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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.17543 |
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| _version_ | 1866909971005833216 |
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| author | Giovagnoli, Davide Merlino, Enzo Maria Moreira, Diego |
| author_facet | Giovagnoli, Davide Merlino, Enzo Maria Moreira, Diego |
| contents | We prove a quantitative inhomogeneous Hopf-Oleinik lemma for viscosity solutions of $$|\nabla u|^αF(D^{2}u)=f $$ and, more generally, for viscosity supersolutions of $|\nabla u|^α\,{M}^-_{λ,Λ}(D^{2}u)\le f$. The result yields linear boundary growth with universal constants depending only on the structural data. We also exhibit a counterexample showing that the Hopf lemma fails for equations that act only in the large-gradient regime (in the sense of Imbert and Silvestre), thereby delineating the scope of our theorem. As applications, we obtain Lipschitz regularity for viscosity solutions of one-phase Bernoulli free boundary problems driven by these degenerate fully nonlinear operators and derive $\varepsilon$-uniform Lipschitz bounds for a one-phase flame propagation model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17543 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A quantitative Hopf-Oleinik lemma for degenerate fully nonlinear operators and applications to free boundary problems Giovagnoli, Davide Merlino, Enzo Maria Moreira, Diego Analysis of PDEs Primary 35J70, 35J60, 35R35, Secondary 35D40, 35R45, 35R50 We prove a quantitative inhomogeneous Hopf-Oleinik lemma for viscosity solutions of $$|\nabla u|^αF(D^{2}u)=f $$ and, more generally, for viscosity supersolutions of $|\nabla u|^α\,{M}^-_{λ,Λ}(D^{2}u)\le f$. The result yields linear boundary growth with universal constants depending only on the structural data. We also exhibit a counterexample showing that the Hopf lemma fails for equations that act only in the large-gradient regime (in the sense of Imbert and Silvestre), thereby delineating the scope of our theorem. As applications, we obtain Lipschitz regularity for viscosity solutions of one-phase Bernoulli free boundary problems driven by these degenerate fully nonlinear operators and derive $\varepsilon$-uniform Lipschitz bounds for a one-phase flame propagation model. |
| title | A quantitative Hopf-Oleinik lemma for degenerate fully nonlinear operators and applications to free boundary problems |
| topic | Analysis of PDEs Primary 35J70, 35J60, 35R35, Secondary 35D40, 35R45, 35R50 |
| url | https://arxiv.org/abs/2512.17543 |