Talk #4
Speaker: Dr. Se-Chan Lee (Korea Institute for Advanced Study)
Title: Time derivative estimates for parabolic $p$-Laplace equations and applications to optimal regularity
Date: Tuesday, October 14, 2025
Time: 16:30-17:30 (KST)
Abstract: In this talk, we establish the boundedness of time derivatives of solutions to parabolic $p$-Laplace equations. Our approach relies on the Bernstein technique combined with a suitable approximation method. As a consequence, we obtain an optimal regularity result with a connection to the well-known $C^{p'}$-conjecture in the elliptic setting. Moreover, we extend our method to deal with global regularity results for both fully nonlinear and general quasilinear degenerate parabolic problems.
Talk #3
Speaker: Prof. Jihoon Ok (Sogang University)
Title: Mean oscillation condition on nonlinear equations and regularity results
Date: Tuesday, September 30, 2025
Time: 16:30-17:30 (KST)
Abstract: We consider general nonlinear elliptic equations of the form
\[
\mathrm{div}\, A(x,Du) = 0 \quad \text{in } \Omega,
\]
where $A:\Omega \times \mathbb{R}^n \to \mathbb{R}^n$ satisfies a quasi-isotropic $(p,q)$-growth condition, which is equivalent the pointwise uniform ellipticity of $A(x,\xi)$ under a suitable $(p,q)$-growth condition. We establish sharp and comprehensive mean oscillation conditions on $A(x,\xi)$ with respect to the $x$ variable to obtain $C^1$- and $W^{1,\gamma}$-regularity results. The results provide new conditions, even in special cases such as $A(x,\xi)=a(x)|\xi|^{p-2}\xi$ and $A(x,\xi)=|\xi|^{p(x)-2}\xi$. This is joint work with Peter H\"ast\"o from University of Helsinki and Mikyoung Lee from Pusan National University.
Talk #2
Speaker: Prof. Minhyun Kim (Hanyang University)
Title: Green function estimates for nonlocal equations
Date: Tuesday, September 16, 2025
Time: 16:30-17:30 (KST)
Abstract: We establish the optimal regularity for solutions to nonlocal elliptic equations with Hölder continuous coefficients in divergence form in bounded $C^{1,\alpha}$ domains. Our proof is based on a delicate higher order Campanato-type iteration at the boundary, which we develop in the context of nonlocal equations and which is quite different from the local theory. As an application of our results, we establish sharp two-sided Green function estimates for the same class of operators.
Talk #1
Speaker: Prof. Sun-Sig Byun (Seoul National University)
Title: Nonlinear Calderón-Zygmund Estimates for Parabolic Equations with Matrix Weights
Date: Tuesday, September 2, 2025
Time: 16:30-17:30 (KST)
Abstract: We investigate nonlinear parabolic equations whose coefficients act like matrix weights, distorting gradients in different directions rather than scaling them uniformly. Such anisotropic effects arise naturally in degenerate or heterogeneous media. Our main result is a Calderón-Zygmund type regularity result: whenever the forcing term has higher integrability, the solution’s gradient inherits the same improved integrability. This extends the classical weighted Calderón-Zygmund theory to a nonlinear, matrix-weighted framework.