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The paper is devoted to the derivation of $\\mathcal{C}^{0,\\alpha}$ estimates that are uniform in the competition parameter $\\beta > 0$, as well as to the regularity of the limiting free-boundary problem obtained for $\\beta \\to + \\infty$.\n  The main novelty of the problem under consideration resides in the non-trivial grouping of the densities: in particular, we assume that"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1506.00800","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-06-02T09:00:52Z","cross_cats_sorted":[],"title_canon_sha256":"8e7939d7f1450386a82e88ee5d9221018c80beda1aaf92525536941bb82b3902","abstract_canon_sha256":"f7d560d0450121a3536f147096bb132142d88e4033740fd08419e8185510b676"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:29.243878Z","signature_b64":"qfLO6hmc9n2MpBS+Mal1xut0x+oagsGr8n0s3uOHN4p8LTVeCa+1aT+xkTGMup1KFvf6yRFbgOctqOJXh96JCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9371f1ae1869a1ee17d603c8551674bc1587039a88e0bdf0c5a66cf7d3f3df6c","last_reissued_at":"2026-05-18T01:01:29.243458Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:29.243458Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"H\\\"older bounds and regularity of emerging free boundaries for strongly competing Schr\\\"odinger equations with nontrivial grouping","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessandro Zilio, Hugo Tavares, Nicola Soave, Susanna Terracini","submitted_at":"2015-06-02T09:00:52Z","abstract_excerpt":"We study regularity issues for systems of elliptic equations of the type \\[\n  -\\Delta u_i=f_{i,\\beta}(x)-\\beta \\sum_{j\\neq i} a_{ij} u_i |u_i|^{p-1}|u_j|^{p+1} \\] set in domains $\\Omega \\subset \\mathbb{R}^N$, for $N \\geq 1$. 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