References & Citations
Mathematics > Analysis of PDEs
Title: The Dirichlet problem with entire data for non-hyperbolic quadratic hypersurfaces
(Submitted on 25 Apr 2024)
Abstract: 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{\pi ^{2}}{4\left( m+ 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.
Submission history
From: Jesus Munarriz Aldaz [view email][v1] Thu, 25 Apr 2024 16:44:45 GMT (12kb)
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