L ∞ 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 and Smart (2012) [24], we derive δ ∞ 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 δ ∞u=Du⊗Du:D 2u+|Du| 2[Du] ⊥δu where [Du] ⊥ is the projection on the null space of Du ⊤. We demonstrate C ∞ solutions having interfaces along which the rank of their gradient is discontinuous and propose a modification with C 0 coefficients which admits varifold solutions. Away from the interfaces, Aronsson maps satisfy a structural property of local splitting to 2 phases, a 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 C 1 diffeomorphisms and singular Aronsson maps. © 2012 Elsevier Inc.