C^{1,{1}/{3}-}$$ very weak solutions to the two dimensional Monge–Ampère equation - Cao, Wentao; Hirsch, Jonas; Inauen, Dominik (Calculus of variations and partial differential equations, 2025)
For any $$θ <\frac{1}{3}$$ θ < 1 3 , we show that very weak solutions to the two-dimensional Monge–Ampère equation with regularity $$C^{1,θ }$$ C 1 , θ are dense in the space of continuous functions. This result is shown by a convex integration scheme involving a subtle decomposition of the defect at each stage. The decomposition diagonalizes the defect and, in addition, incorporates some of the leading-order error terms of the first perturbation, effectively reducing the required amount of perturbations to one.