| Contents |
| Fractional and non-local operators appear in concrete applications
in many fields such as, among the others, optimization, finance,
phase transitions, stratified materials, anomalous diffusion,
crystal dislocation, soft thin films, semipermeable membranes, flame
propagation, conservation laws, ultra-relativistic limits of quantum
mechanics, quasi-geostrophic flows, multiple scattering, minimal
surfaces, materials science and water waves. This is one of the
reason why, recently, non-local fractional problems are widely
studied in the literature.
Aim of this talk will be to present some recent results for nonlocal problems driven by
the fractional Laplace operator $(-\Delta)^s$, which (up to normalization factors) may be defined as
$$-(-\Delta)^s u(x)=
\int_{\mathbb R^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy\,,
\,\,\,\,\, x\in \mathbb R^n\,.$$
These results were obtained through variational and topological methods and extend the validity of some theorems known in the classical case of the Laplacian to the non-local fractional framework. |
|