| Abstract: |
| 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 can not 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 my talk, I will present our recent results on the strong nonlocal-to-local convergence (with rates) for anisotropic kernels with significantly lower regularity assumption compared to previous contributions. These results can, for example, be applied to nonlocal phase-field models such as some anisotropic variants of the Cahn--Hilliard equation. |
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