Title and Abstract

Takwon Kim (KAIST)

Free-Boundary Analysis in the Obstacle Problem and Its Applications

Junkee Jeon (Kyung Hee University)

TBA
TBA

Minhyun Kim (Hanyang University)

Recent advances in nonlocal nonlinear potential theory
The classical nonlinear potential theory has recently been extended to nonlocal nonlinear potential theory, which studies harmonic functions associated with nonlocal nonlinear operators. In this talk, we focus on the harmonic functions solving the nonlocal Dirichlet problem. As in the study of classical Dirichlet problem, the nonlocal Dirichlet problem can be solved by using Sobolev and Perron solutions. We provide several properties of such solutions. This talk is based on joint works with Anders Björn, Jana Björn, Ki-Ahm Lee and Se-Chan Lee.

Se-Chan Lee (Seoul National University)

TBA
TBA

Taehun Lee (KIAS)

Free boundary problems in curvature flows
Free boundary problems, such as ice melting, involve the study of unknown interfaces and have been intensively researched over the past few decades. These problems can be naturally formulated within a geometric setting. In this talk, I will discuss two types of free boundary problems in geometric flow whose hypersurfaces evolve according to their curvatures: (1) Gauss curvature flow with flat sides, and (2) curvature flows with obstacles. The focus will be on the geometric properties and the development of singularities in these problems.

Minkyu Lim (KAIST)

TBA
TBA

Jehan Oh (Kyungpook National University)

Higher integrability for weak solutions to parabolic multi-phase equations
In this talk, we establish a local higher integrability result for the gradient of a weak solution to a parabolic multi-phase equation. To achieve this, we prove parabolic Poincaré type inequalities and reverse Hölder type inequalities for the gradient of a weak solution in each of the different types of intrinsic cylinders. In particular, we formulate a delicate plan of alternatives and stopping time arguments to address the presence of two different transitions.

Jihoon Ok (Sogang University)

Local regularity results for parabolic systems with general growth
We discuss on regularity theory for parabolic systems of the form
u_t - \mathrm{div} A(Du) =0 \quad \text{in }\ \Omega_T=\Omega\times(0,T],
where $u:\Omega_T\to \mathbb{R}^N$, $u=u(x,t)$, is a vector valued function and the nonlinearity $A:\mathbb{R}^{nN}\to \mathbb{R}^{nN}$ satisfies a general Orlicz growth condition characterized by exponents $p$ and $q$, subject to the inequality $\frac{2n}{n+2}<p<q$. It is noteworthy that when if $p<2<q$, the degeneracy of the system remains indeterminate.
This talk focuses on presenting developments in the realm of regularity results concerning  the spatial gradient of solutions of the above system, which include the higher higher integrability, Hölder continuity when $A(\xi)$ satisfies the Uhlenbeck structure, i.e., $A(\xi)=\frac{\varphi'(|\xi|)}{|\xi|}\xi$, and partial Hölder continuity. These results are joint works with Giovanni Scilla and Bianca Stroffolini from University of Naples Federico II, and Peter Hästö from University of Helsinki.

Hyungsung Yun (KIAS)

TBA
TBA

Jaejin Choi (Pusan National Unverisity)

TBA
TBA

Gu gyum Ha (Sogang University)

TBA
TBA

Aesol Jeon (Seoul National University)

TBA
TBA

Bogi Kim (Kyungpook National University)

Regularity for double phase functionals with two modulating coefficients
In this talk, we deal with local minimizers of functionals with non-standard growth conditions and non-uniform ellipticity properties. Additionally, we establish that a local minimizer is Hölder continuous and that its gradient is also Hölder continuous when additional conditions are provided.