Rocky Mountain Journal of Mathematics

Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian

Hussien Abugirda, Birzhan Ayanbayev, and Nikos Katzourakis

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In this paper we consider the PDE system of vanishing normal projection of the Laplacian for C 2 maps u : n Ω N :

[ [ D u ] ] Δ u = 0  in  Ω .

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 p -Laplace system for all p [ 2 , ] . For p = , the -Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial calculus of variations in L . Herein we show that the image of a solution u is piecewise affine if either the rank of D u is equal to one or n = 2 and u has additively separated form. As a consequence we obtain corresponding flatness results for p -Harmonic maps for p [ 2 , ] .

Article information

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

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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

vectorial calculus of variations calculus of variations in $L^{\infty}$ $\infty$-Laplacian $p$-Laplacian rank-one solutions special separated solutions rigidity flatness


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.

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