Extension and lacunas of solutions of linear partial differential equations

Uwe Franken; Reinhold Meise

Annales de l'institut Fourier (1996)

  • Volume: 46, Issue: 2, page 429-464
  • ISSN: 0373-0956

Abstract

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Let K ⊂ Q be compact, convex sets in ℝ n with K ∘ ≠ ∅ 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 ℰ ( K ) of all C ∞ -functions on K extends to a zero solution in ℰ ( Q ) resp. in ℰ ( ℝ n ) . 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 ( D ) with lacunas.

How to cite

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Franken, Uwe, and Meise, Reinhold. "Extension and lacunas of solutions of linear partial differential equations." Annales de l'institut Fourier 46.2 (1996): 429-464. <http://eudml.org/doc/75184>.

@article{Franken1996,
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.},
author = {Franken, Uwe, Meise, Reinhold},
journal = {Annales de l'institut Fourier},
keywords = {Whitney extension of zero-solutions; Phragmén-Lindelöf conditions for algebraic varieties; fundamental solutions with lacunas; continuous linear right inverses for constants coefficient partial differential operators},
language = {eng},
number = {2},
pages = {429-464},
publisher = {Association des Annales de l'Institut Fourier},
title = {Extension and lacunas of solutions of linear partial differential equations},
url = {http://eudml.org/doc/75184},
volume = {46},
year = {1996},
}

TY - JOUR
AU - Franken, Uwe
AU - Meise, Reinhold
TI - Extension and lacunas of solutions of linear partial differential equations
JO - Annales de l'institut Fourier
PY - 1996
PB - Association des Annales de l'Institut Fourier
VL - 46
IS - 2
SP - 429
EP - 464
AB - 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.
LA - eng
KW - Whitney extension of zero-solutions; Phragmén-Lindelöf conditions for algebraic varieties; fundamental solutions with lacunas; continuous linear right inverses for constants coefficient partial differential operators
UR - http://eudml.org/doc/75184
ER -

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