Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian
Hussien A. H. Abugirda, Birzhan Ayanbayev, Nikos Katzourakis · Rocky Mountain Journal of Mathematics · 2020
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 : [ [ 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 , ∞ ] .