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

J. M. Aldaz, H. Render

TL;DR

This work addresses the Dirichlet problem on nonhyperbolic quadratic hypersurfaces in $\mathbb{R}^d$ and studies when data given by entire functions admit entire harmonic extensions. A central ingredient is a sharp spectral lower bound for the quadratic form $\langle x_j^2 f_m, f_m\rangle_{L^2(\mathbb{S}^{d-1})}$ on homogeneous polynomials, obtained via a spherical-harmonic decomposition and Jacobi-polynomial analysis. Combining these estimates with Fischer-decomposition arguments yields existence of entire harmonic solutions for data of sufficiently small order, extending Khavinson–Shapiro-type phenomena to nonhyperbolic quadrics. The approach generalizes earlier $d=2$ results to arbitrary dimensions and provides a unified framework for global regularity of the Dirichlet problem on these domains.

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{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.

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

TL;DR

This work addresses the Dirichlet problem on nonhyperbolic quadratic hypersurfaces in and studies when data given by entire functions admit entire harmonic extensions. A central ingredient is a sharp spectral lower bound for the quadratic form on homogeneous polynomials, obtained via a spherical-harmonic decomposition and Jacobi-polynomial analysis. Combining these estimates with Fischer-decomposition arguments yields existence of entire harmonic solutions for data of sufficiently small order, extending Khavinson–Shapiro-type phenomena to nonhyperbolic quadrics. The approach generalizes earlier results to arbitrary dimensions and provides a unified framework for global regularity of the Dirichlet problem on these domains.

Abstract

We show that for all homogeneous polynomials of degree , in variables, and each , 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.
Paper Structure (4 sections, 7 theorems, 54 equations)

This paper contains 4 sections, 7 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Jacobi polynomials
  3. Proof of Theorem \ref{['arbdimB']}
  4. Proof of Theorem \ref{['arbdimParCyl']}

Key Result

Theorem 2

Let $d\geq 2$, and let $q$ be a nonhyperbolic quadratic polynomial. If $f$ is an entire function of order $\rho < 1 - \beta/2$, then then there is a harmonic entire function of order bounded by $\rho$, which coincides with $f$ on $\{q = 0\}$.

Theorems & Definitions (10)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Theorem 5
  • Lemma 6
  • Theorem 7
  • proof
  • Theorem 8