The Dirichlet problem with entire data for non-hyperbolic quadratic hypersurfaces
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917181247193088 |
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| author | Aldaz, J. M. Render, H. |
| author_facet | Aldaz, J. M. Render, H. |
| contents | We show that for all homogeneous polynomials $
f_{m}$ of degree $m$, in $d$ variables,
and each $j = 1, \dots , d$, we have
\begin{equation*}
\left\langle x_{j}^{2}f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb{S}%
^{d-1}\right) }
\geq
\frac{π^{2}}{4\left( m+ 2 d + 1 \right)^{2}}
\left
\langle
f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb{S}^{d-1}\right) }.
\end{equation*} This result is used to establish the existence of entire harmonic solutions of the Dirichlet problem, when the data are given by entire functions of order sufficiently low on nonhyperbolic quadratic hypersurfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_16735 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Dirichlet problem with entire data for non-hyperbolic quadratic hypersurfaces Aldaz, J. M. Render, H. Analysis of PDEs 35A20 We show that for all homogeneous polynomials $ f_{m}$ of degree $m$, in $d$ variables, and each $j = 1, \dots , d$, we have \begin{equation*} \left\langle x_{j}^{2}f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb{S}% ^{d-1}\right) } \geq \frac{π^{2}}{4\left( m+ 2 d + 1 \right)^{2}} \left \langle f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb{S}^{d-1}\right) }. \end{equation*} This result is used to establish the existence of entire harmonic solutions of the Dirichlet problem, when the data are given by entire functions of order sufficiently low on nonhyperbolic quadratic hypersurfaces. |
| title | The Dirichlet problem with entire data for non-hyperbolic quadratic hypersurfaces |
| topic | Analysis of PDEs 35A20 |
| url | https://arxiv.org/abs/2404.16735 |