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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.15262 |
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| _version_ | 1866908373047312384 |
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| author | Hu, Yuwei Zheng, Jun Tavares, Leandro S. |
| author_facet | Hu, Yuwei Zheng, Jun Tavares, Leandro S. |
| contents | Let $N>2$, $p\in \left(\frac{2N}{N+2},+\infty\right)$, and $Ω$ be an open bounded domain in $\mathbb{R}^N$. We consider the minimum problem
$$
\mathcal{J} (u) := \displaystyle\int_{Ω} \left(\frac{1}{p}| \nabla u| ^p+λ_1\left(1-(u^+)^2\right)^2+λ_2u^+\right)\text{d}x\rightarrow \text{min}
$$
over a certain class $\mathcal{K}$, where $λ_1\geq 0$ and $ λ_2\in \mathbb{R}$ are constants, and $u^+:=\max\{u,0\}$.
The corresponding Euler-Lagrange equation is related to the Ginzburg-Landau equation and involves a subcritical exponent when $λ_1>0$.
For $λ_1\geq 0$ and $ λ_2\in \mathbb{R}$, we prove the existence, non-negativity, and uniform boundedness of minimizers of $\mathcal{J} (u) $. Then, we show that any minimizer is locally $C^{1,α}$-continuous with some $α\in (0,1)$ and admits the optimal growth $\frac{p}{p-1}$ near the free boundary. Finally, under the additional assumption that $λ_2>0$, we establish non-degeneracy for minimizers near the free boundary and show that there exists at least one minimizer for which the corresponding free boundary has finite ($N-1$)-dimensional Hausdorff measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15262 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A minimum problem associated with scalar Ginzburg-Landau equation and free boundary Hu, Yuwei Zheng, Jun Tavares, Leandro S. Analysis of PDEs Let $N>2$, $p\in \left(\frac{2N}{N+2},+\infty\right)$, and $Ω$ be an open bounded domain in $\mathbb{R}^N$. We consider the minimum problem $$ \mathcal{J} (u) := \displaystyle\int_{Ω} \left(\frac{1}{p}| \nabla u| ^p+λ_1\left(1-(u^+)^2\right)^2+λ_2u^+\right)\text{d}x\rightarrow \text{min} $$ over a certain class $\mathcal{K}$, where $λ_1\geq 0$ and $ λ_2\in \mathbb{R}$ are constants, and $u^+:=\max\{u,0\}$. The corresponding Euler-Lagrange equation is related to the Ginzburg-Landau equation and involves a subcritical exponent when $λ_1>0$. For $λ_1\geq 0$ and $ λ_2\in \mathbb{R}$, we prove the existence, non-negativity, and uniform boundedness of minimizers of $\mathcal{J} (u) $. Then, we show that any minimizer is locally $C^{1,α}$-continuous with some $α\in (0,1)$ and admits the optimal growth $\frac{p}{p-1}$ near the free boundary. Finally, under the additional assumption that $λ_2>0$, we establish non-degeneracy for minimizers near the free boundary and show that there exists at least one minimizer for which the corresponding free boundary has finite ($N-1$)-dimensional Hausdorff measure. |
| title | A minimum problem associated with scalar Ginzburg-Landau equation and free boundary |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.15262 |