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We consider entropy solutions to the eikonal equation $|\nabla u|=1$ in twospace dimensions. These solutions are motivated by a class of variationalproblems and fail in general to have bounded variation. Nevertheless they sharewith BV functions, several of their fine properties: we show in particular thatthe set of non-Lebesgue points has co-dimension at least one.

On Lebesgue points of entropy solutions to the eikonal equation