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

Fassina, Martino; Pan, Yifei

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

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

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

Classification : 35J99, 32W99
Mots clés : Elliptic operators, flat functions

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

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Chen, Zhihua; Liu, Yang; Pan, Yifei; Zhang, Yuan Examples of flat solutions to the Cauchy-Riemann equations (2015) (unpublished note)

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

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

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

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

[9] Morrey, Charles B. Jr 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] Morrey, Charles B. Jr 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] Morrey, Charles B. Jr; Nirenberg, Louis 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] Narasimhan, Raghavan Analysis on real and complex manifolds, North-Holland Mathematical Library, 35, North-Holland, 1985

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

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

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

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

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

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

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