April 2020 Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian
Hussien Abugirda, Birzhan Ayanbayev, Nikos Katzourakis
Rocky Mountain J. Math. 50(2): 383-396 (April 2020). DOI: 10.1216/rmj.2020.50.383


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 , ] .


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Hussien Abugirda. Birzhan Ayanbayev. Nikos Katzourakis. "Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian." Rocky Mountain J. Math. 50 (2) 383 - 396, April 2020. https://doi.org/10.1216/rmj.2020.50.383


Received: 30 December 2018; Revised: 10 August 2019; Accepted: 14 August 2019; Published: April 2020
First available in Project Euclid: 29 May 2020

zbMATH: 07210966
MathSciNet: MR4104381
Digital Object Identifier: 10.1216/rmj.2020.50.383

Primary: 35B06 , 35B65 , 35D99 , 49N60 , 49N99

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

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium


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Vol.50 • No. 2 • April 2020
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