Density estimates for a (non)local variational model with degenerate double-well potential
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913915116453888 |
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| author | Dipierro, Serena Farina, Alberto Giacomin, Giovanni Valdinoci, Enrico |
| author_facet | Dipierro, Serena Farina, Alberto Giacomin, Giovanni Valdinoci, Enrico |
| contents | In this paper we provide density estimates for a class of functions which includes all the minimizers of the energy
$\mathcal{E}_s^p(u,Ω):=(1-s)\left(\frac{1}{2}\int_Ω\int_Ω\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\,dx\,dy +\int_Ω\int_{\mathbb{R}^n \setminus Ω}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\,dx\,dy\right)+\int_ΩW(u(x))\,dx,$
where $p\in (1,+\infty)$, $s \in \left(0,1\right)$ and $W$ is a double-well potential with polynomial growth $m\in \left[p,+\infty\right)$ from the minima. The nonlocal estimates obtained are uniform as $s\to1$.
Moreover, making use of a $Γ$-convergence result for $\mathcal{E}_s^p$ as $s\to 1$, we obtain density estimates for the minimizers of the limit energy functional, which takes the form
$\mathcal{E}_1^p(u,Ω):=\frac{K_{n,p}}{2p}\int_Ω \left|\nabla u(x)\right|^p+\int_Ω W(u(x))\,dx,$
for a suitable $K_{n,p}\in (0,+\infty)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_22193 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Density estimates for a (non)local variational model with degenerate double-well potential Dipierro, Serena Farina, Alberto Giacomin, Giovanni Valdinoci, Enrico Analysis of PDEs In this paper we provide density estimates for a class of functions which includes all the minimizers of the energy $\mathcal{E}_s^p(u,Ω):=(1-s)\left(\frac{1}{2}\int_Ω\int_Ω\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\,dx\,dy +\int_Ω\int_{\mathbb{R}^n \setminus Ω}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\,dx\,dy\right)+\int_ΩW(u(x))\,dx,$ where $p\in (1,+\infty)$, $s \in \left(0,1\right)$ and $W$ is a double-well potential with polynomial growth $m\in \left[p,+\infty\right)$ from the minima. The nonlocal estimates obtained are uniform as $s\to1$. Moreover, making use of a $Γ$-convergence result for $\mathcal{E}_s^p$ as $s\to 1$, we obtain density estimates for the minimizers of the limit energy functional, which takes the form $\mathcal{E}_1^p(u,Ω):=\frac{K_{n,p}}{2p}\int_Ω \left|\nabla u(x)\right|^p+\int_Ω W(u(x))\,dx,$ for a suitable $K_{n,p}\in (0,+\infty)$. |
| title | Density estimates for a (non)local variational model with degenerate double-well potential |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.22193 |