Next Talk
Speaker: Dr. Ho-Sik Lee (Bielefeld University)
Title: Self-improving properties and their optimality for nonlocal elliptic equations
Date: Tuesday, November 25, 2025
Time: 16:30-17:30 (KST)
Zoom Link: https://us02web.zoom.us/my/repp.seminar (Meeting ID: 319 319 3319, PW: 112358)
Abstract: The self-improving property is a fundamental regularity result in the theory of linear and nonlinear elliptic equations. For nonlocal elliptic equations, however, establishing this property is significantly more complex than in the local setting. In the first part of this talk, we will discuss several key results and arguments that demonstrate this self-improvement phenomenon. If time permits, we will also present an ongoing project on self-improving properties for equations with fractional Orlicz growth, which is joint work with Kyeong Song (KIAS).
In the second part, we will address the optimality of these self-improving properties. Classically, this optimality was established by Meyers in 1963 through the construction of a counterexample. We will present a nonlocal analogue of Meyers' example, the construction is based on Fourier transform techniques for distributional convolutions. One of the key features of our example is its robustness: it remains valid as the order of the nonlocal operator converges to 2, the order of a classical second-order elliptic operator. This is joint work with Anna Balci, Lars Diening, and Moritz Kassmann (Bielefeld University).
Later Talks
Speaker: Dr. Abhrojyoti Sen (Goethe University Frankfurt)
Title: Recent developments in the regularity theory for parabolic double phase problems
Date: Tuesday, December 9, 2025
Time: 16:30-17:30 (KST)
Abstract: In this talk, we discuss recent regularity results for parabolic double phase equations and systems, focusing on Lipschitz regularity in the presence of gradient nonlinearities. Our approach is based on the celebrated Ishii-Lions method for viscosity solutions. We show that when the modulating coefficient is spatially Lipschitz and its zero set satisfies a mild and natural control condition, then bounded weak solutions to parabolic double phase equations with gradient nonlinearity are locally Lipschitz continuous in space and 1/2 Hölder continuous in time. In the end, we discuss some further open problems.