Asymptotics for Sobolev extremals: the hyperdiffusive case
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916445296787456 |
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| author | Ercole, Grey |
| author_facet | Ercole, Grey |
| contents | Let $Ω$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ For $p>N$ and $1\leq q(p)<\infty$ set \[ λ_{p,q(p)}:=\inf\left\{ \int_Ω\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in W_{0}^{1,p}(Ω)\text{ \ and \ }\int_{Ω}\left\vert u\right\vert ^{q(p)}\mathrm{d}x=1\right\} \] and let $u_{p,q(p)}$ denote a corresponding positive extremal function. We show that if $\lim\limits_{p\rightarrow\infty}q(p)=\infty$, then $\lim\limits_{p\rightarrow\infty}λ_{p,q(p)}^{1/p}=\left\Vert d_{Ω}\right\Vert _{\infty}^{-1}$, where $d_Ω$ denotes the distance function to the boundary of $Ω.$ Moreover, in the hyperdiffusive case: $\lim\limits_{p\rightarrow\infty}\frac{q(p)}{p}=\infty,$ we prove that each sequence $u_{p_{n},q(p_{n})},$ with $p_{n}\rightarrow\infty,$ admits a subsequence converging uniformly in $\overlineΩ$ to a viscosity solution to the problem \[ \left\{ \begin{array} [c]{lll} -Δ_{\infty}u=0 & \text{in} & Ω\setminus M\\ u=0 & \text{on} & \partialΩ\\ u=1 & \text{in} & M, \end{array} \right. \] where $M$ is a closed subset of the set of all maximum points of $d_Ω.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_17103 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotics for Sobolev extremals: the hyperdiffusive case Ercole, Grey Analysis of PDEs 35B40, 35J92, 35J94 Let $Ω$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ For $p>N$ and $1\leq q(p)<\infty$ set \[ λ_{p,q(p)}:=\inf\left\{ \int_Ω\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in W_{0}^{1,p}(Ω)\text{ \ and \ }\int_{Ω}\left\vert u\right\vert ^{q(p)}\mathrm{d}x=1\right\} \] and let $u_{p,q(p)}$ denote a corresponding positive extremal function. We show that if $\lim\limits_{p\rightarrow\infty}q(p)=\infty$, then $\lim\limits_{p\rightarrow\infty}λ_{p,q(p)}^{1/p}=\left\Vert d_{Ω}\right\Vert _{\infty}^{-1}$, where $d_Ω$ denotes the distance function to the boundary of $Ω.$ Moreover, in the hyperdiffusive case: $\lim\limits_{p\rightarrow\infty}\frac{q(p)}{p}=\infty,$ we prove that each sequence $u_{p_{n},q(p_{n})},$ with $p_{n}\rightarrow\infty,$ admits a subsequence converging uniformly in $\overlineΩ$ to a viscosity solution to the problem \[ \left\{ \begin{array} [c]{lll} -Δ_{\infty}u=0 & \text{in} & Ω\setminus M\\ u=0 & \text{on} & \partialΩ\\ u=1 & \text{in} & M, \end{array} \right. \] where $M$ is a closed subset of the set of all maximum points of $d_Ω.$ |
| title | Asymptotics for Sobolev extremals: the hyperdiffusive case |
| topic | Analysis of PDEs 35B40, 35J92, 35J94 |
| url | https://arxiv.org/abs/2404.17103 |