Jae-Hwan Choi (SNU)
Initial trace of solutions to the heat equation
The initial value problem (IVP) for PDEs is a fundamental framework for understanding various phenomena in science and engineering. For many solutions of IVPs that reside in Sobolev spaces, the initial value must be interpreted through the trace operator, which requires a rigorous understanding of its boundedness and properties. In this talk, I will present the initial trace theorem for the heat equation in various function spaces.
Seongmin Jeon (HYU)
Schauder type estimates for degenerate or singular parabolic systems with partially DMO coefficients
We study elliptic and parabolic systems in divergence form with degenerate or singular coefficients. Under the conormal boundary condition on the flat boundary, we establish boundary Schauder type estimates when the coefficients have partially Dini mean oscillation. Moreover, as an application, we achieve higher-order boundary Harnack principles for uniformly parabolic equations with Holder coefficients. This is based on joint work with Hongjie Dong.
Yong-Gwan Ji (KIAS)
Gradient estimates between closely located conductors
If two conductors are closely located, the gradient of the solution may become arbitrarily large as the distance between the conductors tends to zero if the bonding is perfect. Quantitative gradient estimates are important in imaging and the theory of composites and significant progress has been made in this area during the last 25 years. In this talk, I will explain this progress and also discuss cases where the bonding is imperfect. This is based on joint work with Shota Fukushima, Hyeonbae Kang, and Xiaofei Li.
Junha Kim (Ajou University)
On the regularity of solutions to α-SQG equation in a half-plane
In this talk, we consider the α-SQG equation in a half-plane, where α = 0 and α = 1 correspond to the 2D Euler and SQG equations respectively. We prove the local well-posedness of α-SQG in an anisotropic Lipschitz space and the instantaneous blow-up of solutions in Hölder spaces when initial data does not vanish at the boundary. Then, we briefly discuss the case where initial data vanishes at the boundary. This talk is based on joint work with In-Jee Jeong(Seoul National University), Yao Yao(National University of Singapore), and Hideyuki Miura(Institute of Science Tokyo).
Ho-Sik Lee (Bielefeld University)
Gradient regularity for solutions of nonlinear nonlocal equations
Nonlocal equations motivated by the fractional Laplace equation are actively studied nowadays. Especially, regularity theory at the gradient level for solutions of nonlocal equations involving nonlinearity was investigated recently. In this talk, we provide gradient Holder regularity and pointwise gradient estimates for nonlinear nonlocal elliptic/parabolic equations with linear growth, as well as higher differentiability results for solutions of fractional p-Laplace equations. This is the joint work with Prof. Lars Diening (Bielefeld, Germany), Kyeongbae Kim (SNU), and Dr. Simon Nowak (Bielefeld, Germany).
Se-Chan Lee (KIAS)
Homogenization of an obstacle problem with highly oscillating coefficients and obstacles
We develop the viscosity method for the homogenization of an obstacle problem with highly oscillating obstacles. The associated operator, in non-divergence form, is linear and elliptic with variable coefficients. We first construct a highly oscillating corrector, which captures the singular behavior of solutions near periodically distributed holes of critical size. We then prove the uniqueness of a critical value that encodes the coupled effects of oscillations in both the coefficients and the obstacles.
Taehun Lee (KIAS)
Minimal surfaces through the lens of the Allen–Cahn equation
Minimal surfaces, as critical points of the area functional, have been extensively studied for their geometric elegance and connections to various physical phenomena. Interestingly, these minimal surfaces can be approximated by the nodal sets of solutions to the Allen–Cahn equation, a nonlinear partial differential equation rooted in phase transition theory. In this blackboard-style talk, we will explore the relationship between the Allen–Cahn equation and minimal surfaces, focusing on the important problems in each field—the De Giorgi conjecture and the Bernstein problem. Furthermore, we will examine solutions to the Allen–Cahn equation corresponding to leaves of the Hardt–Simon foliation, which are minimal surfaces with asymptotically conical ends. This talk is based on joint work with Kyeongsu Choi (KIAS) and Sanghoon Lee (KIAS).
Sang-Hyuck Moon (UNIST)
A variational approach for the mean field equations on compact surfaces
Given a compact surface (S, g), we prove the existence of a solution for the mean field equation on S. The problem consists of solving a second-order nonlinear elliptic equation with variational structure and exponential nonlinearity. Since the corresponding functional is unbounded from above and from below, we employ topological methods and min-max schemes.
Jinwan Park (Kongju National University)
The regularity of the obstacle problems
In this talk, I will introduce the regularity of the free boundary in the obstacle problem.
The obstacle problems are typical examples of the free boundary problem and arise in porous media, elasto-plasticity, optimal control, and financial mathematics. In the last decades, many properties of the obstacle problems have been studied by L. Caffarelli, K.-A. Lee, A. Figalli, H. Shahgholian, and various researchers. In this talk, I will introduce the regularity theory and other properties in obstacle problems to several operators.
Jinsol Seo (KIAS)
A general theorem on the Lₚ-solvability of Poisson's and the heat equations in non-smooth domains
Poisson's equation and the heat equation are among the most fundamental PDEs. This presentation focuses on the domain problem for these equations, specifically their solvability in non-smooth domains. We review the historical context and previous approaches to this problem, and introduce a general theorem based on localization arguments and the use of superharmonic functions.