The Dirichlet problem with entire data for non-hyperbolic quadratic hypersurfaces

TL;DR

This paper establishes a lower bound inequality for homogeneous polynomials on spheres and uses it to prove the existence of entire harmonic solutions to the Dirichlet problem with entire data on non-hyperbolic quadratic hypersurfaces.

Contribution

It introduces a new inequality for homogeneous polynomials on spheres and applies it to solve the Dirichlet problem with entire data on specific hypersurfaces.

Findings

Proved a lower bound inequality for homogeneous polynomials on spheres.

Established existence of entire harmonic solutions for certain Dirichlet problems.

Applied the inequality to non-hyperbolic quadratic hypersurfaces.

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+ 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.