Adjoint methods for the infinity Laplacian partial differential equation (Q715288)

The authors consider the boundary value problem \[ \begin{cases} -\Delta_\infty u=0 \;\text{ in } U \cr u=g \;\text{ on } \partial U \end{cases} \] where \(\Delta_\infty u= u_{x_i}u_{x_j}u_{x_ix_j}\) and \(g\) is Lipschitz continuous. To study fine properties of certain smooth approximations \({u^\varepsilon}\) to a viscosity solution \(u\) of the above problem, the authors introduce Green's function \({\sigma^\varepsilon}\) for the linearization. Integrating by parts with respect to \({\sigma^\varepsilon}\), they derive various useful integral estimates. It is proved, in particular, the everywhere differentiability of \(u.\)