论文标题
用于整个功能的Fischer分解和抛物线的Dirichlet问题
Fischer decompositions for entire functions and the Dirichlet problem for parabolas
论文作者
论文摘要
令$ p_ {2k} $为$ 2K $的均质多项式，并假设存在$ c> 0 $，$ d> 0 $和$α\ ge 0 $，这样就可以\ begin {equation*} \ left \ left \ langle p_ {2k} f_ {m}，f_ {m} \ right \ rangle_ {l^2（\ mathbb {s}^{d-1}）} \ geq \ geq \ frac {1} {1} f_ {m}，f_ {m} \ rangle _ {\ Mathbb {s}^{d-1}} \ end eNd {equation*}对于所有均质的多项式$ f_ {m f_ {m of度量$ f_ {m。学位的多项式$ j $。本文的主要结果指出，对于任何整个功能，订单的$ f $ $％ρ<\ left（2k-β\右） /α$存在整个功能$ q $和$ρ$的订单的$ h $，以使得\ begin oken \ beken {equation*} f = \ left（p_ {2k} -p_ {2k} -p_p_ p_p_ p_ p_ {β} -p_ {β} - \ \ \ \ \ \ \ \ \ \ \ d DOTS - q+h \ text {and}δ^{h} r = 0。 \ end {equation*}此结果用于确定平面上抛物线域的dirichlet问题的整个谐波解决方案的存在，其数据由小于$ \ frac {1} {2} $的整个订单函数提供。
Let $P_{2k}$ be a homogeneous polynomial of degree $2k$ and assume that there exist $C>0$, $D>0$ and $α\ge 0$ such that \begin{equation*} \left\langle P_{2k}f_{m},f_{m}\right\rangle_{L^2(\mathbb{S}^{d-1})}\geq \frac{1}{C\left( m+D\right) ^{α}}\left\langle f_{m},f_{m}\right\rangle_{\mathbb{S}^{d-1}} \end{equation*} for all homogeneous polynomials $f_{m}$ of degree $m.$ Assume that $P_{j}$ for $j=0, \dots ,β<2k$ are homogeneous polynomials of degree $j$. The main result of the paper states that for any entire function $f$ of order $% ρ<\left( 2k-β\right) /α$ there exist entire functions $q$ and $h$ of order bounded by $ρ$ such that \begin{equation*} f=\left( P_{2k}-P_{β}- \dots -P_{0}\right) q+h\text{ and }Δ^{h}r=0. \end{equation*} This result is used to establish the existence of entire harmonic solutions of the Dirichlet problem for parabola-shaped domains on the plane, with data given by entire functions of order smaller than $\frac{1}{2}$.
