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

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

2011-04-09

arxiv.org/abs/1104.1725v2

Mathematics

Analysis of PDEs

Ann. Mat. Pura Appl

Scientific paper

We study existence, unicity and other geometric properties of the minimizers of the energy functional $$ \|u\|^2_{H^s(\Omega)}+\int_\Omega W(u)\,dx, $$ where $\|u\|_{H^s(\Omega)}$ 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.

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