Abstract/Summary

In this paper we consider the PDE system of vanishing normal projection of the Laplacian for C2 maps u:Rn⊇Ω→RN: [[Du]]⊥Δu=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 Du 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,∞].