A local obstruction for elliptic operators with real analytic coefficients on flat germs

Martino Fassina; Yifei Pan

doi:10.5802/afst.1722

Martino Fassina ¹; Yifei Pan ²

¹ Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
² Department of Mathematical Sciences, Purdue University Fort Wayne, 2101 East Coliseum Boulevard, Fort Wayne, IN 46805, USA

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 5, pp. 1343-1363.

Abstract
Abstract

Let $Ω \subset ℝ^{n}, n \geq 2$ , be an open set. For an elliptic differential operator $L$ on $Ω$ with real analytic coefficients and a point $p \in Ω$ , we construct a smooth function $g$ with the following properties: $g$ is flat at $p$ and the equation $L u = g$ has no smooth local solution $u$ that is flat at $p$ .

Soit $Ω \subset ℝ^{n}, n \geq 2$ , un ensemble ouvert. Pour un opérateur différentiel elliptique $L$ sur $Ω$ avec des coefficients analytiques réels et un point $p \in Ω$ , nous construisons une fonction lisse $g$ avec les propriétés suivantes : $g$ est plat en $p$ et l’équation $L u = g$ n’a pas de solution locale lisse $u$ qui est plate en $p$ .

Received: 2020-04-03
Accepted: 2020-11-05
Published online: 2022-12-08

DOI: 10.5802/afst.1722

Classification: 35J99, 32W99
Keywords: Elliptic operators, flat functions

Author's affiliations:

Martino Fassina ¹; Yifei Pan ²

¹ Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
² Department of Mathematical Sciences, Purdue University Fort Wayne, 2101 East Coliseum Boulevard, Fort Wayne, IN 46805, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{AFST_2022_6_31_5_1343_0,
     author = {Martino Fassina and Yifei Pan},
     title = {A local obstruction for elliptic operators with real analytic coefficients on flat germs},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1343--1363},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 31},
     number = {5},
     year = {2022},
     doi = {10.5802/afst.1722},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1722/}
}

TY  - JOUR
AU  - Martino Fassina
AU  - Yifei Pan
TI  - A local obstruction for elliptic operators with real analytic coefficients on flat germs
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2022
SP  - 1343
EP  - 1363
VL  - 31
IS  - 5
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1722/
DO  - 10.5802/afst.1722
LA  - en
ID  - AFST_2022_6_31_5_1343_0
ER  -

%0 Journal Article
%A Martino Fassina
%A Yifei Pan
%T A local obstruction for elliptic operators with real analytic coefficients on flat germs
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2022
%P 1343-1363
%V 31
%N 5
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1722/
%R 10.5802/afst.1722
%G en
%F AFST_2022_6_31_5_1343_0

Martino Fassina; Yifei Pan. A local obstruction for elliptic operators with real analytic coefficients on flat germs. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 5, pp. 1343-1363. doi : 10.5802/afst.1722. https://afst.centre-mersenne.org/articles/10.5802/afst.1722/

References
Cited by

[1] Émile Borel Sur quelques points de la théorie des fonctions, Ann. Sci. Éc. Norm. Supér., Volume 12 (1895), pp. 9-55 | DOI | Numdam | Zbl

[2] Torsten Carleman Sur un problème d’unicité pour les systèmes d’équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astron. Fysik B, Volume 26 (1939) no. 17, pp. 1-9 | MR | Zbl

[3] Zhihua Chen; Yang Liu; Yifei Pan; Yuan Zhang Examples of flat solutions to the Cauchy-Riemann equations (2015) (unpublished note)

[4] Adam Coffman; Yifei Pan Smooth counterexamples to strong unique continuation for a Beltrami system in $ℂ^{2}$ , Commun. Partial Differ. Equations, Volume 37 (2012) no. 10-12, pp. 2228-2244 | DOI | MR | Zbl

[5] Martino Fassina; Yifei Pan Remarks on the global distribution of the points of finite D’Angelo type (in preparation)

[6] Lars Hörmander Linear partial differential operators, Grundlehren der Mathematischen Wissenschaften, 116, Academic Press Inc., 1963 | DOI | Zbl

[7] Christine Laurent-Thiébaut; Mei-Chi Shaw Solving $\bar{\partial}$ with prescribed support on Hartogs triangles in $ℂ^{2}$ and ${ℂℙ}^{2}$ , Trans. Am. Math. Soc., Volume 371 (2019) no. 9, pp. 6531-6546 | DOI | MR | Zbl

[8] Hans Lewy An example of a smooth linear partial differential equation without solution, Ann. Math., Volume 66 (1957), pp. 155-158 | DOI | MR | Zbl

[9] Charles B. Morrey On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. I. Analyticity in the interior, Am. J. Math., Volume 80 (1958), pp. 198-218 | DOI | MR | Zbl

[10] Charles B. Morrey On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. II. Analyticity at the boundary, Am. J. Math., Volume 80 (1958), pp. 219-237 | DOI | MR | Zbl

[11] Charles B. Morrey; Louis Nirenberg On the analyticity of the solutions of linear elliptic systems of partial differential equations, Commun. Pure Appl. Math., Volume 10 (1957), pp. 271-290 | DOI | MR | Zbl

[12] Raghavan Narasimhan Analysis on real and complex manifolds, North-Holland Mathematical Library, 35, North-Holland, 1985

[13] Louis Nirenberg On Elliptic Partial Differential Equations, Il principio di minimo e sue applicazioni alle equazioni funzionali (Sandro Faedo, ed.) (CIME Summer Schools), Volume 17, Springer, 2011 (reprint of the 1958 original) | DOI

[14] Yifei Pan Unique continuation for Schrödinger operators with singular potentials, Commun. Partial Differ. Equations, Volume 17 (1992) no. 5-6, pp. 953-965 | Zbl

[15] Yifei Pan; Thomas Wolff A remark on unique continuation, J. Geom. Anal., Volume 8 (1998) no. 4, pp. 599-604 | MR | Zbl

[16] Ivan G. Petrovskiĭ Sur l’analyticité des solutions des systèmes d’équations différentielles, Rec. Math. N. S., Volume 47 (1939), pp. 3-70 | Zbl

[17] Murray H. Protter Unique continuation for elliptic equations, Trans. Am. Math. Soc., Volume 95 (1960), pp. 81-91 | DOI | MR | Zbl

[18] Bruce Reznick Homogeneous polynomial solutions to constant coefficient, Adv. Math., Volume 117 (1996) no. 2, pp. 179-192 | DOI | MR | Zbl

[19] Elias M. Stein; Rami Shakarchi Introduction to further topics in analysis, Princeton Lectures in Analysis, 4, Princeton University Press, 2011 | DOI

Cited by Sources: