Regularity of stable solutions to the MEMS problem up to the optimal dimension 6
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908915252330496 |
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| author | Bruera, Renzo Cabre, Xavier |
| author_facet | Bruera, Renzo Cabre, Xavier |
| contents | In this article we address the regularity of stable solutions to semilinear elliptic equations $-Δu = f(u)$ with MEMS type nonlinearities. More precisely, we will have $0\leq u \leq 1$ in a domain $Ω\subset \mathbb{R}^n$ and $f:[0,1)\to (0,+\infty)$ blowing up at $u=1$ and nonintegrable near 1. In this context, a solution $u$ is regular if $u<1$ in all $Ω$ or, equivalently, if $-Δu = f(u)<+\infty$ in $Ω$.
This paper establishes for the first time interior regularity estimates that are independent of the boundary condition that $u$ may satisfy. Our results hold up to the optimal dimension $n=6$ (there are counterexamples for $n\geq 7$) but require a Crandall-Rabinowitz type assumption on the nonlinearity $f$. Our main estimate controls the $L^\infty$ norm of $F(u)$ in a ball, where $F$ is a primitive of $f$, by only the $L^1$ norm of $u$ in a larger ball.
Under the same assumptions, we also give global estimates in dimensions $n\leq 6$ for the Dirichlet problem with vanishing boundary condition, improving previously known results. For $n\leq 2$, we do not need a Crandall-Rabinowitz type assumption and, thus, our global estimate holds for all nonnegative, nondecreasing, convex nonlinearities which blow up at 1 and are nonintegrable near 1. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20916 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity of stable solutions to the MEMS problem up to the optimal dimension 6 Bruera, Renzo Cabre, Xavier Analysis of PDEs In this article we address the regularity of stable solutions to semilinear elliptic equations $-Δu = f(u)$ with MEMS type nonlinearities. More precisely, we will have $0\leq u \leq 1$ in a domain $Ω\subset \mathbb{R}^n$ and $f:[0,1)\to (0,+\infty)$ blowing up at $u=1$ and nonintegrable near 1. In this context, a solution $u$ is regular if $u<1$ in all $Ω$ or, equivalently, if $-Δu = f(u)<+\infty$ in $Ω$. This paper establishes for the first time interior regularity estimates that are independent of the boundary condition that $u$ may satisfy. Our results hold up to the optimal dimension $n=6$ (there are counterexamples for $n\geq 7$) but require a Crandall-Rabinowitz type assumption on the nonlinearity $f$. Our main estimate controls the $L^\infty$ norm of $F(u)$ in a ball, where $F$ is a primitive of $f$, by only the $L^1$ norm of $u$ in a larger ball. Under the same assumptions, we also give global estimates in dimensions $n\leq 6$ for the Dirichlet problem with vanishing boundary condition, improving previously known results. For $n\leq 2$, we do not need a Crandall-Rabinowitz type assumption and, thus, our global estimate holds for all nonnegative, nondecreasing, convex nonlinearities which blow up at 1 and are nonintegrable near 1. |
| title | Regularity of stable solutions to the MEMS problem up to the optimal dimension 6 |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.20916 |