[Sur la non-densité dans une méthode de solutions fondamentales avec des points sources indépendants du temps pour la résolution d’équations paraboliques]
Des combinaisons linéaires de solutions fondamentales avec des points sources indépendants du temps pour la résolution de l’équation de la chaleur sont étudiées. On étudie la question ouverte de savoir si ces combinaisons linéaires génèrent un ensemble dense dans l’espace des fonctions de carrés intégrables sur la limite latérale d’un cylindre espace-temps et on montre que la réponse à cette question est négative. L’indépendance linéaire de l’ensemble des solutions fondamentales est démontrée. Il est souligné à la fin pour un cas particulier que de telles combinaisons linéaires sont linéairement indépendantes et denses dans l’espace des fonctions de carrés intégrables définies sur la partie supérieure (où le temps est fixe) de la limite du cylindre espace-temps.
Linear combinations of fundamental solutions to the parabolic heat equation with source points fixed in time is investigated. The open problem whether these linear combinations generate a dense set in the space of square integrable functions on the lateral boundary of a space-time cylinder, is settled in the negative. Linear independence of the set of fundamental solutions is shown to hold. It is outlined at the end, for a particular example, that such linear combinations constitute a linearly independent and dense set in the space of square integrable functions on the upper top part (where time is fixed) of the boundary of this space-time cylinder.
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DOI : 10.5802/crmath.204
B. Tomas Johansson 1
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@article{CRMATH_2021__359_6_733_0,
author = {B.~Tomas Johansson},
title = {On non-denseness for a method of fundamental solutions with source points fixed in time for parabolic equations},
journal = {Comptes Rendus. Math\'ematique},
pages = {733--738},
publisher = {Acad\'emie des sciences, Paris},
volume = {359},
number = {6},
year = {2021},
doi = {10.5802/crmath.204},
zbl = {07390654},
language = {en},
}
TY - JOUR AU - B. Tomas Johansson TI - On non-denseness for a method of fundamental solutions with source points fixed in time for parabolic equations JO - Comptes Rendus. Mathématique PY - 2021 SP - 733 EP - 738 VL - 359 IS - 6 PB - Académie des sciences, Paris DO - 10.5802/crmath.204 LA - en ID - CRMATH_2021__359_6_733_0 ER -
%0 Journal Article %A B. Tomas Johansson %T On non-denseness for a method of fundamental solutions with source points fixed in time for parabolic equations %J Comptes Rendus. Mathématique %D 2021 %P 733-738 %V 359 %N 6 %I Académie des sciences, Paris %R 10.5802/crmath.204 %G en %F CRMATH_2021__359_6_733_0
B. Tomas Johansson. On non-denseness for a method of fundamental solutions with source points fixed in time for parabolic equations. Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 733-738. doi : 10.5802/crmath.204. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.204/
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