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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2501.13022 |
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| _version_ | 1866910794757701632 |
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| author | da Silva, João Vítor Rossi, Julio Salort, Ariel |
| author_facet | da Silva, João Vítor Rossi, Julio Salort, Ariel |
| contents | We study regularity issues and the limiting behavior as $p\to\infty$ of nonnegative solutions for elliptic equations of $p-$Laplacian type ($2 \leq p< \infty$) with a strong absorption: $$
-Δ_p u(x) + λ_0(x) u_{+}^q(x) = 0 \quad \text{ in } \quad Ω\subset \mathbb{R}^N, $$ where $λ_0>0$ is a bounded function, $Ω$ is a bounded domain and $0\leq q<p-1$. When $p$ is fixed, such a model is mathematically interesting since it permits the formation of dead core zones, i.e, a priori unknown regions where non-negative solutions vanish identically. First, we turn our attention to establishing sharp quantitative regularity properties for $p-$dead core solutions. Afterwards, assuming that $\ell \:=\lim_{p \to \infty} q(p)/p \in [0, 1)$ exists, we establish existence for limit solutions as $p\to \infty$, as well as we characterize the corresponding limit operator governing the limit problem. We also establish sharp $C^γ$ regularity estimates for limit solutions along free boundary points, that is, points on $ \partial \{u>0\} \cap Ω$ where the sharp regularity exponent is given explicitly by $γ= \frac{1}{1-\ell}$. Finally, some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density, porosity and convergence of the free boundaries are proved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13022 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity properties for $p-$dead core problems and their asymptotic limit as $p \to \infty$ da Silva, João Vítor Rossi, Julio Salort, Ariel Analysis of PDEs We study regularity issues and the limiting behavior as $p\to\infty$ of nonnegative solutions for elliptic equations of $p-$Laplacian type ($2 \leq p< \infty$) with a strong absorption: $$ -Δ_p u(x) + λ_0(x) u_{+}^q(x) = 0 \quad \text{ in } \quad Ω\subset \mathbb{R}^N, $$ where $λ_0>0$ is a bounded function, $Ω$ is a bounded domain and $0\leq q<p-1$. When $p$ is fixed, such a model is mathematically interesting since it permits the formation of dead core zones, i.e, a priori unknown regions where non-negative solutions vanish identically. First, we turn our attention to establishing sharp quantitative regularity properties for $p-$dead core solutions. Afterwards, assuming that $\ell \:=\lim_{p \to \infty} q(p)/p \in [0, 1)$ exists, we establish existence for limit solutions as $p\to \infty$, as well as we characterize the corresponding limit operator governing the limit problem. We also establish sharp $C^γ$ regularity estimates for limit solutions along free boundary points, that is, points on $ \partial \{u>0\} \cap Ω$ where the sharp regularity exponent is given explicitly by $γ= \frac{1}{1-\ell}$. Finally, some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density, porosity and convergence of the free boundaries are proved. |
| title | Regularity properties for $p-$dead core problems and their asymptotic limit as $p \to \infty$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.13022 |