Variational Problems for Maps and the Aronsson PDE System

Abstract. By employing Aronsson’s Absolute Minimizers of L ∞ functionals, we prove that Absolutely Minimizing Maps u: Rn − → RN solve a “tangential” Aronsson PDE system. By following Sheffield-Smart [SS], we derive ∆ ∞ with respect to the dual operator norm and show that such maps miss information along a hyperplane when compared to Tight Maps. We recover the lost term which causes non-uniqueness and derive the complete Aronsson system which has discontinuous coefficients. In particular, the Euclidean ∞-Laplacian is ∆∞u = Du ⊗Du: D2u + |Du|2[Du]⊥∆u where [Du] ⊥ is the projection on the null space of Du>. We exibit C ∞ solutions having interfaces along which the rank of their gradient is discontinuous and propose a modification with C0 coefficients which admits varifold solutions. Away from the interfaces, Aronsson Maps satisfy a structural property of local splitting to 2 phases, an horizontal and a vertical; horizontally they possess gradient flows similar to the scalar case and vertically solve a linear system coupled by a scalar Hamilton Jacobi PDE. We also construct singular∞-Harmonic local C1 diffeomorphisms and singular Aronsson Maps