Seminario del Departamento de
Ecuaciones Diferenciales y Análisis Numérico

Fecha : 27 de abril de 2016
Hora  : 11:30
Lugar : Seminario del Departamento (Fac. de Matemáticas, 3a. planta, módulo 34)

Marc Briane

(INSA de Rennes, Francia)

Resumen

A gradient field DU, respectively a divergence free field J, in R^d is said to be isotropically realizable if there exists a positive isotropic conductivity A such that div(A DU)=0, respectively curl(1/A J)=0, in R^d. A regular gradient field is isotropically realizable in R^d if it does not vanish in R^d. The construction of an admissible conductivity is based on the gradient flow. When the gradient field is periodic, the non-vanishing condition is not sufficient to ensure the isotropic realizability with a periodic conductivity. The isotropic realizability of a regular divergence free field J in R^3 is more delicate. In connection with Frobenius' theorem a necessary condition of isotropic realizability is that J and curl J are orthogonal in R^3. The isotropic realizability of J holds true in R^3 if (J, curl J, J x curl J) makes an orthogonal basis of R^3. The construction of an admissible conductivity is based on a three-time dynamical system flowing along the orthogonal directions (J, curl J, J x curl J).