Workshop Indam

Third Trilateral Meeting “Australia -Taiwan - Italia”

Università di Roma “La Sapienza”, January 16-20,2006

Titles and Abstracts

mappaItalia Taiwan

Francis Clarke (Institutuniversitaire de France et Université de Lyon 1): Regularity of solutions in the calculus of variations: anextension of the Hilbert-Haar approach

We study the problem of minimizing

Ω [ F(Du(x)) + G(x,u(x)) ]dx
over the functions u in W1,1(Ω) that assume given boundaryvalues φ on Γ :=∂ Ω. The Lagrangian F and thedomain Ω are assumed convex. A new type ofhypothesis on the boundary function φ isintroduced: the lower (or upper) bounded slopecondition. This condition is less restrictive than the classicalbounded slope condition of Hartman, Nirenberg and Stampacchia, and allowsconsideration of nonsmooth boundary data in a Hilbert-Haar type of context.It turns out to have significant consequences for the regularity of thesolution u. In particular, it implies that u is locally Lipschitz in a certain uniform manner. Incertain cases, as when Γ is a polyhedron orelse of class C1,1, it implies inaddition a global Hölder condition.}

Australian Delegation

Philip Broadbridge (Australian Mathematical Sciences Institute,Melbourne): Exact Solution of Nonlinear BoundaryValue Problems for Surface Diffusionphil@amsi.org.au14-20

Curvature-driven surface diffusion on crystalline surfaces is modeled bythe nonlinear 4th order Mullins equation. There is a class of weaklyanisotropic nonlinear models that is fully integrable. Exact solutions areconstructed for development of a grain boundary groove and for smoothing ofan initial ramp dislocation. For the grooving problem, a piecewisenonlinear but solvable model can be made arbitrarily close to the isotropicnonlinear Mullins equation. The solution shows that unlike in the linearmodel, the groove depth remains bounded as the dihedral angle approachesvertical. At a dislocation point of infinite curvature, the quasilinearMullins model should be extended to a fully nonlinear model to account forGibbs-Thompson

Alan Carey (AustralianNational University, Canberra): Spectral flowAlan.Carey@maths.anu.edu.au

I will discuss some aspects of recent research on the analytic approach tospectral flow for unbounded self adjoint Fredholm operators.

Florica Cirstea (AustralianNational University, Canberra): Blow-up solutions of elliptic equations with competingregularly/rapidly varying nonlinearities

We present some general uniqueness results dealing with positive solutionsof

Δu + au = b(x)f(u)   in   Ω

such that lim d(x) → 0 u(x) =∞, where Ω is a smooth boundeddomain in RN and d(x) := dist(x, ∂Ω). We assume that b is a continuous nonnegative function on the closureof Ω, a is a realnumber and f is locally Lipschitz continuous on[0, ∞) with f(u)/u increasing for u > 0. We analysethe competing case when b(x) vanishes on ∂Ω at a regular rate (e.g., [d(x)]α with α > 0) or rapid rate (e.g., exp(-[d(x)]q) with q <0), while f varies regularly at ∞ with index p > 1(that is limu → ∞f(λu)/f(u)= λp, for every λ >0). The extreme case when p = ∞ willalso be considered. We emphasize the links between the asymptotic behaviourof the blow-up solution and the variation speeds of f and b.

Norman Dancer (University ofSydney): Finite Morse indexsolutions of some supercritical problems

We discuss finite Morse index solutions of supercritical ellipic problemson unbounded domains and their application to supercritical problems onbounded domains.

Joseph Grotowski (Universityof Queensland, Brisbane): 2-dimensional Harmonic map heat flow vs. 4-dimensionalYang-Mills heat flow

The two geometric flows in the title are the gradient flows associated tocertain energy functionals. In the considered dimension, (i.e. dimension 2for the harmonic map heat flow, dimension 4 for the Yang Mills heat flow),the associated energy functional is (locally) conformally invariant, i.e.the dimension is critical. In this talk we discuss similarities anddifferences between the flows in certain symmetric situations.

Neil Trudinger (AustralianNational University, Canberra): Hessian Measures for Non-commuting VectorFields

We consider the weak continuity of the 2-Hessian operator on non-commutingvector fields satisfying the Hormander condition.

Taiwanise Delegation

Chiun-Chuan Chen (NationalTaiwan University, Taipei): Variational structure of travelling waves inreaction-diffusion equationschchchen@math.ntu.edu.tw

When the wave speed is zero in a bistable system, there is a naturalvariational setting for the problem of finding a travelling wave. If thewave speed is not zero, it is difficult to use variational method to find asolution. In this talk, we are going to discuss the possible variationalstructure for the case with non-zero wave speed.

Chi-Kun Lin (NationalChen-Kung University, Tainan): Homogenization and memory effect of the Maxwellsystem

The purpose of this paper is to study thememory (or nonlocal) effect induced by homogenization of theMaxwell equations. The memory kernel is described by the Volterraintegral equation. Furthermore, it can be characterized explicitlyin terms of Young's measure, and the kinetic formulation of thehomogenized equation is also obtained. The kinetic formulationallows us to obtain the homogenization of the energy density andthe associated conservation law with the Poynting vector. Theinteresting interaction phenomenon of the microscopic andmacroscopic scales is also discussed and the memory effectexplains qualitatively something about irreversibility.

Chin-Cheng Lin (NationalCentral University, Taoyuan): Weighted boundedness of Marcinkiewiczintegral

The Marcinkiewicz integral defined by

formula

is essentially a Littlewood-Paley g-function,which plays a very important role in harmonic analysis. In the talk, wefirst survey the Lp boundedness andweighted Lp boundedness of theMarcinkiewicz integral, and then introduce other recent results concerningthe Marcinkiewicz integral with rough kernel and its new development.

Ching-Lung Lin (Chung ChengUniversity, Chia-Yi): Threespheres inequalities for a two-dimensional elliptic system and itsapplicationcllin@math.ccu.edu.tw15-22 double

In this paper we prove three spheres inequalities for a two-dimensionalstrongly elliptic system. We then give an application of these threespheres inequalities to the inverse problem of identifying cavities bypartial boundary measurements.

Tai-Chia Lin (National TaiwanUniversity, Tamsui): Spikesin Two-component Systems of Nonlinear Schrodinger Equations with TrappingPotentialstclin@math.ntu.edu.tw

Here we study the ground state solution of a coupled system of nonlinearSchrodinger equations with external trap potentials. The coupled system isa model which describes a binary mixture of Bose-Einstein condensates. Dueto Feshbach resonance, the coupled system can be transformed into asingular perturbed problem with a small parameter. As the parameter goes tozero, the existence and the asymptotic behavior of ground state solutionscan be proved, and spikes can be found in these ground state solutions.Furthermore, we may figure out how trap potentials affect the locations andconfigurations of spikes.

Fon-Che Liu (TamkangUniversity): Weak-LpInequality and LusinPropertymaliufc@math.sinica.edu.tw15-21single

A survey will be given of various forms of Lusin property, among which avery general form for Calderón-Zygmund clesses of functions will beemphasized. Related weak-Lp inequalities will also be discussed.

Tai-Ping Liu (AcademiaSinica, Taipei): Multi-dimensional Euler equations in gasdynamicsliu@math.stanford.edu

Multi-dimensional Euler equations in gas dynamics possess very rich andcomplex wave phenomena. There are definite experimental and computationalresults, but preciously little analytical understanding. In fact, to theembarassments of the workers in the field, the classical book ofCourant-Friedrichs “Supersonic Flows and Shock Waves ”, writtenin 1948, is still mostly updated. Due to the obvious difficulties of thepresence of shock waves, one of the main efforts to identify physicallyimportant problems that have simple scaling properties. Two such problemshave already reported in Courant-Friedrichs. One is to find the criterionof the transition of the regular-Mach shock reflections for a shock wavemoving toward a ramp. The other is the determination of the phsicality ofthe weak-strong shock reflections for supersonic flows around a wedge. Wewill formulate these problems and report on the recent progresses on them.For these problems, the scaling analysis reduces the dimensions and theflows become self-similar. This simplifies the problem. On the other hands,the equations in self-smilarity coordinates are of mixed types, even thoughthe original Euler equations are hyperbolic. To avoid the issue of highlyunstable vortex sheets and to concentrate on the study of shock andacoustic waves, one considers the potential equations. In self-similarcoordinates, there are interesting goemetric consequences, such as thebasic Ellipticity Principle. The question of well-posedness of boundaryvalue problem for degenerate elliptic equation with the free boundary ofshock waves and degenerate sonic circles is also very interesting. Newissues arise for elliptic PDEs.

Shih-Hsien Yu (The CityUniversity of Hong Kong): Singular limits in fluid and kinetictheorymashyu@cityu.edu.hk

There are several fundamental singular limit problems such as the zerodissipation limits for coservation laws and the zero mean free path limitfor Boltzmann equation. The limits are singular because of the presence ofinitial, boundary and shock layers. We plan to discuss the recent analysisin dealing with some of these limits.

Italian Delegation

Emilio Acerbi (Università di Parma): Recent results on the p-laplacian

The p-laplacian operator and its non standardversion, the p(x)-laplacian, have been the objectof an intensive study in the last years. In particular, several regularityresults have been obtained, concerning the integrability properties ofsolutions both in the elliptic and parabolic cases.

This talk will illustrate some recent results obtained in collaborationwith G. Mingione, the newest of which are two Calderón-Zygmundtype theorems, covering the following statement, where the exponent p may be even below 2: IfF is in Lqloc for some q > p then the solutions uof the parabolic system of PDEs

ut - div(|Du|p-2Du) = div(|F|p-2F)

are such that Du belongs to Lqloc.

The technique employed contains some fundamental elements derived from asimilar theory developed for the elliptic p(x)-laplacian.

Antonio Ambrosetti (SISSA,Trieste): NonlinearScrhoedinger equations: Recent results and new trends

We survey some recent results on NLS. Some new trends on this field willalso be discussed.

Luigi Ambrosio (ScuolaNormale, Pisa): HamiltonianODE's in the Wasserstein space of probability measures

In this talk we describe a recent joint work with W.Gangbo. We consider aHamiltonian H on P2(R2d), the set of probabilitymeasures with finite quadratic moments on the phase space R2d, which is ametric space when endowed with the Wasserstein distance W2. We study the initial value problem t/dt + ∇ (Jd vtμt) = 0, where Jdis the canonical symplectic matrix, μ0 is prescribed, vt is a tangent vector to P2(R2d) at μt, and belongs to ∂H(μt), the subdifferential ofH at &mut.

Concerning existence of solutions, two methods for constructing solutionsof the evolutive system are provided. The first one concerns only the casewhere μ0 is absolutely continuous.It ensures that μt remainsabsolutely continuous and vt = ∇H(μt) is the element of minimal norm in ∂ H(μt). The second method handlesany initial measure μ0. If wefurthermore assume that H is λ-convex, proper and lower semicontinuous onP2(R2d), we prove that the Hamiltonianis preserved along any solution of our evolutive system.

Concerning uniqueness of solutions, the problem is still largely open andrecent progress on “generic” uniqueness results in thefinite-dimensional case will be described.

Vieri Benci (Università di Pisa): Existence of Vortices in the Abelian Gauge Theory

We consider the “gauge theory” functional

J(ψ,A) = <dA,dAψ > + <dAA,dAA> + W(ψ)

where A = i &Sigma Aj dxjis a connection form with values in u(1)=iR and W is asuperquadratic function which satisfies suitable assumptions.

For physical reasons, we assume N=4 and that themetric < . ,. > isthe Minkowsky metric. A vortex is a critical point of the above functionalstationary in time and with cylindrical symmetry.

In this seminar we will discuss the problems related to the study of theabove functional and we will sketch the proof of the existence of vortices.

Piermarco Cannarsa (Università di Roma “Tor Vergata”):Perimeter estimates for attainablesets of control systems

The attainable set at time T, from a given closedset of a Euclidean space, is a well known object in control theory. Here,such a set is investigated for the symmetric system

x'(t) = f(x(t))u(t),

where u(t) takes its values in the closed unitball. A recent result obtained in collaboration with H. Frankowskaguarantees that, for f(x) smooth andnondegenerate, the attainable set at time T >0 satisfies a uniform interior sphere condition. As a consequence ofthis property, the attainable set has finite perimeter. Such an informationwill be further analyzed following the approach of a joint work with P.Cardaliaguet where estimates for the perimeter and volume growth ofattainable sets have recently been obtained.

Ermanno Lanconelli (Università di Bologna): A Sphere Theorem for a Class of Reinardt Domains withConstant Gauss-Levi Curvature

We present the following result, obtained in collaboration with J. Hounie:a bouded Reinhard domain of Cn+1 withconstant Gauss-Levi curvature and with rotational symmetry in twocomplemantary sets of variables is a ball.

Piero Marcati (Università dell'Aquila): Asymptotic problems for hyperbolic systems and theirrelations with the incompressible 3-D Navier-Stokes equation

We describe various singular limits for hyperbolic problems related to theincompressible 3-D Navier-Stokes equation. We are going to providehyperbolic approximations for the weak solutions in the class of Leray. Thelimiting properties are deduced by the use of dispersive estimates ofStrichartz type combined with nonlinear waves interaction estimates for thebilinear terms. These approximations are motivated by considerations fromphysics and numerical analysis, in particular there are relations with twoimportant methods. The former is an hyperbolic version the so called “quasicompressibility” method investigated for instance by J.L. Lions, Temam andrecently by Nochetto, the latter is a time relaxation limit of diffusivetype related to old numerical results by Garabedian and Chorin. Theseresults have been obtained in collaboration with D. Donatelli.

Luigi Orsina (Università di Roma “La Sapienza”): Semilinear elliptic equations with exponentialnonlinearities

We discuss existence, nonexistence and stability results for solutions ofsemilinear elliptic equations involving exponential nonlinearities. Moreprecisely, we will deal with the equation

-Δu + (eu - 1) = μ,

with μ a bounded Radon measure, givingconditions on the measure in order to have existence of solutions. The maintools will be a sharp estimate on solutions of linear elliptic equationswith data in Morrey spaces and a decomposition result for Radon measureswith respect to the Hausdorff measure. We will also deal with the equation

-Δun = eun,

with a boundedness assumption in a Morrey space on the sequence eun, proving aconcentration-compactness result of Brezis-Merle type.