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 $u:{\mathbb{R}}^n\supseteq \Omega ⟶{\mathbb{R}}^N$: \[ [\![\mathrm{D} u]\!]^\bot Δu = 0 \ \, \text{ in }Ω. \] 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 $p\in [2,\infty ]$. For $p=\infty$, the $\infty$-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial Calculus of Variations in $L^{\infty }$. 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, $p\in [2,\infty ]$.

15 pages, 2 figures