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Convergence of bounded solutions of nonlinear parabolic problems on a bounded interval

Monday, November 4th, 2013

Thursday 7th November, 1400, MC113 (C26)

Dr Daniel Hauer

from

University of Sydney

will speak on

Convergence of bounded solutions of nonlinear parabolic problems on a bounded interval: the singular case

In this talk we prove that for every latex-image-2 and for every continuous function latex-image-1 which is Lipschitz continuous in the second variable, uniformly with respect to the first one , each bounded solution of the one-dimensional heat equation
latex-image-2
with homogeneous Dirichlet boundary conditions converges as latex-image-1 to a stationary solution. The proof follows an idea of Matano which is based on a comparison principle. Thus, a key step is to prove a comparison principle on non-cylindrical open sets.

On a free boundary problem for the curvature flow with driving force

Monday, November 4th, 2013

Thursday 7th November, 1300, MC113 (C26)

Professor Hiroshi Matano

from

University of Tokyo

will speak on

On a free boundary problem for the curvature flow with driving force

This talk is concerned with a free boundary problem associated with the curvature dependent motion of planar curves in the upper half plane whose two endpoints slide along the horizontal axis with prescribed fixed contact angles. The first main result is on the classification of solutions; every solution falls into one of the three categories, namely, area expanding, area bounded and area shrinking types. We then study in detail the asymptotic behavior of solutions in each category. Among other things we show that solutions are asymptotically self-similar both in the area expanding and the area shriknking cases, while solutions converge to either a stationary solution or a traveling wave in the area bounded case. Thus the renormalized curve converges to some profile in each of the three cases, but the proof of the convergence is totally different among the three cases.

This is joint work with Jong-Shenq Guo, Masahiko Shimojo and Chang-Hong Wu.

On Monge-Ampere type equations and near field optics

Saturday, October 19th, 2013

Thursday 24th October, 1400, MC113 (C26)

Professor Neil Trudinger

will speak on

On Monge-Ampere type equations and near field optics

We report on a general theory of Monge-Ampere type partial differential equations which are given by prescribing the Jacobians of vector mappings of one jets. Such PDEs arise in near field geometric optics and we consider some examples together with the underlying convexity and ensuing regularity theories.

Bubbling Solutions for Chern-Simons Model in a Torus

Friday, September 27th, 2013

27th September 2013, 1530 MC113

Associate Professor Shusen Yan

from

University of New England

will speak on

Bubbling Solutions for Chern-Simons Model in a Torus

In this talk, I will present some results on the existence of different types of bubbling solutions for Chern-Simons model in torus. I will also talk about the exact number of solutions for this problem. (This is joint work with C.-S. Lin in NTU.)

System of Elliptic Equations

Thursday, May 19th, 2011

At 2pm on Tuesday, 24th May, 2011

Prof. Daomin Cao

from the Chinese Academy of Science

will speak in MC206 (Building C26)

on a

System of Elliptic Equations.

Abstract:

In this talk the speaker will consider a system of elliptic equations arising from physics. Previous results on the existence of solutions will be presented. The speaker
will also introduce the results on the existence of ground state in his joint work with I.L.Chern and J.C.Wei.

BOUNDARY BLOW UP ELLIPTIC PROBLEMS WITH SIGN-CHANGING WEIGHTS: EXISTENCE AND MULTIPLICITY OF POSITIVE SOLUTIONS

Thursday, May 12th, 2011

On Tuesday, 17th May, 2011

JORGE GARCIA-MELIAN
Universidad de La Laguna

will speak on

BOUNDARY BLOW UP ELLIPTIC PROBLEMS WITH SIGN-CHANGING WEIGHTS: EXISTENCE AND MULTIPLICITY OF POSITIVE SOLUTIONS

at 2pm in MC206.

Abstract. We consider an elliptic boundary blow-up problem in a bounded smooth domain of R^N, and show that as a parameter in the equation varies, the solution set form a smooth curve in a suitable function space, and the curve has a turning point at a critical value of the parameter. The proofs are mainly based on continuation methods.

Rearrangement Inequalities for Principal Eigenvalues of Integral Operators and Applications

Monday, April 4th, 2011

25th March 2011, 1530 MC206

Professor Xing Liang

from

University of Science and Technology of China

will speak on

Rearrangement Inequalities for Principal Eigenvalues
of Integral Operators and Applications

Abstract:

In this talk, some new rearrangement inequalities for principal eigenvalues of nonsymmetric integral operators will be introduced and the applications on spreading speeds for monostable spatially periodic reaction-diff usion equations and integral diff erence equations will be considered.

Quantizable Kähler Geometry, Sasakian Geometry, Conical Kähler Geometry and Locally Conformal Kähler Geometry

Wednesday, November 17th, 2010

18th November 2010, 1100 MC206

Emeritus Professor Dmitri Alekseevski

from

University of Edinburgh

will speak on

Quantizable Kähler Geometry, Sasakian Geometry, Conical Kähler Geometry and Locally Conformal Kähler Geometry

We discuss relations between these four types of geometries. In particular, we associate with a quantizable Kähler manifold M a principal bundle SM with the structure group A=S1 or R and a contact form, θ, which, together with the complex structure, J, of M, defines a Sasakian structure on S. The Riemannian cone C(S) over S has a canonical Kähler conical structure and its quotient carries a Vaisman locally conformal Kähler structure. We will discuss under which conditions this construction can be inverted. We consider also the homogeneous case and indicate some applications. This is joint work with V. Cortes, H. Hasegawa and Y. Kamishima

Elliptic equations with singular lower order terms

Tuesday, April 13th, 2010

13th April 2010, 1500 MC206

Professor Gary Lieberman

from

Iowa State University

will speak on

Elliptic equations with singular lower order terms

The classical theory of elliptic equations assumes that the lower order terms are all bounded (or are in some Lp space with p sufficiently large). In this talk, we examine equations with singular lower order terms. Although some elements of the classical theory can be carried over to this case, several interesting new phenomena occur. In some cases, the equation has only one bounded solution in a given bounded domain. In other cases, there are solutions for arbitrary Dirichlet data but only one smooth solution.

Representation of Functions in A(−∞) by Dirichlet Series and Applications

Monday, March 1st, 2010

2nd March 2010, 1500 MC206

Professor Le Hai Khoi

from

NTU Singapore

will speak on

Representation of Functions in A−∞ by Dirichlet Series and Applications

Consider a space A−∞(Ω) of holomorphic functions in a bounded convex domain Ω of Cn, with the so-called polynomial growth near the boundary ∂Ω. Introducing a space AΩ of entire functions with certain growth condition, we establish the mutual duality of A−∞(Ω) and A−∞. As one of applications of the obtained duality, a possibility of Ω representation of functions from both spaces A−∞(Ω) and A−∞Ω in a form of Dirichlet series is given. These results are due to Abanin A.V. and myself.