Long-range phase coexistence models with degenerate potentials

This survey offers an overview of recent advances in nonlocal phase transition problems, modeled by Ginzburg–Landau type energies of the form $$\frac{1}{4}{\iint }_{{\mathbb{R}}^{2n}\setminus ({\mathbb{R}}^n\setminus \mathrm{\Omega }{)}^2}\frac{\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+2s}}dxdy+{\int }_{\mathrm{\Omega }}W(u(x))dx\mathrm{.}$$ Here,~$W$ is a smooth and possibly $\mathit{\text{degenerate}}$ double well potential, with a polynomial control on its second derivatives near the wells. The emphasis is on qualitative properties of minimizers and critical points of the energy functional.