Abstract
Consider a function u defined on ℛ n , except, perhaps, on a closed set of potential singularities 𝒮. Suppose that u solves the eikonal equation ‖Du‖ = 1 in the pointwise sense on ℛ n \\𝒮, where Du denotes the gradient of u and ‖·‖ is a norm on ℛ n with the dual norm ‖·‖*. For a class of norms which includes the standard p-norms on ℛ n , 1 < p < ∞, we show that if 𝒮 has Hausdorff 1-measure zero and n ≥ 2, then u is either affine or a “cone function,” that is, a function of the form u(x) = a ± ‖x − z‖*.
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