Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T20:09:40.687Z Has data issue: false hasContentIssue false
  • English
  • Français

PLANAR IMMERSIONS WITH PRESCRIBED CURL AND JACOBIAN DETERMINANT ARE UNIQUE

Part of: Partial differential equations

Published online by Cambridge University Press:  08 October 2021

ANTHONY GRUBER Open the ORCID record for ANTHONY GRUBER [Opens in a new window]
ANTHONY GRUBER*
Affiliation:
Department of Scientific Computing, Florida State University, 400 Dirac Science Library, Tallahassee, FL 32306-4120, USA
*
e-mail: agruber@fsu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that immersions of planar domains are uniquely specified by their Jacobian determinant, curl function and boundary values. This settles the two-dimensional version of an outstanding conjecture related to a particular grid generation method in computer graphics.

Keywords

prescribed curlprescribed Jacobian determinantuniqueness problemgrid generation

MSC classification

Primary: 35A02: Uniqueness problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 1 , August 2022 , pp. 126 - 131
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Castillo, J. E., ‘A discrete variational grid generation method’, SIAM J. Sci. Comput. 12(2) (1991), 454–468.CrossRefGoogle Scholar
Chen, X., Numerical Construction of Diffeomorphism and the Applications to Grid Generation and Image Registration, PhD Thesis (University of Texas at Arlington, Arlington, 2016). http://hdl.handle.net/10106/26119.Google Scholar
Garanzha, V. A., Kudryavtseva, L. N. and Utyuzhnikov, S. V., ‘Variational method for untangling and optimization of spatial meshes’, J. Comput. Appl. Math. 269 (2014), 24–41.Google Scholar
Hestenes, D. and Sobczyk, G., Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, Fundamental Theories of Physics, 5 (Springer Science and Business Media, 2012).Google Scholar
Kohan, M., ‘Pointwise equality of codimension-zero immersions’, Mathematics Stack Exchange. https://math.stackexchange.com/q/4205739 (version: 2021-07-23).Google Scholar
Liao, G., Cai, X., Liu, J., Luo, X., Wang, J. and Xue, J., ‘Construction of differentiable transformations’, Appl. Math. Lett. 22(10) (2009), 1543–1548.CrossRefGoogle Scholar
Lounesto, P., Clifford Algebras and Spinors, London Mathematical Society Lecture Note Series, 286 (Cambridge University Press, Cambridge, 2001).Google Scholar
Steinberg, S. and Roache, P. J., ‘Variational grid generation’, Numer. Methods Partial Differential Equations 2(1) (1986), 71–96.Google Scholar
Wu, T., Liu, X., An, W., Huang, Z. and Lyu, H., ‘A mesh optimization method using machine learning technique and variational mesh adaptation’, Chinese J. Aeronaut. (to appear). Published online (4 July 2021).Google Scholar
Zhou, Z., Chen, X., Cai, X. X. and Liao, G., ‘Uniqueness of transformation based on Jacobian determinant and curl-vector’, Preprint, 2017, arXiv:1712.03443.Google Scholar
Zhou, Z. and Liao, G., ‘Construction of diffeomorphisms with prescribed Jacobian determinant and curl’, Preprint, 2021, arXiv:2105.09302.Google Scholar
Zhu, Y., Zhou, Z., Liao, G., Yang, Q. and Yuan, K., ‘Effects of differential geometry parameters on grid generation and segmentation of MRI brain image’, IEEE Access 7 (2019), 68529–68539.CrossRefGoogle Scholar