. Lin, T.C. Lin, and W.W. Lin: Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates. Phys. D 196, 341–361 (2004)

M. Conti, S. Terracini, G. Verzini, A variational problem for the spatial segregation ofreaction–diffusion systems, Indiana Univ. Math. J. 54, 3 (2005) 779–815

M. Conti, S. Terracini, G. Verzini, On a class of optimal partition problems related tothe Fucık spectrum and to the monotonicity formulae, Calc. Var. Partial Differential Equations 22,1 (2005) 45–72

M. Conti, S. Terracini, G. Verzini,Asymptotic estimates for the spatial segregation ofcompetitive systems, Adv. Math. 195 (2005), no. 2, 524-560.

M. Conti, S. Terracini, G. Verzini, Uniqueness and Least Energy Property for StronglyCompeting Systems, Interfaces and Free Boundaries, 8 (2006), 437-446.

E.N. Dancer and Z. Zhang, Dynamics of Lotka-Volterra competition systems with largeinteraction, Journal of Differential Equations, 182(2) (2002), 470- 489.

E.N. Dancer, K. Wang and Z. Zhang, Uniform Holder estimate for singularly perturbedparabolic systems of Bose-Einstein Condensates and competing species, Journal of DifferentialEquations, 251 (2011) 2737-2769.

49 More references

E.N. Dancer, K. Wang and Z. Zhang, Dynamics of Strongly Competing Systems withMany Species, Transactions of the American Mathematical Society, 364 (2012) 961-1005.E.N. Dancer, K. Wang and Z. Zhang, The limit equation for the Gross-Pitaevskii

equations and S. Terracini’s conjecture, Journal of Functional Analysis, 262 (2012), no. 2, 1087-1131

N. Garofalo and F.-H. Lin Monotonicity properties of variational integrals, Ap weights andunique continuation. Indiana Univ. Math. J. 35 (1986), no. 2, 245–268.

R. Hardt, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, N. Nadirashvili,Critical sets of solutions to elliptic equations. J. Differential Geom. 51, p. 359-373 (1999).

F. Lin, Nodal sets of solutions of elliptic and parabolic equations. Comm. Pure Appl. Math.44 (1991), no. 3, 287–308.

B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Holder bounds for nonlinearSchrodinger systems with strong competition Comm. Pure Appl. Math., 63 (2010), no.3, 267-302.

B. Noris, H. Tavares, S. Terracini and G. Verzini, Convergence of minimax andcontinuation of critical points for singularly perturbed systems, JEMS, to appear Available onlineat: http://arxiv.org/abs/0910.5317

H. Tavares and S. Terracini, Regularity of the nodal set of segregated critical configurationsunder a weak reflection law, Calc. var. PDEs, to appear

K. Wang, sl On the De Giorgi type conjecture for an elliptic system modeling phase separationpreprint 2012

50 Asymptotic limits of a system of Gross-Pitaevskii equations

Consider the following system of nonlinear Schrodinger equations

−∆ui + λiui = ωiu

3i − βui

∑j 6=i βiju

2j

ui ∈ H10(Ω), ui > 0 in Ω.

i = 1, . . . , h,

in a smooth bounded domain Ω ⊂ RN , N = 2, 3. Such type of systems arises in the theory ofBose-Einstein condensation in multiple spin states. Here we consider βij = βji 6= 0 (which gives avariational structure to the problem) and take λi, ωi ∈ R and β ∈ (0,+∞) large. The existence ofsolutions for β large is still an open problem for some choices of λi, ωi.

One of the many interesting questions about these systems is the asymptotic study of its solutionsas β → +∞ (which represents an increasing of the interspecies scattering length) and study of theregularity of the limiting profiles. From the uniform Holder bounds theorem we know that:

C0,α– bounds (for all 0 < α < 1) for any given L∞–bounded family of solutions Uβ =(u1,β, . . . , uh,β) of the system;

the possible limit configurations U = limβ→+∞Uβ are Lipschitz continuous.

As a byproduct, we have

Theorem 22 Let U be a limit as β → +∞ of a family Uβ of L∞–bounded solutions of thesystem. Then the conclusion of Theorem 19 holds.

All the required assumptions are satisfied for such limiting profiles, with fi(x, s) = fi(s) =ωis

3 − λis, except for the weak reflection law. The procedure to verify it is the following: definingan approximated energy associated with system - which has a variational structure-,

Eβ(r) =1

rN−2

∫Br(x0)

(|∇Uβ|2 − 〈F (Uβ), Uβ〉

)+

∫Br(x0)

2β∑i<j

u2i,βu

2j,β

by a direct calculation it holds

E ′β(r) =2

rN−2

∫∂Br(x0)

(∂νUβ)2 dσ +2

rN−1

∫Br(x0)

∑i

fi(ui,β)〈∇ui,β, x− x0〉+

+1

rN−1

∫Br(x0)

(N − 2)〈F (Uβ), Uβ〉 −1

rN−2

∫∂Br(x0)

〈F (Uβ), Uβ〉 dσ+

+4−NrN−1

∫Br(x0)

β∑i<j

u2i,βu

2j,β +

∫∂Br(x0)

β∑i<j

u2i,βu

2j,β dσ.

We know the following facts:

there holds strong convergence Uβ → U in H1 ∩ C0,α(Ω) for every 0 < α < 1,

and∫

Ω β∑

i<j u2i,βu

2j,β → 0.

Hence, as β → +∞, we prove that U satisfies the weak reflection law.

51 Lotka-Volterra competitive interactions with symmetric competition rates

Consider the following Lotka-Volterra model for the competition between h different species.

−∆ui = fi(ui)− βui

∑j 6=i ai,juj in Ω,

ui ≥ 0 in Ω, ui = ϕi on ∂Ω.

with Ω ⊂ RN a smooth bounded domain and ϕi positive W 1,∞(∂Ω)–functions with disjointsupports. We focus on the asymptotic study of solutions as β → +∞. It is not difficult to showthat all the possible H1–limits U of a given sequence of solutions Uββ>0 (as β → +∞) belong tothe class

S(Ω) =

(u1, . . . , uh) ∈(H1(Ω)

)h: ui ≥ 0 in Ω, ui · uj = 0 if i 6= j and −∆ui ≤ fi(ui),

−∆(ui −∑j 6=i

aijajiuj) ≥ fi(x, ui(x))−

∑j 6=i

aijajifj(x, uj)

.

Theorem 23 Let U ∈ S, then if aij = aji, ∀i, j the conclusion of Theorem 19 holds.

52 Regularity of interfaces in optimal partition problems related to the firsteigenvalue

Next we consider some optimal partition problems involving eigenvalues. For any integer h ≥ 0, wedefine the set of h–partitions of Ω as

Bh = (ω1, . . . , ωh) : ωi measurable , |ωi ∩ ωj| = 0 for i 6= j and ∪i ωi ⊆ Ω .Consider the following optimization problems: for any positive real number p ≥ 1,

Lh,p := infBh

(1

h

h∑i=1

(λ1(ωi))p

)1/p

,

and, for p = +∞ we find the limiting problem

Lh := infBh

maxi=1,...,h

(λ1(ωi)),

where λ1(ω) denotes the first eigenvalue of −∆ in H10(ω) in a generalized sense. We refer to the

papers Conti, Verzini, T. and Helffer, Hoffmann-Ostenhof, T., for a more detailed description ofthese problems.

Our theorem applies to suitable multiples of the eigenfunctions associated with the optimal par-tition. More precisely, we proved that

let p ∈ [1,+∞) and let (ω1, . . . , ωh) ∈ Bh be any minimal partition associated with Lh,p andlet (φi)i be any set of positive eigenfunctions normalized in L2 corresponding to (λ1(ωi))i. Thenthere exist ai > 0 such that the functions ui = aiφi verify in Ω, for every i = 1, . . . , h, thedifferential inequalities (in the distributional sense):

−∆ui ≤ λ1(ωi)ui and −∆(ui−∑

j 6=i uj) ≥ λ1(ωi)ui−∑

j 6=i λ1(ωi)uj;

and:

let (ω1, . . . , ωh) ∈ Bh be any minimal partition associated with Lh and let (φi)i be any set ofpositive eigenfunctions normalized in L2 corresponding to (λ1(ωi))i. Then there exist ai ≥ 0,not all vanishing, such that the functions ui = aiφi verify in Ω, for every i = 1, . . . , h, thedifferential inequalities (in the distributional sense):

−∆ui ≤ Lhui and −∆(ui−∑

j 6=i uj) ≥ Lh(ui−∑

j 6=i uj).

In particular the functions U = (u1, . . . , uh) and U = (u1, . . . , uh) belong to S(Ω). As conse-quence, we have the following result:

Theorem 24 Let (ω1, . . . , ωh) ∈ Bh be any minimal partition and let Γ be the union of theinterfaces; then the conclusion of Theorem 19 holds.

53 Extremality conditions for partitions involving higher eigenvalues

We would like to attack the optimal partition problem for higher eigenvalues (k ≥ 2):

L = min1

h

(h∑i=1

λk(ωi)

).

What are the extremality conditions? Hadamard domain variation requires simple eigenvalues.Introduce the penalized functional:

Eβ(u1, · · · , uh) =

∫Ω

∑i

|∇ui|2 + β∑i 6=j

|ui|2|uj|2

with constraints

∫Ω

|ui|2 = 1 ∀ i = 1, · · · , h .

As β +∞, critical points of Eβ converge to pairs of segregated eigenfunctions.

Problem: How to define an appropriate critical level for the penalized functional?

Existence of the minimal partition has been proved by Bucur–Buttazzo.

54 Eigenvales and eigenfunctions

Suppose that u is an eigenfunction of the Dirichlet realization of the Schrodinger operator H(Ω) =−∆ + V in Ω attached to the eigenvalue Λ :

−∆u + V (x)u = Λu .

Introduce the Rayleigh quotient:

R(u) =

∫Ω |∇u|

2 + V (x)u2∫Ω u

2

By adding a constant we can assume R > 0. We know that eigenvalues and eigenfunctions arecritical levels and points of R.

λ1 = minu∈H1

0(Ω)\0R(u)

55 Fisher-Courant Theorem and Courant nodal Theorem

λk = minEk⊂H

10(Ω)

dim(Ek)=k

maxu∈Ek\0

R(u)

From this variational characterization, it easily follows that if λ = λk is the kth eigenvalue thenthe number of nodal domains µ(u) of the corresponding eigenfunction is at most k.

Λ = λk =⇒ µ(u) ≤ k .

Definition 1 If Λ = λk and

µ(u) = k .

Then we say that u is Courant-sharp.

For one dimensional problems we infer from the Sturm oscillation principle that Courant sharpnessholds for every k.

56 Optimal partition problems related to eigenvalues

Next we consider some optimal partition problems involving eigenvalues. For any integer h ≥ 0, wedefine the set of h–partitions of Ω as

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

Consider the following optimization problems: for any positive real number p ≥ 1,

Lh,p := infBh

(1

h

h∑i=1

(λ1(ωi))p

)1/p

,

and, for p = +∞ we find the limiting problem

Lh := infBh

maxi=1,...,h

(λ1(ωi)),

where λ1(ω) = minR(u) : u ∈ H10(Ω) \ 0 , u ≡ 0 a.e. on Ω \ ω.

57 Spectral partitions

Consider an eigenfunction, and the associated nodal partition. how can we detect whether itpossesses any minimization property?

We easily understand that the most appropriate function of the eigenvalue to be considered is:

Lh := infBh

maxi=1,...,h

(λ1(ωi)),

Indeed, a straightforward consequence of the Fisher–Courant theorem are:

λ2 = L2 .

and, for every k ≥ 3,

λk ≤ Lk .

58 Oscillation principles and partitions

For one-dimensional problems we have, for every k,

λk = Lk .

On the other hand, in more space dimensions, the k-th eigenfunction may possess less than knodal domains.

Given k, we denote by Lk the smallest eigenvalue (if any) whose eigenspace contains an eigen-function with k nodal domains (Lk = +∞ if no such an eigenfunction exists).

In general, an easy consequence of the Courant nodal theorem for connected domains is that

λk ≤ Lk .

So we have the inequalities

λk ≤ Lk ≤ Lk , ∀k.

Can we characterize the equality cases?

Theorem 25 (Helffer, Hoffman-Ostenhof, T, 2009–2010) Suppose Ω ⊂ RN regular. Ifeither λk = Lk or Lk = Lk then

λk = Lk = Lk .

In addition, one can find in the eigenspace associated to λk an eigenfunction uk havingextactly k nodal domains.

The k–th eigenfunction has k nodal domains (i.e. is sharp with respect to the Courant nodalTheorem) if and only if the associated nodal k–partition is optimal.

As a consequence, every time we know (for instance for the symmetries of the problem) that thesecond eigenvalue is degenerate, then the minimal spectral 3-partition has necessarily a nontrivialclustering point.

59 Regularity of minimal partitions

Definition 2 We will say that a regular closed set has the equal angle crossing property if thehalf curves cross with equal angle at each critical point of N and also at the boundary togetherwith the boundary.

Theorem 26 In dimension N = 2, any k- minimal partition is regular and strong. Moreoverit satisfies the equal angle crossing property.

In general, there is no reason for a minimal partition to be unique. For example, a multiplicity ofminimizers exist in presence of symmetries. However, we will show that the uniqueness propertyalways holds for subpartitions of a given minimal partition. More precisely, we have

Theorem 27 Let D be a minimal k-partition relative to Lk(Ω). Let Ω′ ⊂ Ω be connected andD′ ⊂ D be any subpartition of D into 1 ≤ k′ < k elements such that

Ω′ =⋃ωi∈D′

ωi .

Then

Lk(Ω) = Lk′(Ω′)

and this last is uniquely achieved.

60 Graphs

We associate with the optimal partition the adjacency graph:

λk = Lk ⇐⇒ the graph is bipartite.

61 Pleijel result

A classical result of Pleijel (1956), implies (here V ≡ 0):

Theorem 28 The equality

λk = Lk

holds for finitely many k’s.

We have a universal lower bound for Lk.

Proposition 3 If Ω ⊂ R2, then

Lk(Ω) ≥ πJ2

|Ω|k ,

where |Ω| denotes the area of Ω and J is the smallest positive 0 of the Bessel function J0:J ∼ 2.4048 . . . .

The proof is actually a side result of the proof by Pleijel. The main point is that for any domainω, one has, as a consequence of the Faber-Krahn Inequality, for the Dirichlet Laplacian :

λ1(ω) ≥λ1(B1/

√π)

|ω|.

The lowest eigenvalue for the disk of radius 1/√π is known to be :

λ1(B1/√π) = πJ2 ,

and therefore:

λ1(ω) ≥ J2

|ω|.

If u is a minimizer of Lk then we have for any of these nodal domains ωi :

λ1(ωi) ≥πJ2

|ωi|That we rewrite in the form

|ωi| ≥πJ2

Lk.

Summing over i, we get

|Ω| ≥ kπJ2

Lk.

So we have shown that for the minimizer corresponding k nodal domains we have

Lk ≥ kπJ2

|Ω|.

While the Weyl formula says that

λk ∼ k4π

|Ω|, as k → +∞. .

Note that the fundamental point for Pleijel’s argument is that

λ(B1) > 4 .

62 Limiting behavior for many components

Proposition 4 In the case of the Laplacian and if Ω is regular, we have

lim supk→+∞

Lk(Ω)

k≤ λ1(Hx1)/|Ω| ,

where Hx1 is the regular hexagon of area 1.

For the proof, we just use a (non strong) partition of Ω by equal hexagons of area |Ω|k . We conjecturethat

Lk(Ω)

k' λ1(Hx1)/|Ω| , as k → +∞. .

63 The case of the sphere

We consider the Laplace-Beltrami operator on the two-sphere.

Conjecture 2 (Bishop 1992) The minimal 3-partition for 13(∑3

i=1 λ1(Di)) corresponds tothe Y-partition, whose boundary is given by the intersection of S2 with the three half-planesdefined respectively by φ = 0, 2π

3 ,−2π

3

The conjecture can be restated as

L3,1(S2) =15

4

and also

L3,p(S2) = L3(S2) =15

4,∀p

Bishop’s Conjecture was motivated by the analysis of the properties of harmonic functions inconic sets. A reference paper in this context is that by Friedland-Hayman. It is proved there thatthe optimal two-partition is achieved by the two half spheres.

64 Uniqueness for L3 in two dimensions

Theorem 29 (Helffer, Hoffman-Ostenhof, T) Any minimal spectral 3-partition of S2 is(up to a rotation) obtained by the Y-partition. Hence

L3(S2) =15

4.

Consider a homogeneous function in R3 of the form

u(x) = rαg(θ, φ)

which is harmonic outside its nodal set: −∆u = 0 , u > 0 and such that the nodal set dividesthe sphere in three parts, then

α(α + 1) ≥ L3(S2) .

Hence our theorem implies that α ≥ 3/2.

65 Ideas of the proof:

first, minimal partitions on S2 in three parts exist and share the same properties as for planardomains: regularity and equal angle meeting property. Hence the nodal set is a finite union ofarcs.

because the second eigenvalue of the Laplace-Beltrami operator is singular (has nontrivial mul-tiplicity), then the minimal 3-partition cannot be a nodal partition.

use Euler’s formula and deduce that the nodal line s of a minimal 3-partitions consists exactlytwo points x1 and x2 and three arcs joining these two points.

use Borsuk (or Ljusternik-Schnirelman) theorem to prove that the nodal set contains a pair ofantipodal points.

the next point is that any minimal 3-partition which contains two antipodal points in its bound-ary can be lifted to a symmetric 6-partition on the double covering S2

C.

finally, the last point is to show that on the double covering a minimal symmetric 6-partition isnecessarily the lifting on the double covering of the Y-partition.

– use the knowledge of the spectrum of the Laplace-Beltrami on the double covering and classifyall odd and even spectrum.

– use again the characterization of the eigenvalues whose nodal partition is minimal: this holdsif and only the minimal eigenvalue whose nodal partition has k nodal domain is the minimalone,

66 References

B. Helffer, T. Hoffmann-Ostenhof and S. Terracini, Nodal domains and spectralminimal partitions Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 101–138.

B. Helffer, T. Hoffmann-Ostenhof and S. Terracini, On Spectral Minimal Partitions:the Case of the Sphere, to appear in Around the Research of Vladimir Mazya III. Analysis andApplications Ari Laptev (Ed.) International Mathematical Series, Vol. 13, Springer, 2010

B. Helffer, T. Hoffmann-Ostenhof and S. Terracini, Nodal minimal partitions indimension 3. Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 617–635.