Formula omitted]-regularity in the Dirichlet problem for [formula omitted

Abstract

We answer the much sought after question on regularity of the viscosity solution u to the Dirichlet problem for the infinity Laplacian Δ ∞ in x = ( x 1 , … , x n ) ∈ R n ( n ≥ 1 ) with Lipschitz boundary data on ∂ U of the open set U (whether u is C 1 ( U ) ), that in fact u has Hölder regularity C ( 1 , 1 / 3 ) ( U ) . Furthermore, if each of the first partials u x j never vanishes in U ̄ (a coordinate dependent condition) then 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 u purely in terms of boundary integrals involving only first order derivatives of u on the boundary.