Section: New Results

Applied calculus of variations

In [34], B. Merlet et al. study a variational problem which models the behavior of topological singularities on the surface of a biological membrane in $P_{\beta }$-phase (see [112]). 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 $\epsilon >0$. B. Merlet et al. 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.

In [27], B. Merlet et al. consider the branched transportation problem in 2D associated with a cost per unit length of the form $1+\beta \theta$ where $\theta$ denotes the amount of transported mass and $\beta >0$ is a fixed parameter (notice that the limit case $\beta =0$ corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, they introduce a family of functionals $\left({\left\{{ℱ}_{\epsilon }\right\}}_{\epsilon >0}\right)$ which approximate the above branched transport energy. They justify rigorously the approximation by establishing the equicoercivity and the $\Gamma$-convergence of $\left\{{ℱ}_{\epsilon }\right\}$ as $\epsilon \downarrow 0$. The functionals are modeled on the Ambrosio–Tortorelli functional and are easy to optimize in practice. Numerical evidences of the efficiency of the method are presented.

In [28], B. Merlet et al. establish new results on the approximation of $k$-dimensional surfaces ($k$-rectifiable currents) by polyhedral surfaces with convergence in $h$-mass and with preservation of the boundary (the approximating polyhedral surface has the same boundary as the limit). This approximation result is required in the convergence study of [29].

In [29], B. Merlet et al. consider a generalization of branched transportation in arbitrary dimension and codimension: minimize the $h$-mass of some oriented $k$-dimensional branched surface in ${𝐑}^n$ with some prescribed boundary. Attached to the surface is a multiplicity $m\left(x\right)$ which is not necessarily an integer and is a conserved quantity (Kirchhoff current law is satisfied at branched points). The $h$-mass is defined as the integral of a cost $h\left(\right|m\left(x\right)\left|\right)$ over the branched surface. As usual in branched transportation, the cost function is a lower-semicontious, sublinear increasing function with $h\left(0\right)=0$ (for instance $h\left(m\right)=\sqrt{1+a{m}^2}$ if $m\ne 0$ and $h\left(0\right)=0$). For numerical purpose, it is convenient to approximate the measure defined by the $k$-dimensional surfaces by smooth functions in ${𝐑}^n$. In this spirit, B. Merlet et al. propose phase field approximations of the branched surfaces and of their energy in the spirit of the Ambrosio–Tortorelli functional. The convergence of these approximations towards the original $k$-dimensional branched transportation problem is established in the sense of $\Gamma$-convergence. Next, considering the cost $h\left(m\right)=\sqrt{1+a{m}^2}$ and sending $a$ to 0, a phase field approximation of the Plateau problem is obtained. Numerical experiments show the efficiency of the method. These numerical results are exceptional as they are obtained without any guess on the topology of the minimizing $k$-surface (as opposed to methods based on parameterizations of the $k$-surface).

In [33], [62], B. Merlet et al. study a family of functionals penalizing oblique oscillations. These functionals naturally appear in some variational problems related to pattern formation and are somewhat reminiscent of those introduced by Bourgain, Brezis and Mironescu to characterize Sobolev functions. More precisely, for a function $u$ defined on a tensor product ${\Omega }_1\times {\Omega }_2$, the family of functionals ${\left\{E_{\epsilon }\left(u\right)\right\}}_{\epsilon >0}$ that we consider vanishes if $u$ is of the form $u\left(x_1\right)$ or $u\left(x_2\right)$. We prove the converse property and related quantitative results. In particular, we describe the fine properties of functions with ${sup}_{\epsilon }{E}_{\epsilon }\left(u\right)<\infty$ by showing that roughly, such $u$ is piecewise of the form $u\left(x_1\right)$ or $u\left(x_2\right)$ on domains separated by lines where the energy concentrates. It turns out that this problem naturally leads to the study of various differential inclusions and has connections with branched transportation models.