Calculus of Variations and Geometric Measure Theory

G. Palatucci - O. Savin - E. Valdinoci

Local and global minimizers for a variational energy involving a fractional norm

created by palatucci on 18 Apr 2011
modified on 29 Jul 2013


Published Paper

Inserted: 18 apr 2011
Last Updated: 29 jul 2013

Journal: Ann. Mat. Pura Appl.
Volume: 192
Number: 4
Pages: 673-718
Year: 2013
Doi: 10.1007/s00526-013-0656-y


We study existence, unicity and other geometric properties of the minimizers of the energy functional $$ \
2{Hs(\Omega)}+\int\Omega W(u)\,dx, $$ where $\
_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$ norm of $u$ and $W$ is a double-well potential. We also deal with the solutions of the related fractional elliptic Allen-Cahn equation on the entire space $\mathbb{R}^n$.

The results collected here will also be useful for forthcoming papers, where the second and the third author will study the $\Gamma$-convergence and the density estimates for level sets of minimizers.

Keywords: phase transitions, fractional Laplacian, Nonlocal energy, Gagliardo norm