{
  "abstract": "In this manuscript, we analyze a two-phase free boundary problem of Alt-Phillips-type driven by fully nonlinear elliptic equations with lower-order ingredients. More precisely, we consider equations of the type \\[ F(x,D^{2}u)+\\mathscr{H}(x,Du)=\\mathscr{F}(x,u^{+},u^{-}) \\quad \\text{in } B_1 \\subset \\mathbb{R}^{n}, \\] where the Hamiltonian term exhibits mixed linear and nonlinear gradient dependence, namely \\[ \\mathscr{H}(x,ξ):= \\langle \\mathcal{B}(x),ξ\\rangle+\\varrho(x)|ξ|^σ, \\quad 0< σ\\leq2, \\,\\,\\, σ\\neq 1, \\] and the reaction is governed by a two-phase power-type nonlinearity of semilinear-type \\[ \\mathscr{F}(x,u^{+},u^{-}):=\\mathfrak{g}(x)\\big[(u^{+})^{m}-(u^{-})^{m}\\big] \\quad \\text{for} \\quad 0<m<1. \\] Under suitable structural assumptions on the coefficients and the operator $F$, we develop a robust analytical framework to capture the fine properties of solutions near the free boundary. In particular, we establish improved growth estimates at higher-order singular nodal points, points where both phases meet and the solution exhibits critical degeneracy. The results are based upon a fine blow-up argument, stability and a Liouville-type theorem. We further prove quantitative non-degeneracy results, ensuring that solutions detach from zero at a controlled rate on each phase. As a consequence of our analysis, we derive a Liouville-type theorem for global solutions within this class. Our results extend and unify several previously known scenarios, and remain new even in the presence of gradient-dependent nonlinearities of order ($σ\\neq 1$), thereby encompassing models with nonlinear drift and absorption effects arising in phase transition and reaction-diffusion phenomena.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23216",
  "author_total": 4,
  "authors": [
    {
      "name": "Junior da Silva Bessa"
    },
    {
      "name": "João Vitor da Silva"
    },
    {
      "name": "Yuwei Hu"
    },
    {
      "name": "Mayra Soares"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3x8a2bwc8ytts.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23216",
  "fulltext_url": null,
  "id": "3x8a2bwc8ytts",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23216",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7214016506",
  "pdf_url": "https://arxiv.org/pdf/2609.23216",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3x8a2bwc8ytts.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23216",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23216"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A fully nonlinear two-phase Alt-Phillips problem with gradient nonlinearities",
  "title_text": "A fully nonlinear two-phase Alt-Phillips problem with gradient nonlinearities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/3x8a2bwc8ytts",
  "venue": "arXiv",
  "versions": []
}