Heteroclinic connections for fractional Allen-Cahn equations with degenerate potentials
Francesco De Pas, Serena Dipierro, Mirco Piccinini, Enrico Valdinoci
TL;DR
The paper analyzes a nonlocal Ginzburg-Landau-type energy with a possibly degenerate double-well potential and general kernels modeled on the fractional Laplacian. It develops a robust variational framework, barrier methods, and 1D limiting arguments to establish existence and (in 1D) uniqueness (up to translations) of a nontrivial class A minimizer $u^{(0)}$ solving $L_K u = W'(u)$ with $u(x)\to\pm1$ as $x\to\pm\infty$, monotonicity, and precise tail decay rates that depend on the potential exponents. In the symmetric case, the minimizer is odd, and both minimizers and their derivatives exhibit optimal decay, with a finite renormalized energy $\mathcal{G}(u^{(0)})<\infty$. The results extend prior work to degenerate potentials and general kernels by leveraging barrier arguments, convexity-type estimates on $W$, and a detailed 1D limiting analysis relevant to nonlocal phase transitions and crystal dislocation models.
Abstract
We investigate existence, uniqueness and asymptotic behavior of minimizers of a family of non-local energy functionals of the type $$ \frac{1}{4}\iint_{\mathbb{R}^{2n}\setminus (\mathbb{R}^n \setminus Ω)^2}|u(x)-u(y)|^2 K(x-y) \,dx dy + \int_ΩW(u(x)) \,dx. $$ Here, $W$ is a possibly degenerate double well potential with a polynomial control on its second derivative near the wells. Also, ${K}$ belongs to a wide class of measurable kernels and is modeled on that of the fractional Laplacian.