{"paper":{"title":"Transition Layer for the Heterogeneous Allen-Cahn Equation","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrea Malchiodi, Fethi Mahmoudi, Juncheng Wei","submitted_at":"2007-02-28T13:31:59Z","abstract_excerpt":"We consider the equation $\\e^{2}\\Delta u=(u-a(x))(u^2-1)$ in $\\Omega$, $\\frac{\\partial u}{\\partial \\nu} =0$ on $\\partial \\Omega$, where $\\Omega$ is a smooth and bounded domain in $\\R^n$, $\\nu$ the outer unit normal to $\\pa\\Omega$, and $a$ a smooth function satisfying $-1<a(x)<1$ in $\\ov{\\Omega}$. We set $K$, $\\Omega_+$ and $\\Omega_-$ to be respectively the zero-level set of $a$, {a>0} and {a<0}. Assuming $\\nabla a \\neq 0$ on $K$ and $a\\ne 0$ on $\\partial \\Omega$, we show that there exists a sequence $\\e_j \\to 0$ such that the above equation has a solution $u_{\\e_j}$ which converges uniformly t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0702878","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}