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M[§e|æIáÆRegularity for convex envelope and degenerateMonge-Ampere equations

6401¿

z 9:2010:00

,R§uÀÆLattice structure of 3D surface superconductivityand magnetic potentials with singularities

6401¿

10:00-10:20

4° 10:2011:00

U1§®ÆExterior problem for parabolic Monge-Ampere e-quations

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ö§®ÆA variational approach for self-similar solutions ofthe generalized curve-shortening problem

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ol²§þ°ÏÆ&ICU BoulderÆSome preliminary study of non-negative solutionsto fractional Laplace equations

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r§4ïÆ&IUtah²áÆCoupled nonlinear Schrodinger equations withmixed coupling

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R§u¥ÆMultiple solutions for logarithmic Schrodinger e-quations

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4în§ÄÑÆNew existence results for a Schrodinger systemwith non-constant coefficients

6401¿

10:00-10:20

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²§H®ÆVariational PDE based segmentation and registra-tion

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H_§uÆSolutions for fractional operator problem via localPohozaev identities

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Exterior problem for parabolic Monge-Ampere equations

U1§®Æ

We use Perron method to prove the existence of exterior problem for

a kind of parabolic Monge-Ampere equation −ut detD2u = f(x, t) with

prescribed asymptotic behavior at infinity outside a bowl area in Rn+1− ,

where f is a perturbation of 1 at infinity. This report is based on the joint

work with Shuyu Gong and Ziwei Zhou.

5Dirac§±ÏÅ

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0ïÄþfnØ¥Ñy5Dirac§,#?Ð.AO´

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õ­5.

Solutions for fractional operator problem via local Pohozaev

identities

H_§uÆ

We consider the following fractional Schrodinger equation involving

critical exponent:(−∆)su+ V (|y′|, y′′)u = u2∗s−1 in RN ,u > 0, y ∈ RN ,

(P )

where s ∈ (12, 1), (y′, y′′) ∈ R2 ×RN−2, V (|y′|, y′′) is a bounded nonnega-

tive function with a weaker symmetry condition. We prove the existence

16 u¥Æ

of infinitely many solutions for the above problem by a finite dimensional

reduction method combining various Pohazaev identies.

A variational approach for self-similar solutions of the generalized

curve-shortening problem

ö§®Æ

Consider the generalized curve shortening problem:

∂γ

∂t= Φ(θ)|k|σ−1kN, σ > 0, θ ∈ S1,

where γ(·, t) is a planar curve, k(·, t) is its curvature with respect to the

unit normal N = −(cos θ, sin θ), and Φ is a positive function depending on

θ. Assuming that γ(·, t) is convex, then in terms of the support function

w(t, θ), the equation becomes

∂w

∂t=

−Φ(θ)

(wθθ + w)σ, θ ∈ S1. (1)

A self-similar solution of (1) is of the form w(θ, t) = ξ(t)u(θ), and u is a

solution of

uθθ + u =a(θ)

up+1, θ ∈ S1 (2)

with a(θ) = Φ1σ (θ), p + 1 = 1

σ . In this talk we will present a variational

approach for the π-periodic solutions of equation (2) with p > 2.

Some preliminary study of non-negative solutions to fractional

Laplace equations

ol²§þ°ÏÆ&ICU BoulderÆ

In this talk, we present some recent work on solutions to fractional

Laplace equations with singularities. First, we establish Bocher type the-

u¥Æ 17

orems on a punctured ball via distributional approach. Then, we develop

a few interesting maximum principles on a punctured ball. Our distribu-

tional approach only requires the basic L1loc-integrability. Furthermore,

several basic lemmas are introduced to unify the treatments of Laplacian

and fractional Laplacian. This is a joint work with C. Liu, Z. Wu and H.

Xu.

New existence results for a Schrodinger system with non-constant

coefficients

4în§ÄÑÆ

Assume Vj, µj, β ∈ C(Rn), n = 1, 2, 3. For the Schrodinger system

of two equations with non-constant coefficients−∆u+ V1(x)u = µ1(x)u3 + β(x)v2u, in Rn,

−∆v + V2(x)v = µ2(x)v3 + β(x)u2v, in Rn,

u(x)→ 0, v(x)→ 0, as |x| → ∞,

we present some new results on the existence of a positive solution and the

existence of multiple positive solutions. This is joint work with Haidong

Liu.

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Lattice structure of 3D surface superconductivity and magnetic

potentials with singularities

,R§uÀÆ

In this talk we discuss lattice structure of the solutions of 3D mag-

netic Schrodinger equation, which could start the second step towards

understanding the surface superconducting states of 3D type 2 supercon-

ductors in an applied magnetic field. We shall also examine the effects of

magnetic potentials with singularities on superconductivity, which exhibit

analogy with the magnetic Aharonov-Bohm effect. The first part is joint

work with Soren Fournais and Jean-Philippe Miqueu, and the second part

is joint work with Ayman Kachmar.

Multiple solutions for logarithmic Schrodinger equations

R§u¥Æ

We discuss the following logarithmic Schrodinger equation

−∆u+ V (x)u = u log u2, x ∈ RN .

The classical variational methods cannot be applied directly, because the

corresponding energy functional is not well defined in H1(RN ). By using

direction derivative and constrained minimization method, we prove the

existence of positive and sign-changing solutions in H1(RN ) under differ-

ent types of potential. Moreover, if the potential is radially symmetric,

we also construct infinitely many nodal solutions in H1r (RN ).

Regularity for convex envelope and degenerate Monge-Ampere

equations

M[§e|æIáÆ

u¥Æ 19

In this talk, we discuss the regularity for convex envelope and degen-

erate Monge-Ampere equations. Given a function v in a domain Ω, the

convex envelope is defined by u(x) = supL(x) : L ≤ v in Ω. For given

α ∈ (0, 1], we prove that u ∈ C1,α(Ω) if ∂Ω ∈ C2+2α, v ∈ C2+2α(Ω),

and Ω is uniformly convex. By example we show that these conditions

are optimal. We also prove the global C1,α regularity for the Dirichlet

problem of the degenerate Monge-Ampere equation detD2u = f(x) in Ω,

u = φ on , assuming that φ, ∂Ω ∈ C2,β(Ω) and Ω is uniformly convex, for

some β > 0. Again all these conditions are sharp. We will also make a

brief survey on the regularity for the Monge-Ampere equations.

Coupled nonlinear Schrodinger equations with mixed coupling

r§4ïÆ&IUtah²áÆ

We discuss work on existence and qualitative property of positive so-

lutions for coupled nonlinear Schrodinger equations. Depending upon the

system being attractive or repulsive, solutions may tend to be component-

wisely synchronized or segregated. We report recent work on the effect of

mixed coupling for which coexistence of synchronization and segregation

may occur. These are joint work with J. Byeon and Y. Sato, S. Peng and

Y. Wang, respectively.

Variational PDE based segmentation and registration

²§H®Æ

Image segmentation and registration are always challenging tasks in

image processing. In this talk, we will discuss two variational PDE mod-

els. One is concerned a region inhomogeneity active contour which is to

quantify intensity inhomogeneity and convert it to be a useful feature to

110 u¥Æ

improve the segmentation results. Another one is a BD based registration

algorithm which is formulated in a variational framework by supposing

the displacement field to be a function of bounded deformation. The nu-

merical experiments indicate that the proposed models perform well than

some of the state-of-art models.

u¥Æ 111

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2 ù¬ ¥IÆêÆXÚÆïÄ ïÄ dmcao@amt.ac.cn

3 z ÉÇÆ Ç chenhua@whu.edu.cn

4 §âl =²Æ Ç chengxy@lzu.edu.cn

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19 o°f ®Æ BÇ hgli@bnu.edu.cn

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24 Ü) S Æ BÇ weisheng.niu@gmail.com

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26 Û+ u¥Æ BÇ pluo@mail.ccnu.edu.cn

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28 ,R uÀÆ Ç xbpan@math.ecnu.edu.cn

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47 xp¸ u¥Æ Ç gfzheng@mail.ccnu.edu.cn

48 ±t ÉÇnóÆ Ç hszhou@wipm.ac.cn

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