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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.21899 |
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| _version_ | 1866915217493983232 |
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| author | Alcantara, Claudemir da Silva, João Vitor Sá, Ginaldo |
| author_facet | Alcantara, Claudemir da Silva, João Vitor Sá, Ginaldo |
| contents | In this manuscript, we investigate geometric regularity estimates for problems governed by quasi-linear elliptic models in non-divergence form, which may exhibit either degenerate or singular behavior when the gradient vanishes, under strong absorption conditions of the form: \[ |\nabla u(x)|^γ Δ_p^{\mathrm{N}} u(x) = f(x, u) \quad \text{in} \quad B_1, \] where $γ> -1$, $p \in (1, \infty)$, and the mapping $u \mapsto f(x, u) \lesssim \mathfrak{a}(x) u_{+}^m$ (with $m \in [0, γ+ 1)$) does not decay sufficiently fast at the origin. This condition allows for the emergence of plateau regions, i.e., a priori unknown subsets where the non-negative solution vanishes identically. We establish improved geometric $\mathrm{C}^κ_{\text{loc}}$ regularity along the set $\mathscr{F}_0 = \partial \{u > 0\} \cap B_1$ (the free boundary of the model) for a sharp value of $κ\gg 1$, which is explicitly determined in terms of the structural parameters. Additionally, we derive non-degeneracy results and other measure-theoretic properties. Furthermore, we prove a sharp Liouville theorem for entire solutions exhibiting controlled growth at infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21899 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric regularity estimates for quasi-linear elliptic models in non-divergence form with strong absorption Alcantara, Claudemir da Silva, João Vitor Sá, Ginaldo Analysis of PDEs In this manuscript, we investigate geometric regularity estimates for problems governed by quasi-linear elliptic models in non-divergence form, which may exhibit either degenerate or singular behavior when the gradient vanishes, under strong absorption conditions of the form: \[ |\nabla u(x)|^γ Δ_p^{\mathrm{N}} u(x) = f(x, u) \quad \text{in} \quad B_1, \] where $γ> -1$, $p \in (1, \infty)$, and the mapping $u \mapsto f(x, u) \lesssim \mathfrak{a}(x) u_{+}^m$ (with $m \in [0, γ+ 1)$) does not decay sufficiently fast at the origin. This condition allows for the emergence of plateau regions, i.e., a priori unknown subsets where the non-negative solution vanishes identically. We establish improved geometric $\mathrm{C}^κ_{\text{loc}}$ regularity along the set $\mathscr{F}_0 = \partial \{u > 0\} \cap B_1$ (the free boundary of the model) for a sharp value of $κ\gg 1$, which is explicitly determined in terms of the structural parameters. Additionally, we derive non-degeneracy results and other measure-theoretic properties. Furthermore, we prove a sharp Liouville theorem for entire solutions exhibiting controlled growth at infinity. |
| title | Geometric regularity estimates for quasi-linear elliptic models in non-divergence form with strong absorption |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.21899 |