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The distance function and defect energy

Several energies measuring jump discontinuities of a unit length gradient field are considered and are called defect energies. The main example is a total variation I(φ) of the hessian of a function φ in a domain. It is shown that the distance function is the unique minimiser of I(φ) among all non-negative Lipschitz solutions of the eikonal equation |grad φ| = 1 with zero boundary data, provided that the domain is a two-dimensional convex domain. An example shows that the distance function is not a minimiser of I if the domain is noncovex. This suggests that the selection mechanism by I is different from that in the theory of viscosity solutions in general. It is often conjectured that the minimiser of a defect energy is a distance function if the energy is formally obtained as a singular limit of some variational problem. Our result suggests that this conjecture is very subtle even if it is true.