Adjoint Methods for the Infinity Laplacian Partial Differential Equation
Lawrence Craig Evans, Charles K. Smart · Archive for Rational Mechanics and Analysis · 2011
To study fine properties of certain smooth approximations $${u^\varepsilon}$$ to a viscosity solution u of the infinity Laplacian partial differential equation (PDE), we introduce Green’s function $${\sigma^\varepsilon}$$ for the linearization. We can then integrate by parts with respect to $${\sigma^\varepsilon}$$ and derive various useful integral estimates. We are, in particular, able to use these estimates (i) to prove the everywhere differentiability of u and (ii) to rigorously justify interpreting the infinity Laplacian equation as a parabolic PDE.