An Explicit Solution of the Lipschitz Extension Problem

Building Lipschitz extensions of functions is a problem of classical analysis. Extensions are not unique: the classical results of Whitney and McShane provide two explicit examples. In certain cases there exists an optimal extension, which is the solution of an elliptic partial differential equation, the infinity Laplace equation. In this work, we find an explicit formula for a suboptimal extension, which is an improvement over the Whitney and McShane extensions: it can improve the local Lipschitz constant. The formula is found by solving a convex optimization problem for the minimizing extensions at each point. This work extends a previous solution for domains consisting of a finite number of points, which has been used to build convergent numerical schemes for the infinity Laplace equation, and in Image Inpainting applications.