Nonlocal equations with various kernels
A talk in the Nonlocal Equations: Analysis and Numerics series by
Jihoon Ok from Seoul
| Abstract: | Nonlocal equations with p-growth is modeled by $P.V. \int_{\mathbb{R}^n} |u(x) - u(y)|^{p-2} (u(x) - u(y)) k(x,y) dy = 0$, $x \in \Omega$ where $1 < p < \infty$ and $\Omega \subset \mathbb{R}^n$ with $n \geq 2$. If the kernel $K : \mathbb{R}^n \times \mathbb{R}^n \rightarrow [0, \infty)$ satisfies the following s-order uniform ellipticity condition for some $s \in (0, 1)$: $\dfrac{1}{\Lambda} \dfrac{1}{|x-y|^{n+sp}} \leq K(x,y) \leq \Lambda \dfrac{1}{|x-y|^{n+sp}}$ where $\Lambda \geq 1$ is a constant, then weak solutions of these nonlocal equations are Hölder continuous and satisfy the Harnack inequality. Our main interest is to investigate conditions on kernels that do not satisfy the above uniform ellipticity but still lead to these regularity results. In this talk, I will introduce two classes of such kernels. The first class consists of degenerate or singular kernels associated with the Muckenhoupt class. The second class covers general kernels that include both the above uniformly elliptic cases and borderline cases as s approaches 1. This is based on joint works with Linus Behn, Lars Diening, and Julian Rolfes (Bielefeld University), and Kyeong Song (KIAS, Seoul). Within the CRC this talk is associated to the project(s): A7 |