Two-dimensional Jacobians det and Det for bounded variation functions and applications
The paper deals with the comparison in dimension two between the strong Jacobian determinant det and the weak (or distributional) Jacobian determinant Det. Restricting ourselves to dimension two, we extend the classical results of Ball and Müller as well as more recent ones to bounded variation vect...
| Autores: | , |
|---|---|
| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:dnet:idus________::9b6636cdc9c98b20f0330adbc1ac889f |
| Acceso en línea: | https://hdl.handle.net/11441/186085 https://doi.org/10.1007/s13163-024-00496-3 |
| Access Level: | acceso abierto |
| Palabra clave: | Jacobian determinants Det and det Bounded variation function ODE’s flow Minimization under constraint Polyconvex energy |
| Sumario: | The paper deals with the comparison in dimension two between the strong Jacobian determinant det and the weak (or distributional) Jacobian determinant Det. Restricting ourselves to dimension two, we extend the classical results of Ball and Müller as well as more recent ones to bounded variation vector-valued functions, providing a sufficient condition on a vector-valuedU in BV(Ω)2 such that the equality det(∇U) =Det(∇U) holds either in the distributional sense on Ω, or almost-everywhere in Ω when U is in W1,1(Ω)2. The key-assumption of the result is the regularity of the Jacobian matrix-valued ∇U along the direction of a given non vanishing vector field b ∈ C1(Ω)2, i.e. ∇Ub is assumed either to belong to C0(Ω)2 with one of its coordinates in C1(Ω), or to belong to C1(Ω)2. Two examples illustrate this new notion of twodimensional distributional determinant. Finally, we prove the lower semicontinuity of a polyconvex energy defined for vector-valued functions U in BV(Ω)2, assuming that the vector field b and one of the coordinates of ∇Ub lie in a compact set of regular vector-valued functions. |
|---|