arXiv subjects

B. Merlet

Publications and source records attributed to B. Merlet.

2 recordsLinked to original sources

arXiv2019

Non-convex functionals penalizing simultaneous oscillations along independent directions: rigidity estimates

We study a family of non-convex functionals $\{\mathcal{E}\}$ on the space of measurable functions$u: \Omega_1\times\Omega_2 \subset \mathbb{R}^{n_1}\times\mathbb{R}^{n_2} \to \mathbb{R}$. These functionals vanish on the non-convex subset $S(\Omega_1\times\Omega_2)$ formed by functions of the form $u(x_1,x_2)=u_1(x_1)$ or $u(x_1,x_2)=u_2(x_2)$. We investigate under which conditions the converse implication "$\mathcal{E}(u) = 0 \Rightarrow u \in S(\Omega_1\times\Omega_2)$" holds. In particular, we show that the answer depends strongly on the smoothness of u. We also obtain quantitative versions of this implication by proving that (at least for some parameters) $\mathcal{E}(u)$ controls in a strong sense the distance of $u$ to $S(\Omega_1\times\Omega_2)$.

math.AP↗

arXiv2016

A simple phase-field approximation of the Steiner problem in dimension two

In this paper we 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 (notice that the limit case $\alpha = 0$ corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals $(\{F_\epsilon\}_{\epsilon>0})$ which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the $\Gamma$-convergence of $\{F_\epsilon\}$ as $\epsilon \downarrow 0$. Our functionals are modeled on the Ambrosio-Tortorelli functional and are easy to optimize in practice. We present numerical evidences of the efficiency of the method.

math.AP↗