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

Abstract

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

arXiv commentAnn. Mat. Pura Appl

References in corpus (3)

• Gamma-convergence of nonlocal perimeter functionals
• Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions
• Regularity results for nonlocal equations by approximation

Cited by in corpus (3)

• Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions
• Density estimates for a variational model driven by the Gagliardo norm
• Some Energy Estimates for Stable Solutions to Fractional Allen-Cahn Equations