SPECIAL SESSION 29 129

Special Session 29: Self-Organized Behavior of Nonlinear EllipticEquations and Pattern Formation of Strongly Interacting Systems

Susanna Terracini, University of Milano-Bicocca, ItalyJun-cheng Wei, Chinese University of Hong Kong, Hong Kong

Several phenomena can be described by a number of densities (of mass, population, probability, ...) dis-tributed in a domain and subject to di↵usion, reaction, and either cooperative or competitive interaction.Whenever the interaction is the prevailing mechanism, one can reasonably expect pattern formation and, inthe competitive case, that the several densities can not coexist and tend to segregate hence determining apartition of the domain.

We focus on the analysis of the qualitative properties of solutions of systems of semilinear elliptic equations,whenever the interaction is not negligible with respect to di↵usion (e.g. when the parameter describing thecompetition diverges to infinity). Prototype of variational systems are those of Gross-Pitaevski equations,describing the motion of a Bose-Einstein condensated quantum gas with di↵erent spin states. Strong com-petition appears in association with diverging interspecific scattering lengths. The limiting behaviour isknown for the ground state stationary solutions and solitary waves: the wave amplitudes segregate, that is,their supports tend to be disjoint (phase separation). Of course, the partition becomes the main object ofinvestigation both from the analytical point of view than from the geometric, with emphasis on the mul-tiple points of multiple intersection. When the system possesses a variational structure, one can associatean optimal partition problem with the ground states. Conversely, one can regard at optimal partitions re-lated to linear or nonlinear eigenvalues as limits of competing systems as the competition parameter diverges.

A key role is played by the entire solutions to systems, with polynomial interactions. In the case of an eigen-value, the variational properties of the nodal partition associated with an eigenfunction is deeply connectedto the number of nodal domains it is possible to link the minimizing property of the nodal partition to thatof being sharp with respect to the Courant’s nodal Theorem.

On a prescribed mean curvature equation inmodeling MEMS

Nicholas BrubakerUniversity of Delaware, USAbrubaker@math.udel.eduAlan E. Lindsay, John A. Pelesko

Mathematical models of the form Mu = f(u;�),where M is the nonparametric mean curvature oper-ator and � is a nonlinear eigenvalue, have been stud-ied since the early 1800s, beginning with the work ofThomas Young and Pierre-Simon Laplace. Recently,it has been shown that when f(u;�) = �(1 + u)�2

the aforementioned model describes the shape of anelectrostatically deflected thin, elastic membrane. Inthis talk, we look at the contrast between this par-ticular prescribed mean curvature equation and itsstandard approximation �u = f(u).

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Solutions with point singularities for a MEMSequation with fringing field

Juan DavilaUniversidad de Chile, Chilejdavila@dim.uchile.cl

We construct solutions of the equation

��u =�(1 + |ru|2)

(1� u)2, 0

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Singularity of eigenfunctions at the junctionof shrinking tubes

Veronica FelliUniversity of Milano-Bicocca, Italyveronica.felli@unimib.itSusanna Terracini

Consider two domains connected by a thin tube:it can be shown that, generically, the mass of a giveneigenfunction of the Dirichlet Laplacian concentratesin only one of them. The restriction to the other do-main, when suitably normalized, develops a singular-ity at the junction of the tube, as the channel sectiontends to zero. Our main result states that, undera nondegeneracy condition, the normalized limitingprofile has a singularity of order N � 1, where N isthe space dimension. We give a precise descriptionof the asymptotic behavior of eigenfunctions at thesingular junction, which provides us with some im-portant information about its sign near the tunnelentrance. More precisely, the solution is shown to beone-sign in a neighborhood of the singular junction.In other words, we prove that the nodal set does notenter inside the channel.

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130 9th AIMS CONFERENCE – ABSTRACTS

Solutions for a semilinear elliptic equationsinvolving critical exponents.

Ignacio GuerraUniversidad de Santiago de Chile, Chileignacio.guerra@usach.clJuncheng Wei

Let ⌦ be a bounded domain in RN . We considerthe problem

��u = up + �uq, u > 0 in ⌦

u = 0 in @⌦,

where p = (N + 2)/(N � 2). We describe large so-lutions for this problem, and we obtain precise esti-mates for � = �(c) where c := max

⌦u and tends to

infinity. This depends strongly in the dimension Nand the value of q. The case q = 1 is the well knowBrezis-Nirenberg problem. We also study a charac-terization of the point in ⌦ where solution concen-trates as c!1.

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Hot-spot solutions in a model of crime

Theodore KolokolnikovDalhousie, Canadatkolokol@gmail.comMichael Ward, Juncheng Wei

We study a model of crime developed recently byShort and collaborators at UCLA. This model ex-hibits hot-spots of crime – localized areas of highcriminal activity. In a certain asymptotic limit, weuse singular perturbation theory to construct theprofile of these hot-spots and then study their sta-bility.

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Concentration behaviour in biharmonic equa-tions of MEMS.

Alan LindsayUniversity of Arizona, USAalindsay@math.arizona.eduJ. Lega, M.J. Ward, K.B. Glasner

A Micro electro-mechanical systems (MEMS) capac-itor is a microscopic device consisting of two platesheld opposite one other. The lower plate is immobilewhile the upper plate is fixed along its edges butfree to deflect in the presence of an electric poten-tial towards the lower plate. The deflecting upperplate may reach a stable equilibrium, however, if theapplied potential exceeds a threshold, known as thepull-in voltage, the upper plate will touchdown onthe lower plate. When certain physical assumptionsare applied, the deflection of the upper surface canbe modeled as a fourth order PDE with a singular

non-linearity. It is shown that the model capturesthe pull-in instability of the device and provides aprediction of the pull-in voltage. When the pull-involtage is exceeded, the equations develop a finitetime singularity (FTS) and it is demonstrated thatthis FTS forms on multiple isolated points or on acontinuous set of points. The touchdown set is pre-dicted for certain domains by means of asymptoticexpansions. We also illustrate a new model whichregularizes the infinite electric field and allows thedynamics to be continued beyond the initial FTS.

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The e↵ects of alliances in competing speciesmodels

Julian Lopez-GomezComplutense University of Madrid, SpainLopez Gomez@mat.ucm.esM. Molina-Meyer

We analyze the e↵ects of spatial refuges in hetero-geneous competing species systems as well as thee↵ects of strategic alliances in competitive environ-ments. The spatial refuges under strong competitionenhance segregation and, hence, diversity. The facil-itative e↵ects increase dramatically the stability ofthe ecosystems.

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Triple-junctions in a strong interaction limitof a three-component system

Hideki MurakawaKyushu University, Japanmurakawa@math.kyushu-u.ac.jpHirokazu Ninomiya

We consider a three-component reaction-di↵usionsystem with a reaction rate parameter, and investi-gate its singular limit as the reaction rate tends toinfinity. The limit problem is described by a nonlin-ear cross-di↵usion system. The system is regardedas a weak form of a free boundary problem whichpossesses three types of free boundaries. Triple junc-tion points appear at the intersection of the threeinterfaces. Furthermore, the dynamics is governedby a system of equations in each region separated bythe free boundaries.

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SPECIAL SESSION 29 131

Convergence of minimax and continuation ofcritical points for singularly perturbed sys-tems

Benedetta NorisUniversita’ degli Studi di Milano-Bicocca, Italybenedettanoris@gmail.comHugo Tavares, Susanna Terracini, GianmariaVerzini

In the recent literature, the phenomenon of phaseseparation for binary mixtures of Bose–Einstein con-densates can be understood, from a mathematicalpoint of view, as governed by the asymptotic limit ofthe stationary Gross–Pitaevskii system

8<

:

��u+ u3 + �uv2 = �u��v + v3 + �u2v = µvu, v 2 H1

0 (⌦), u, v > 0,

as the interspecies scattering length � goes to +1.For this system we consider the associated energyfunctionals J� , � 2 (0,+1), with L2–mass con-straints, which limit J1 (as � ! +1) is stronglyirregular. For such functionals, we construct multi-ple critical points via a common minimax structure,and prove convergence of critical levels and optimalsets. Moreover we study the asymptotics of the crit-ical points.

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Nonnegative solutions of elliptic equationsand their nodal structure

Peter PolacikUniversity of Minnesota, USApolacik@math.umn.edu

We consider the Dirichlet problem for a of classnonlinear elliptic equations on reflectionally symmet-ric bounded domains. We are mainly interested innonnegative, nonzero solution which are not strictlypositive. We will discuss the existence of such solu-tions and their symmetry properties, including thesymmetry of their nodal set.

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A double bubble solution in a ternary systemwith long range interaction

Xiaofeng RenGeorge Washington Unviersity, USAren@gwu.eduJuncheng Wei

We consider a ternary system of three constituents,a model motivated by the triblock copolymer theory.The free energy of the system consists of two parts:an interfacial energy coming from the boundariesseparating the three constituents, and a longer range

interaction energy that functions as an inhibitor tolimit micro domain growth. We show that a per-turbed double bubble exists as a stable solution ofthe system. Each bubble is occupied by one con-stituent. The third constituent fills the complementof the double bubble. The location of the doublebubble is determined by the Green’s function of theLaplace operator on the sample domain.

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Optimal partition problems involving Lapla-cian eigenvalues

Hugo TavaresUniversity of Lisbon, Portugalhugorntavares@gmail.comSusanna Terracini

Given a bounded domain ⌦ ✓ RN , N � 2, and apositive integer m 2 N, consider the following opti-mal partition problem:

inf{mX

i=1

�k(!i) : !i ⇢ ⌦ open 8i,

!i \ !j = ; whenever i 6= j},

where �k(!) denotes the k-th eigenvalue of �� inH1

0 (!). Approximating this problem by a system ofelliptic equations with competition terms, we showthe existence of regular optimal partitions. More-over, multiplicity of sign-changing solutions for theapproximating system is obtained.

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Extremality conditions for optimal partitions

Susanna TerraciniUniversity of Milano-Bicocca, Italysusanna.terracini@unimib.it

We deal with the free boundary problem associatedwith optimal partitions related with linear and non-linear eigenvalues. We are concerned with extremal-ity conditions and the regularity of the interfaces.These properties are then linked with extremalityconditions of the nodal set of eigenfunctions and thenumber of their nodal components.

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132 9th AIMS CONFERENCE – ABSTRACTS

Natural constraints in variational methodsand superlinear Schroedinger systems

Gianmaria VerziniPolitecnico di Milano, Italygianmaria.verzini@polimi.itB. Noris

For a C2-functional J defined on a Hilbert spaceX, we consider the set

N = {x 2 A : projVxrJ(x) = 0},

where A ⇢ X is open and Vx ⇢ X is a closed lin-ear subspace, possibly depending on x 2 A. Westudy su�cient conditions for a constrained criti-cal point of J restricted to N to be a free criticalpoint of J , providing a unified approach to di↵er-ent natural constraints known in the literature, suchas the Birkho↵-Hestenes natural isoperimetric condi-tions and the Nehari manifold. As an application,we prove multiplicity of solutions to a class of super-linear Schroedinger systems on singularly perturbeddomains.

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Qualitative properties and existence resultsfor an nonlinear elliptic system

Juncheng WeiChinese University of Hong Kong, Hong Kongwei@math.cuhk.edu.hkH. Berestycki, TC Lin, S. Terracini, KeleiWang, CY Zhao

We consider the following nonlinear ellitpic system

�U = UV 2, �V = V U2, U, V > 0 in Rn

We first prove the existence, asymmetry, stabilityand local uniqueness of the one-dimensional solution.Then we prove the two-dimensional De Giorgi Con-jecture and Stability Conjecture, under some growthcondition. Finally we construct solutions with anypolynomial growth at +1.

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Spiky patterns in a consumer chain model

Matthias WinterBrunel University, Englandmatthias.winter@brunel.ac.ukJuncheng Wei

We study a consumer chain model which is basedon Schnakenberg type kinetics. This model is re-alistic in a predator-prey context if cooperation ofpredators is prevalent in the system. The systemalso serves as a model for a sequence of irreversibleautocatalytic reactions in a container which is in

contact with a well-stirred reservoir. In this modelthere is a middle predator feeding on the prey and afinal predator feeding on the middle predator. Thismeans that the middle predator plays a hybrid role:it acts as both predator and prey. We will considertwo cases: (i) Final predator has smaller di↵usiv-ity than the rest. (ii) Middle predator has smallerdi↵usivity than the rest.

We will rigorously prove the existence of spikypatterns for this system. We will also study theirstability properties.

The analytical results are confirmed by numericalcomputations. Biological applications are discussed.

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Monge-Ampere equations on exterior do-mains

Lei ZhangUniversity of Florida, USAleizhang@ufl.eduJiguang Bao, Haigang Li

We consider the Monge-Ampere equationdet(D2u) = f where f is a positive continuous func-tion in Rn and is a perturbation of a positive con-stant at infinity. First we prove that every globallydefined solution is close to a parabola plus a loga-rithmic term in two dimensional spaces and is closeto a parabola in higher dimensional spaces. Thenwe show that given any prescribed asymptotic be-havior mentioned above, there exists a unique globalsolution corresponding to it. Finally we solve the ex-terior Dirichlet problem with given function on theboundary of a bounded convex set and a prescribedasymptotic behavior at infinity.

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Regularity results for boundary partitionproblems

Alessandro ZilioPolitecnico di Milano, Italyalessandro.zilio@mail.polimi.itSusanna Terracini, Gianmaria Verzini

We analyze the regularity of solutions to bound-ary partition problems. In particular, we considera system of elliptic equations arising as constrainedminima of a certain energy functional, under thecondition that the solutions have traces with disjointsupport on some portion of the boundary. Thanks tothe link between boundary dynamics and fractionaldi↵usion processes, our results extend also to segre-gation problems involving fractional powers of thelaplace operator.

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