Optimal decay of heteroclinic solutions of the fractional Allen-Cahn equation with a degenerate potential
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| Format: | Preprint |
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2026
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| _version_ | 1866915923539001344 |
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| author | De Pas, Francesco Dipierro, Serena Valdinoci, Enrico |
| author_facet | De Pas, Francesco Dipierro, Serena Valdinoci, Enrico |
| contents | We refine the asymptotic estimates for minimizers of a class of nonlocal energy functionals of the form \[ \frac{1}{4} \iint_{\R^{2n} \setminus (\R^n \setminus Ω)^2} \snr{u(x) - u(y)}^2 K(x - y) \,dx\,dy + \int_ΩW(u(x)) \,dx, \] as originally studied in~\cite{DPDV}, and we prove the optimality of our improved bounds.
Here, $W$ denotes a possibly \emph{degenerate} oscillatory double-well potential, satisfying a polynomial control on its second derivative near the wells. The kernel~$K$ belongs to a broad class of measurable functions and is modeled on the one of the fractional Laplacian. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_06721 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal decay of heteroclinic solutions of the fractional Allen-Cahn equation with a degenerate potential De Pas, Francesco Dipierro, Serena Valdinoci, Enrico Analysis of PDEs We refine the asymptotic estimates for minimizers of a class of nonlocal energy functionals of the form \[ \frac{1}{4} \iint_{\R^{2n} \setminus (\R^n \setminus Ω)^2} \snr{u(x) - u(y)}^2 K(x - y) \,dx\,dy + \int_ΩW(u(x)) \,dx, \] as originally studied in~\cite{DPDV}, and we prove the optimality of our improved bounds. Here, $W$ denotes a possibly \emph{degenerate} oscillatory double-well potential, satisfying a polynomial control on its second derivative near the wells. The kernel~$K$ belongs to a broad class of measurable functions and is modeled on the one of the fractional Laplacian. |
| title | Optimal decay of heteroclinic solutions of the fractional Allen-Cahn equation with a degenerate potential |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.06721 |