Extension and lacunas of solutions of linear partial differential equations

Franken, Uwe; Meise, Reinhold

Annales de l'Institut Fourier, Tome 46 (1996) no. 2, pp. 429-464.

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

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

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