| Abstract |
Self-improving properties are a kind of fundamental regularity results in the theory of elliptic equations. However, for nonlocal elliptic equations, establishing this property is significantly more comlicated than in the local setting. In the first part of this talk, we will discuss several results and arguments that demonstrate this self-improving property. 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 by constructing a counterexample. We will present a nonlocal analogue of Meyers' example. The construction the example is based on Fourier transform techniques for distributional convolutions. One of the key feature 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). |
