Section: New Results

Variational modeling and analysis

Bose-Einstein condensates are a unique way to observe quantum effects at a (relatively) large scale. The fundamental state​s​ of such condensates ​are​ obtained as minimizer​s​ of a Gross-Pitaievskii functional.​ In [33], M. Goldman and B. Merlet consider the case of a two component Bose-Einstein condensate in the strong segregation regime (the energy favors spatial segregation of the two different Boson species). They identify two different regimes in the strong segregation and small healing length limit. In one of these regimes, the relevant limit is an interesting weighted isoperimetric problem which explains some of the numerical simulations of [70].

In [32], B. Merlet et al. consider the branched transportation problem in 2D associated with a cost per unit length of the form $1+\alpha m$ where $m$ denotes the amount of transported mass and $\alpha >0$ is a fixed parameter (the limit case $\alpha =0$ corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, they introduce a family of functionals (${\left\{F_{\epsilon }\right\}}_{\epsilon >0}$) which approximate the above branched transport energy. They justify rigorously the approximation by establishing the equicoercivity and the $\Gamma$-convergence of $F_{\epsilon }$ as $\epsilon \downarrow 0$. These functionals are modeled on the Ambrosio-Tortorelli functional and are easy to optimize in practice (the algorithm amounts to perform repetitively the alternate optimization of two quadratic functionals). Numerical evidences of the efficiency of the method are presented.