Section: New Results

Calculus of variations applied to Image processing, physics and biology

In [23], Benoît Merlet et al. consider the branched transportation problem in dimension two with a cost of transport per unit length of path of the form $f_a\left(m\right)=a+m$ where $a>0$ is fixed and $m$ is the flux along the path. As usual in branched transportation, an admissible transport is represented as a vector measure with prescribed divergence $\sum m_j{\delta }_{x_j}-\sum m_l^{\text{'}}{\delta }_{y_l}$ (the $x_j$ representing the sources and the $y_k$ the sinks). The paper introduces a family of functionals ${\left\{F_{\epsilon }^a\right\}}_{\epsilon >0}$ and the authors establish that this family of functionals approximate the branched transportation energy in the sense of $\Gamma$-convergence. The energy $F_{\epsilon }^a$ is modeled on the Ambrosio-Tortorelli functional and is easy to optimize in practice (using dual formulation for the constraints and alternate direction optimization). In [48], the same authors extend their previous work to functionals defined on $k$-currents: the objects are no more lines that transport masses but $k$-dimensional surfaces transporting a given quantity of $(k-1)$-dimensional objects. The ambient space is now of any dimension $n$. A new family of approximate energies ${\left\{F_{\epsilon }^a\right\}}_{\epsilon >0}$ is introduced and a $\Gamma$-convergence analysis is performed in the limit $\epsilon \downarrow 0$. The limit objects are now $k$-currents with prescribed boundary, the limit functional controls both their masses (the total flux) and sizes ($k$-dimensional volume of the object). In the limit $a\downarrow 0$, the limit energy is the $k$-volume of the object so that these energies can be used for the numerical optimization of the size of $k$-currents with prescribed boundary. Although rather theoretical, the works [23], [48] are motivated by an image reconstruction issue: how to recover the contours of partially masked objects in an image.

In [26], Michael Goldman and Benoît Merlet study the strong segregation limit for mixtures of Bose-Einstein condensates modelled by a Gross-Pitaievskii functional. They study the behavior of minimizers of the Hamiltonian. First, they show that in the presence of a trapping potential, for different intracomponent strengths, the Thomas-Fermi limit is sufficient to determine the shape of the minimizers. Then they study the case of asymptotically equal intracomponent strengths: at leading order the two phases are then undistinguishable, the authors extract the next order and show that the relevant limit optimization problem is a weighted isoperimetric problem. Then, they study the minimizers, proving radial symmetry or symmetry breaking for different values of the parameters. Eventually, they show that in the absence of a confining potential, even for non-equal intracomponent strengths, one needs to study a related isoperimetric problem to gain information about the shape of the minimizers.

In [49], Michael Goldman, Benoît Merlet and Vincent Millot study a variational problem which models the behavior of topological singularities on the surface of a biological membrane in $P_{\beta }$-phase (see  [92]). The problem combines features of the Ginzburg-Landau model in 2D and of the Mumford-Shah functional. As in the classical Ginzburg-Landau theory, a prescribed number of point vortices appear in the moderate energy regime; the model allows for discontinuities, and the energy penalizes their length. The novel phenomenon here is that the vortices have a fractional degree $1/m$ with $m$ prescribed. Those vortices must be connected by line discontinuities to form clusters of total integer degrees. The vortices and line discontinuities are therefore coupled through a topological constraint. As in the Ginzburg-Landau model, the energy is parameterized by a small length scale $\epsilon >0$. The authors perform a complete $\Gamma$-convergence analysis of the model as $\epsilon \downarrow 0$ in the moderate energy regime. Then, they study the structure of minimizers of the limit problem. In particular, the line discontinuities of a minimizer solve a variant of the Steiner problem.