Next Talk
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)
Zoom Link: https://us02web.zoom.us/my/repp.seminar (Meeting ID: 319 319 3319, PW: 112358)
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.
Later Talks
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.
Speaker: Prof. Kyungkeun Kang (Yonsei University)
Title: TBA
Date: Wednesday, October 29, 2025
Time: 16:30-17:30 (KST)
Abstract: TBA