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math.AP2026
Sharpness of the Osgood Criterion for the Continuity Equation with Divergence-free Vector Fields
Roberto Colombo, Anuj Kumar
For any modulus of continuity that fails the Osgood condition, we construct a divergence-free velocity field for which the associated ODE admits at least two…
math.AP2025
A convex integration scheme for the continuity equation past the Sobolev embedding threshold
Maria Colombo, Roberto Colombo, Anuj Kumar
We introduce a convex integration scheme for the continuity equation in the context of the Di Perna-Lions theory that allows to build incompressible vector fields in $C_{t}W^{1,p}_…
math.AP2024
Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian
Matteo Carducci, Roberto Colombo
We establish generic regularity results of free boundaries for solutions of the obstacle problem for the fractional Laplacian . We prove that, for almost every obstacle, th…