Speaker
Description
The goal of nonlocal-to-local convergence is to show that certain singular, nonlocal convolution-type integral operators converge to a local differential operator as the convolution kernel concentrates at zero. This can be a useful tool in the physical justification of mathematical models (e.g., the Cahn-Hilliard equation), especially when a desired local differential operator cannot be derived by microscopic laws.
The nonlocal-to-local convergence of convolution operators with radially symmetric (i.e., isotropic) kernels having $W^{1,1}$-regularity is already very well understood. However, the assumption of $W^{1,1}$-regularity is too strong for many applications. Also, in some situations (e.g., crystallization phenomena), convolution kernels are not radially symmetric but merely even (i.e., anisotropic).
In this talk, I will present very recent results from a collaboration with H. Abels and C. Hurm concerning the strong nonlocal-to-local convergence with convergence rates for anisotropic kernels, which merely need to have a significantly lower regularity than $W^{1,1}$. Also applications to Cahn--Hilliard type phase-field models will be discussed.