Abstract
We prove a result on unique continuation at characteristic boundary points for the Kohn Laplacian on the Heisenberg group. The work is related to the paper by H. S. Shapiro (Exposition Math 13:247–275, 1995).
Similar content being viewed by others
Data availability
There is no conflict of interest.
References
Baouendi, M.S., Rothschild, L.: A local Hopf lemma and unique continuation for harmonic functions. Int. Math. Res. Not. 8, 245–251 (1993)
Baouendi, M.S., Rothschild, L.: Harmonic functions satisfying weighted signed conditions on the boundary. Annales de l’institut Fourier 43, 1311–1318 (1993)
Berhanu, S.: Boundary unique continuation for a class of elliptic equations. Am. J. Math. 143(3), 783–810 (2021)
Berhanu, S.: Boundary unique continuation for the Laplace equation and the biharmonic operator, Communications in Analysis and Geometry, to appear
Berhanu, S.: A local Hopf lemma and unique continuation for elliptic equations. Adv. Math. 389, 107912 (2021)
Berhanu, S.: Boundary unique continuation for elliptic real analytic differential operators, preprint
Berhanu, S., Hounie, J.: A local Hopf lemma and unique continuation for the Helmholtz equation. Comm. PDEs 43, 448–466 (2018)
Birindelli, I., Cutri, A.: A semi-linear problem for the Heisenberg Laplacian. Rend. Sem. Mat. Univ. Padova 94, 137–153 (1995)
Folland, G.B.: A fundamental solution for the subelliptic operator. Bull. Am. Math. Soc. 79, 373–376 (1973)
Folland, G.B., Stein, E.M.: Estimates for the \(\overline{\partial }_{b}\) complex and analysis on the Heisenberg group. Comm. Pure Appl. Math. 27, 429–522 (1974)
Huang, X., Krantz, S.: A unique continuation problem for holomorphic mappings. Comm. Partial Differ. Equ. 18, 241–263 (1993)
Huang, X.J., Krantz, S.G., Ma, D., Pan, Y.: A Hopf lemma for holomorphic functions and applications. Complex Variables Theory Appl. 26, 273–276 (1995)
Lakner, M.: Finite order vanishing of boundary values of holomorphic mappings. Proc. AMS 112, 521–527 (1991)
Monticelli, D.D.: Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators. J. Eur. Math. Soc. 12, 611–654 (2010)
Monti, R., Morbidelli, D.: Kelvin transform for Grushin operators and critical semilinear equations. Duke Math. J. 131, 167–202 (2006)
Martino, V., Tralli, G.: On the Hopf-Oleinik lemma for degenerate elliptic equations at characteristic points. Calc. Variat. Partial Differ. Equ. 55, 1–20 (2016)
Niu, P., Han, Y., Han, J.: A Hopf type lemma and a CR type inversion for the generalized Greiner operator. Canad. Math. Bull. 47, 417–430 (2004)
Shapiro, H.S.: Notes on a theorem of Baouendi and Rothschild. Exposition. Math. 13(2–3), 247–275 (1995)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
There are no conflict of interest in this article.
Additional information
Dedicated to the memory of Harold S. Shapiro.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Work supported in part by NSF DMS 1855737.
Rights and permissions
About this article
Cite this article
Berhanu, S. A local Hopf lemma for the Kohn Laplacian on the Heisenberg group. Anal.Math.Phys. 12, 72 (2022). https://doi.org/10.1007/s13324-022-00682-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13324-022-00682-w