Calculus of Variations and Geometric Measure Theory

L. De Luca - M. Morini - M. Ponsiglione - E. Spadaro

Parabolic $\alpha$-Riesz flows and limit cases $\alpha\to 0^+$, $\alpha\to d^-$

created by deluca on 16 Jun 2023



Inserted: 16 jun 2023
Last Updated: 16 jun 2023

Year: 2023


In this paper we introduce the notion of parabolic $\alpha$-Riesz flow, for $\alpha\in(0,d)$, extending the notion of $s$-fractional heat flows to negative values of the parameter $s=-\frac{\alpha}{2}$. Then, we determine the limit behaviour of these gradient flows as $\alpha \to 0^+$ and $\alpha \to d^-$.

To this end we provide a preliminary $\Gamma$-convergence expansion for the Riesz interaction energy functionals. Then we apply abstract stability results for uniformly $\lambda$-convex functionals which guarantee that $\Gamma$-convergence commutes with the gradient flow structure.