{"paper":{"title":"An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Joaquim Serra, Xavier Cabre","submitted_at":"2015-10-06T14:40:25Z","abstract_excerpt":"We study nonlinear elliptic equations for operators corresponding to non-stable L\\'evy diffusions. We include a sum of fractional Laplacians of different orders. Such operators are infinitesimal generators of non-stable (i.e., non self-similar) L\\'evy processes. We establish the regularity of solutions, as well as sharp energy estimates. As a consequence, we prove a 1-D symmetry result for monotone solutions to Allen-Cahn type equations with a non-stable L\\'evy diffusion. These operators may still be realized as local operators using a system of PDEs ---in the spirit of the extension problem o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.01596","kind":"arxiv","version":2},"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"}