An Overdetermined Neumann boundary value problem with a general driving force
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913833093693440 |
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| author | Minlend, Ignace Aristide Wu, Jing |
| author_facet | Minlend, Ignace Aristide Wu, Jing |
| contents | In this paper, we prove the existence of a family of non trivial compact subdomains $Ø$ in the manifold $\mathcal{M}=\R^N\times \R/2π\Z, N\geq 2$ for which the overdetermined Neumann boundary value problem
\begin{align}\label{Neumann1}
\left \{
\begin{aligned}
$-\D w&=μg(w) && \qquad \text{in $ Ω$,}$
\frac{\partial w}{\partialη} &=0 &&\qquad \text{on $\partial Ω,$}
w&=c\ne 0 &&\qquad \text{on $\partial Ω$,}
\end{aligned}
\right. \end{align}
admits solutions for some $μ> 0$ and a $C^{1, α}$ function $g:\R \rightarrow \R.$ The domains we construct have nonconstant principal curvature, and therefore are not isoparametric nor homogeneous. The argument we develop applies for both linear and non-linear functions $g$. By this, we generalise a recent result obtained by Fall, Weth and the first named author in \cite{Fall-MinlendI-Weth4}, where the overdetermined Neumann eigenvalue problem for the Laplacian was considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07063 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Overdetermined Neumann boundary value problem with a general driving force Minlend, Ignace Aristide Wu, Jing Analysis of PDEs 2020: 35J57, 35J66, 35N25, 35J25, 35R35, 58J55 In this paper, we prove the existence of a family of non trivial compact subdomains $Ø$ in the manifold $\mathcal{M}=\R^N\times \R/2π\Z, N\geq 2$ for which the overdetermined Neumann boundary value problem \begin{align}\label{Neumann1} \left \{ \begin{aligned} $-\D w&=μg(w) && \qquad \text{in $ Ω$,}$ \frac{\partial w}{\partialη} &=0 &&\qquad \text{on $\partial Ω,$} w&=c\ne 0 &&\qquad \text{on $\partial Ω$,} \end{aligned} \right. \end{align} admits solutions for some $μ> 0$ and a $C^{1, α}$ function $g:\R \rightarrow \R.$ The domains we construct have nonconstant principal curvature, and therefore are not isoparametric nor homogeneous. The argument we develop applies for both linear and non-linear functions $g$. By this, we generalise a recent result obtained by Fall, Weth and the first named author in \cite{Fall-MinlendI-Weth4}, where the overdetermined Neumann eigenvalue problem for the Laplacian was considered. |
| title | An Overdetermined Neumann boundary value problem with a general driving force |
| topic | Analysis of PDEs 2020: 35J57, 35J66, 35N25, 35J25, 35R35, 58J55 |
| url | https://arxiv.org/abs/2405.07063 |