{"paper":{"title":"Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xavier Cabre","submitted_at":"2011-02-15T15:58:19Z","abstract_excerpt":"We establish the uniqueness of a saddle-shaped solution to the diffusion equation $-\\Delta u = f(u)$ in all of $\\mathbb{R}^{2m}$, where $f$ is of bistable type, in every even dimension $2m \\geq 2$. In addition, we prove its stability whenever $2m \\geq 14$. Saddle-shaped solutions are odd with respect to the Simons cone ${\\mathcal C} = \\{(x^1,x^2) \\in \\mathbb{R}^m \\times \\mathbb{R}^m : |x^1|=|x^2| \\}$ and exist in all even dimensions. Their uniqueness was only known when $2m=2$. On the other hand, they are known to be unstable in dimensions 2, 4, and 6. Their stability in dimensions 8, 10, and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.3111","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"}