| Abstract: |
| The $p$-fractional capacitary problem provides a natural nonlocal counterpart of classical capacity for arbitrary compact sets. A central issue is the behavior at infinity of the associated equilibrium potentials and its relation with the scaling of the energy.
In this talk I will first discuss the decay at infinity in the subcritical regime. I will then turn to the critical threshold, where the picture changes dramatically. I will show that the corresponding fractional capacity vanishes, revealing a sharp degeneracy of the critical regime. This contrast highlights the special role of the threshold case and the need for a different potential-theoretic perspective at criticality. |
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