Extension and lacunas of solutions of linear partial differential equations

Franken, Uwe ; Meise, Reinhold

Annales de l'Institut Fourier, Tome 46 (1996), p. 429-464 / Harvested from Numdam

Résumé

Soient $K\subset Q$ des ensembles compacts, convexes dans ${\Bbb R}^n$ tel que ${\mathrel {\mathop {\hspace{0.0pt}K}\limits ^{\circ }}}\ne \emptyset$ et soit $P(D)$ un opérateur linéaire aux dérivées partielles à coefficients constants. On donne plusieurs conditions qui sont équivalentes au fait que chaque zéro-solution de $P(D)$ dans l’espace ${\cal E}(K)$ des fonctions $C^\infty $ sur $K$ au sens de Whitney a une extension comme zéro-solution dans ${\cal E}(Q)$ ou dans ${\cal E}({\Bbb R}^n)$. Des caractérisations intéressantes sont une condition du type de Phragmén-Lindelöf sur la variété de $P$ dans ${\Bbb C}^n$ et une condition pour des solutions élémentaires pour $P(D)$ avec lacunes.

Abstract

Let $K\subset Q$ be compact, convex sets in ${\Bbb R}^n$ with ${\mathrel {\mathop {\hspace{0.0pt}K}\limits ^{\circ }}}\ne \emptyset$ and let $P(D)$ be a linear, constant coefficient PDO. It is characterized in various ways when each zero solution of $P(D)$ in the space ${\cal E}(K)$ of all $C^\infty $-functions on $K$ extends to a zero solution in ${\cal E}(Q)$ resp. in ${\cal E}({\Bbb R}^n)$. The most relevant characterizations are in terms of Phragmén-Lindelöf conditions on the zero variety of $P$ in ${\Bbb C}^n$ and in terms of fundamental solutions for $P(D)$ with lacunas.

Détail

Publié le : 1996-01-01

MR 97h:35005 | Zbl 0853.35022

DOI : https://doi.org/10.5802/aif.1520

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