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 $C^2$ maps $u : $\mathbb{R}$^n \supseteq \Omega \longrightarrow $\mathbb{R}$^N$: $$[\text{\hspace{0.17em}⁣}[\mathrm{D}u]\text{\hspace{0.17em}⁣}{]}^{\mathrm{\perp }}\mathrm{\Delta }u=0\text{ in }\mathrm{\Omega }\mathrm{.}$$ 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 $p\in [2,\mathrm{\infty }]$. For $p=\mathrm{\infty }$, the $\mathrm{\infty }$-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial Calculus of Variations in $L^{\mathrm{\infty }}$. Herein we show that the image of a solution $u$ is piecewise affine if either the rank of $\mathrm{D}u$ is equal to one or $n=2$ and $u$ has the additively separated form $u(x,y)=f(x)+g(y)$. As a consequence we obtain corresponding flatness results for the images of $p$-Harmonic maps, $p\in [2,\mathrm{\infty }]$.