We study the microlocal analyticity of solutions of the nonlinear equation
where is complex-valued, real analytic in all its arguments and holomorphic in . We show that if the function is a solution, and or if is a solution, , , and , then . Here denotes the analytic wave-front set of and Char is the characteristic set of the linearized operator. When , we prove a more general result involving the repeated brackets of and of any order.
Nous étudions l’analyticité microlocale des solutions de l’équation non linéaire
où est une fonction analytique réelle, à valeurs complexes, et holomorphe en . Nous montrons que si est une solution de classe , et , ou si est une solution de classe , , et , alors . Ici, désigne le front d’onde analytique de et l’ensemble caractéristique de l’opérateur linéarisé. Quand , nous démontrons un résultat plus général faisant intervenir les crochets des opérateurs et de tout ordre.
Accepted:
DOI: 10.5802/aif.2463
Classification: 35A18, 35B65, 35F20
Keywords: Analytic wave-front set, linearized operator
Author's affiliations:
@article{AIF_2009__59_4_1267_0, author = {Berhanu, Shif}, title = {On microlocal analyticity of solutions of first-order nonlinear {PDE}}, journal = {Annales de l'Institut Fourier}, pages = {1267--1290}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {4}, year = {2009}, doi = {10.5802/aif.2463}, zbl = {1195.35011}, mrnumber = {2566960}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2463/} }
TY - JOUR TI - On microlocal analyticity of solutions of first-order nonlinear PDE JO - Annales de l'Institut Fourier PY - 2009 DA - 2009/// SP - 1267 EP - 1290 VL - 59 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2463/ UR - https://zbmath.org/?q=an%3A1195.35011 UR - https://www.ams.org/mathscinet-getitem?mr=2566960 UR - https://doi.org/10.5802/aif.2463 DO - 10.5802/aif.2463 LA - en ID - AIF_2009__59_4_1267_0 ER -
Berhanu, Shif. On microlocal analyticity of solutions of first-order nonlinear PDE. Annales de l'Institut Fourier, Volume 59 (2009) no. 4, pp. 1267-1290. doi : 10.5802/aif.2463. https://aif.centre-mersenne.org/articles/10.5802/aif.2463/
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