Differential Geometry and PDE Seminar
Yunhee Euh (NIMS)
Riemannian curvature functionals on 4-manifolds
ABSTRACT.
In this talk, we study the critical points of several Riemannian functionals in dimension 4.
We introduce the curvature identity deduced from Chern-Gauss-Bonnet theorem for a 4-dimensional closed manifold. As its application, we discuss the critical metrics of
the squared L2-norm functionals of the curvature tensor, the Ricci tensor and the scalar curvature respectively
and describe the relations among their critical metrics.
Kyungha Hwang (UNIST)
Well-posedness and ill-posedness for the cubic fractional schrodinger equations
ABSTRACT.
In this talk, we discuss the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrodinger equations
with Levy indices 1 < α < 2. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally
well-posed in Hs for s ≥ (2-α)⁄4. This is shown via a trilinear estimate in Bourgain's Xs,b space. We also
show that non-periodic equations are ill-posed in Hs for (2-3α)⁄4(α+1) < s < (2-α)⁄4
in the sense that the flow map is not locally uniformly continuous. This is joint work with Prof. Yonggeun Cho, Prof. Soonsik Kwon and Prof. Sanghyuk Lee.
Hyung Jun Choi (POSTECH)
Corner singularity for the stationary Navier-Stokes equations with no-slip boundary condition on polygons
ABSTRACT.
Corners of the (non-convex) polygon raise the singularity. Especially it makes some numerical schemes slow or wrong in applied mathematics,
and concentrates stresses, grow cracks, etc. in mechanical or electrical engineering. I will talk about the corner singularities of the
incompressible Navier-Stokes equations with no-slip boundary condition and derive their corner singularity expansion. To show the importance
of dealing with the corner singularities, I will briefly show some numerical examples.
Jinmyoung Seok (Kyonggi Univ.)
Results on stationary states for Chern-Simons-Schrödinger equations
ABSTRACT.
The Chern-Simons theory was proposed in the 1980's to explain electromagnetic phenomena of anyon physics such as the high temperature
super-conductivity or the fractional quantum Hall effect. In this talk, I will introduce the nonlinear Schrödinger equation coupled
with the Chern-Simons gauge fields, proposed by Jackiw and Pi in 1990 and present recent results about the existence and nonexistence
of the stationary states. We will see that there is no nontrivial stationary state if the Chern-Simons coupling constant λ is
less than 1 and there is a stationary state with a vortex point of arbitrary order N if λ > 1. If λ = 1, it turns out
that every stationary state is gauge equivalent to a solution of the first order self-dual system.
Younghun Hong (Univ. of Texas, Austin)
Unconditional uniqueness of the cubic Gross-Pitaevskii hierarchy with low regularity.
ABSTRACT.
First, we briefly review the derivation of nonlinear Schrodinger equation (NLS) from N-body linear Schrodinger equation via the cubic
Gross-Pitaevskii (GP) hierarchy, which is an infinite system of coupled linear equations. Such a derivation was established by the
seminal works of Erdos-Schlein-Yau. In the derivation, the most involved part is the proof of unconditional uniqueness of solutions
to GP hierarchy. Recently, Chen-Hainzl-Pavlovic-Seiringer gave a simpler alternative proof of uniqueness via the quantum de Finetti
theorem. Adapting this new approach, we established the unconditional uniqueness of solutions to the GP hierarchy in a low regularity
Sobolev type space. Precisely, we reduce the
regularity requirement down to the currently known regularity requirement for unconditional uniqueness of solutions to NLS.
This is a joint work with Kenneth Taliaferro and Zhihui Xie at UT Austin.
Hyenkyun Woo ((KIAS)
L∞-norm and its applications in image processing
ABSTRACT.
In this talk, I will show the usefulness of L∞-norm in image processing.
1. Speckle (multiplicative noise) naturally appear in variouscoherent imaging systems, such as synthetic aperture radarand ultrasound.
Due to the strong interference phenomena incoherent imaging systems, it is hard to identify the valuableobjects from the captured noisy data.
In this talk, weintroduce framework for total variation based specklereduction problems. The framework is based on m-th roottransformation
and linearized proximal alternatingminimization algorithm.
2. In this talk, we introduce L∞-norm based new low rank promoting regularization framework, that is asymmetric soft
regularization framework for robust nonnegative matrix factorization (NMF). The main advantage of the proposed low rank enforcing ASR
framework is that it is less sensitive to the rank selecting regularization parameters since we use soft regularization framework,
instead of using the conventional hard constraints such as nuclear norm, gamma2-norm, or rank itself in matrix factorization.
Note that, this talk is based on the talk at SIAM Conf. on Imaging Sci. and SIAM Conf. on Optimization.
Inbo Sim (Univ. of Ulsan)
On p(x)-Laplace problems
ABSTRACT.
In this talk, I will introduce p(x)-Laplace problems which are subject to Dirichelt boundary condition
and its suitable function space $W^{1,p(x)}(\Omega)$ and compare it with p-Laplace problems in the following subjects;
(1) positivity of the first eigenvalue,
(2) uniqueness of positive solutions,
(3) concave-convex type nonlinearity problems.
Neal Bez (Saitama Univ.)
Optimal versions of the trace theorem on the sphere
ABSTRACT.
The classical trace theorem for the sphere says that if $f$ is
a function on $\mathbb{R}^d$ with at least half a derivative in $L^2(\
mathbb{R}^d)$ then $f$ has a well-defined restriction to the sphere in $
L^2(\mathbb{S}^{d-1})$. In this talk, I will show how to prove optimal
versions of this result by obtaining a characterisation of the extremal
functions $f$. I will also discuss various optimal extensions and
generalisations, including an extension to $L^p(\mathbb{S}^{d-1})$, and
generalisations to Sobolev spaces which incorporate angular regularity
on the sphere.
Seick Kim (Yonsei Univ.)
On a linearized Monge-Ampere type equation on certain Riemannian manifolds
ABSTRACT.
We study a version of linearized Monge-Ampere type equation naturally arising from optimal transport problems with the Riemannian distance squared cost function.
The source and target measures of the transportation are assumed only to be bounded above and below. This assumption requires the linearized equation to be
defined only weakly, and to be degenerate elliptic. We assume nonnegative cross-curvature and log-concavity of the Jacobian determinant of the exponential map.
We obtain a Harnack type inequality and Holder estimates for the solution. This extends the result of Caffarelli and Gutierrez given for the linearized
equation of the classical Monge-Ampere equation on the Euclidean space, to the round sphere and the products of round spheres and the Euclidean spaces.
*This is joint work with Young-Heon Kim.
Seok Bae Yun (Sungkyunkwan University)
Spatially homogeneous Boltzmann equation for relativistic particles
ABSTRACT.
In this talk, we are concerned with the spatially homogeneous theory of the relativistic Boltzmann equation, which is a fundamental equation describing the time evolution of the phase space
distribution of relativistic particles. More precisely, we concentrate on the problem of the propagation of uniform $L^{\infty}$ bound. The key idea is to reformulate
the gain term of the collision operator to separate the dependence of the post-collisional velocity variables on the spherical part and the velocity part.
This is a joint work with Robert Strain.
Yunhee Euh (NIMS)
Critical metrics for quadratic Riemannian functionals on 4-dimensional manifolds
ABSTRACT.
Let $M$ be a 4-dimensional oriented smooth manifold and $\mathfrak{M}(M)$ be the space of all Riemannian metrics on $M$. We denote by $\mathcal{A}$,
$\mathcal{B}$ and $\mathcal{C}$ the squared $L^2$-norm functionals of the curvature tensor, the Ricci tensor and the scalar curvature, respectively.
Then the Euler characteristic $\chi(M)$ of a 4-dimensional compact, oriented Riemannian manifold $M=(M, g)$ is given by the following generalized
Gauss-Bonnet formula:
$$\chi(M) = \frac{1}{32\pi^2}\big(\mathcal{A}(g)-4\mathcal{B}(g)+\mathcal{C}(g)\big)$$
for any $g\in \mathfrak{M}(M)$.
Chun-Siung Hsia (National Taiwan University)
On semilinear elliptic equations involving Sobolev and Sobolev-Hardy critical nonlinearities with singularities on the boundary
ABSTRACT.
We consider a series of elliptic partial differential equations arising from the study of the extremal functions for
Caffarelli-Kohn-Nirenberg (CKN)'s inequalities. By assuming the curvature of the singularity on the boundary be negative/positive,
Ghoussoub and Kang showed the existence of positive solutions for the Dirichlet/Neuman problem, respectively. Recently, in
collaboration with Lin and Wadade, we improved the existence theorems for the Dirichlet problems.
TOPICS Differential Geometry - Geometric Analysis - PDE ORGANIZERS Young-Jun CHOI - Korea Institute for Advanced Study Soyeun JUNG - CMC, Korea Institute for Advanced Study Hyunsuk KANG - Korea Institute for Advanced Study Youngwoo KOH - Korea Institute for Advanced Study Keomkyo SEO - Sookmyung Women's Univ. Jinmyoung SEOK - Kyonggi University How to visit KIAS Directions to KIAS English - Korean
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