Rocky Mountain Journal of Mathematics
- Rocky Mountain J. Math.
- Volume 50, Number 2 (2020), 383-396.
Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian
In this paper we consider the PDE system of vanishing normal projection of the Laplacian for maps :
This system has discontinuous coefficients and geometrically expresses the fact that the Laplacian is a vector field tangential to the image of the mapping. It arises as a constituent component of the -Laplace system for all . For , the -Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial calculus of variations in . Herein we show that the image of a solution is piecewise affine if either the rank of is equal to one or and has additively separated form. As a consequence we obtain corresponding flatness results for -Harmonic maps for .
Rocky Mountain J. Math., Volume 50, Number 2 (2020), 383-396.
Received: 30 December 2018
Revised: 10 August 2019
Accepted: 14 August 2019
First available in Project Euclid: 29 May 2020
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35B06: Symmetries, invariants, etc. 35B65: Smoothness and regularity of solutions 35D99: None of the above, but in this section 49N60: Regularity of solutions 49N99: None of the above, but in this section
Abugirda, Hussien; Ayanbayev, Birzhan; Katzourakis, Nikos. Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian. Rocky Mountain J. Math. 50 (2020), no. 2, 383--396. doi:10.1216/rmj.2020.50.383. https://projecteuclid.org/euclid.rmjm/1590739277