26 research outputs found
A historical account on characterizations of C1-manifolds in Euclidean spaces by tangent cones
Abstract A historical account on characterizations of C 1 -manifolds in Euclidean spaces by tangent cones is provided. Old characterizations of smooth manifold (by tangent cones), due to Valiron (1926, 1927) and Severi (1929, 1934) are recovered; modern characterizations, due to Gluck (1966, 1968) and Tierno (1997) are restated. All these results are consequences of the Four-cones coincidence theorem due to [1]
Classical flows of vector fields with exponential or sub-exponential summability
We show that vector fields whose spatial derivative satisfies a
Orlicz summability condition have a spatially continuous representative and are
well-posed. For the case of sub-exponential summability, their flows satisfy a
Lusin (N) condition in a quantitative form, too. Furthermore, we prove that if
satisfies a suitable exponential summability condition then the flow
associated to has Sobolev regularity, without assuming boundedness of . We then apply these results to the representation and Sobolev
regularity of weak solutions of the Cauchy problem for the transport and
continuity equations.Comment: 35 page