## Rocky Mountain Journal of Mathematics

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

#### Abstract

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

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

Source
Rocky Mountain J. Math., Volume 50, Number 2 (2020), 383-396.

Dates
Revised: 10 August 2019
Accepted: 14 August 2019
First available in Project Euclid: 29 May 2020

https://projecteuclid.org/euclid.rmjm/1590739277

Digital Object Identifier
doi:10.1216/rmj.2020.50.383

Mathematical Reviews number (MathSciNet)
MR4104381

Zentralblatt MATH identifier
07210966

#### Citation

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

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