C(1,1/3)C(1,1/3)-regularity in the Dirichlet problem for Δ∞

We answer the much sought after question on regularity of the viscosity solution uu to the Dirichlet problem for the infinity Laplacian Δ∞ in x=(x1,…,xn)∈Rnx=(x1,…,xn)∈Rn (n≥1n≥1) with Lipschitz boundary data on ∂U∂U of the open set UU (whether uu is C1(U)C1(U)), that in fact uu has Hölder regularity C(1,1/3)(U)C(1,1/3)(U). Furthermore, if each of the first partials uxjuxj never vanishes in Ū (a coordinate dependent condition) then u∈C(1,1)(U)u∈C(1,1)(U). The methods that we employ are distinctly different from what is generally practiced in the viscosity methods of solution, and include ‘action’ of boundary distributions, Lebesgue differentiation and regularization near the boundary and a definition of product of distributions not satisfying the Hörmander condition on their wavefront sets, while representing the first partial derivatives of uu purely in terms of boundary integrals involving only first order derivatives of uu on the boundary.