We continue the analysis of a family of energies penalizing oscillations in oblique directions: they apply to functions $u(x_1,x_2)$ with $x_l\in\mathbb{R}^{n_l}$ and vanish when $u(x)$ is of the form $u_1(x_1)$ or $u_2(x_2)$. We mainly study the rectifiability properties of the defect measure $\nabla_1\nabla_2u$ of functions with finite energy. The energies depend on a parameter $\theta\in(0,1]$ and the set of functions with finite energy grows with $\theta$. For $\theta<1$ we prove that the defect measure is $(n_1-1,n_2-1)$-tensor rectifiable in $\Omega_1\times\Omega_2$. We first get the result for $n_1=n_2=1$ and deduce the general case through slicing using White's rectifiability criterion. When $\theta=1$ the situation is less clear as measures of arbitrary dimensions from zero to $n_1+n_2-1$ are possible. We show however, in the case $n_1=n_2=1$ and for Lipschitz continuous functions, that the defect measures are $1\,$-rectifiable. This case bears strong analogies with the study of entropic solutions of the eikonal equation.

This paper deals with a variant of the optimal transportation problem. Given f ∈ L^1( ℝ^d, [0,1]) and a cost function c ∈ C(ℝ^d ×ℝ^d) of the form c(x,y)=k(y-x) , we minimise c dγ among transport plans γ whose first marginal is f and whose second marginal is not prescribed but constrained to be smaller than 1-f . Denoting by Υ (f) the infimum of this problem, we then consider the maximisation problem sup{Υ (f): f = m } where m > 0 is given. We prove that maximisers exist under general assumptions on k, and that for k radial, increasing and coercive these maximisers are the characteristic functions of the balls of volume m.

. We prove a rigidity result for k-rectifiable sets E in Rn (that is, up to an Hk-negligible set, E is covered by a countable union of k-manifolds of class C1). Given some decompositions k = k1 + k2, n = n1 + n2, we consider the following properties. L1(x) xL2(x) for some k1-plane L1(x) subset of Rn1 and some k2-plane L2(x) subset of E2 subset of Rn2 such that E1 is k1-rectifiable and E2 is k2-rectifiable (we say that E is (k1, k2)-rectifiable). We always have (2) =double right arrow (1). We establish a partial converse: if A = ALE for some normal rectifiable G-flat k-chain A, then (1) =double right arrow (2) in the sense that A = ALE1 x E2 with E1, E2 as in (2). In the proof we introduce the new groups of tensor flat chains (or (k1, k2)chains) in Rn1 x Rn2 generalizing Fleming's G-flat chains. The other main tool is White's rectifiable slices theorem.

Motivated by some models of pattern formation involving an unoriented director field in the plane, we study a family of unoriented counterparts to the Aviles-Giga functional. We introduce a nonlinear curl operator for such unoriented vector fields as well as a family of even entropies which we call 'trigonometric entropies'. Using these tools, we show two main theorems which parallel some results in the literature on the classical Aviles-Giga energy. The first is a compactness result for sequences of configurations with uniformly bounded energies. The second is a complete characterization of zero-states, that is, the limit configurations when the energies go to 0. These are Lipschitz continuous away from a locally finite set of points, near which they form either a vortex pattern or a disclination with degree 1/2. The proof is based on a combination of regularity theory together with techniques coming from the study of the Ginzburg-Landau energy. Our methods provide alternative proofs in the classical Aviles-Giga context.

In this paper, we establish a C^1,α -regularity theorem for almost-minimizers of the functional ℱ_ε ,γ=P-γ P_ε , where γ∈ (0,1) and P_ε is a nonlocal energy converging to the perimeter as ε vanishes. Our theorem provides a criterion for C^1,α -regularity at a point of the boundary which is uniform as the parameter ε goes to 0. Since the two terms in the energy are of the same order when ε is small, we are considering here much stronger nonlocal interactions than those considered in most related works. As a consequence of our regularity result, we obtain that, for ε small enough, volume-constrained minimizers of ℱ_ε ,γ are balls. For small ε , this minimization problem corresponds to the large mass regime for a Gamow-type problem where the nonlocal repulsive term is given by an integrable kernel G with sufficiently fast decay at infinity.

We introduce the notion of set-decomposition of a normal G-flat chain. We show that any normal rectifiable $G$-flat chain admits a decomposition in set-indecomposable sub-chains. This generalizes the decomposition of sets of finite perimeter in their ``measure theoretic'' connected components due to Ambrosio, Caselles, Masnou and Morel. It can also be seen as a variant of the decomposition of integral currents in indecomposable components by Federer.As opposed to previous results, we do not assume that G is boundedly compact. Therefore we cannot rely on the compactness of sequences of chains with uniformly bounded N-norms. We deduce instead the result from a new abstract decomposition principle. As in earlier proofs a central ingredient is the validity of an isoperimetric inequality. We obtain it here using the finiteness of some h-mass to replace integrality.

We are interested in a reduced model for corrosion of iron, in which ferric cations and electrons evolve in a fixed oxide layer subject to a self-consistent electrostatic potential. Reactions at the boundaries are modeled thanks to Butler-Volmer formulas, whereas the boundary conditions on the electrostatic potential model capacitors located at the interfaces between the materials. Our model takes inspiration in existing papers, to which we bring slight modifications in order to make it consistent with thermodynamics and its second principle. Building on a free energy estimate, we establish the global in time existence of a solution to the problem without any restriction on the physical parameters, in opposition to previous works. The proof further relies on uniform estimates on the chemical potentials that are obtained thanks to Moser iterations. Numerical illustrations are finally provided to highlight the similarities and the differences between our new model and the one previously studied in the literature.

In this note we extend White's deformation theorem for G-flat chains to the setting of G-flat tensor chains. As a corollary we obtain that the groups of normal tensor chains identify with some subgroups of normal chains. Moreover the corresponding natural group isomorphisms are isometric with respect to norms based on the coordinate slicing mass. The coordinate slicing mass of a k-chain is the integral of the mass of its 0-slices along all coordinate-planes of codimension k. The fact that this quantity is equivalent to the usual mass is not straightforward. To prove it, we use the deformation theorem and a partial extension of the restriction operator defined for all chains (not only of finite mass). On the contrary, except in some limit or degenerate cases, the whole groups of tensor chains and of finite mass tensor chains do not identify naturally with subgroups of chains.

Motivated by Gamow's liquid drop model in the large mass regime, we consider an isoperimetric problem in which the standard perimeter P(E) is replaced by P(E)-γP ε (E), with 0<γ<1 and P ε a nonlocal energy such that P ε (E)→P(E) as ε vanishes. We prove that unit area minimizers are disks for ε>0 small enough. More precisely, we first show that in dimension 2, minimizers are necessarily convex, provided that ε is small enough. In turn, this implies that minimizers have nearly circular boundaries, that is, their boundary is a small Lipschitz perturbation of the circle. Then, using a Fuglede-type argument, we prove that (in arbitrary dimension n≥2) the unit ball in ℝ n is the unique unit-volume minimizer of the problem among centered nearly spherical sets. As a consequence, up to translations, the unit disk is the unique minimizer. This isoperimetric problem is equivalent to a generalization of the liquid drop model for the atomic nucleus introduced by Gamow, where the nonlocal repulsive potential is given by a radial, sufficiently integrable kernel. In that formulation, our main result states that if the first moment of the kernel is smaller than an explicit threshold, there exists a critical mass m 0 such that for any m>m 0 , the disk is the unique minimizer of area m up to translations. This is in sharp contrast with the usual case of Riesz kernels, where the problem does not admit minimizers above a critical mass.

This work is part of a general study on the long-term safety of the geological repository of nuclear wastes. A diffusion equation with a moving free boundary in one dimension is introduced and studied. The model describes some mechanisms involved in corrosion processes at the surface of carbon steel canisters in contact with a claystone formation. The main objective of the paper is to prove the existence of weak solutions to the problem which are maximal in time. For this, a time semidiscrete minimizing movements scheme based on a Wasserstein-like distance is introduced. The existence of solutions to the scheme is proved. Then, using a priori estimates, it is shown that as the time step goes to zero these solutions converge up to extraction towards a maximal weak solution to the free boundary model.

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 $S(\Omega_1\times\Omega_2)$.

Abstract Let h : ℝ → ℝ + {h:\mathbb{R}\to\mathbb{R}_{+}} be a lower semicontinuous subbadditive and even function such that h ⁢ ( 0 ) = 0 {h(0)=0} and h ⁢ ( θ ) ≥ α ⁢ | θ | {h(\theta)\geq\alpha|\theta|} for some α > 0 {\alpha>0} . If T = τ ⁢ ( M , θ , ξ ) {T=\tau(M,\theta,\xi)} is a k-rectifiable chain, its h-mass is defined as 𝕄 h ⁢ ( T ) := ∫ M h ⁢ ( θ ) ⁢ 𝑑 ℋ k . \mathbb{M}_{h}(T):=\int_{M}h(\theta)\,d\mathcal{H}^{k}. Given such a rectifiable flat chain T with 𝕄 h ⁢ ( T ) < ∞ {\mathbb{M}_{h}(T)<\infty} and ∂ ⁡ T {\partial T} polyhedral, we prove that for every η > 0 {\eta>0} , it decomposes as T = P + ∂ ⁡ V {T=P+\partial V} with P polyhedral, V rectifiable, 𝕄 h ⁢ ( V ) < η {\mathbb{M}_{h}(V)<\eta} and 𝕄 h ⁢ ( P ) < 𝕄 h ⁢ ( T ) + η {\mathbb{M}_{h}(P)<\mathbb{M}_{h}(T)+\eta} . In short, we have a polyhedral chain P which strongly approximates T in h-mass and preserves the homological constraint ∂ ⁡ P = ∂ ⁡ T {\partial P=\partial T} . When h ′ ⁢ ( 0 + ) {h^{\prime}(0^{+})} is well defined and finite, the definition of the h-mass extends as a finite functional on the space of finite mass k-chains (not necessarily rectifiable). We prove in this case a similar approximation result for finite mass k-chains with polyhedral boundary. These results are motivated by the study of approximations of 𝕄 h {\mathbb{M}_{h}} by smoother functionals but they also provide explicit formulas for the lower semicontinuous envelope of T ↦ 𝕄 h ⁢ ( T ) + 𝕀 ∂ ⁡ S ⁢ ( ∂ ⁡ T ) {T\mapsto\mathbb{M}_{h}(T)+\mathbb{I}_{\partial S}(\partial T)} with respect to the topology of the flat norm.

We study a variational model which 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 small 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 $m\geq 2$ 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 $\varepsilon>0$. We perform a complete $\Gamma$-convergence analysis of the model as $\varepsilon\downarrow0$ in the small energy regime. We then study the structure of minimizers of the limit problem. In particular, we show that the line discontinuities of a minimizer solve a variant of the Steiner problem. We finally prove that for small $\varepsilon>0$, the minimizers of the original problem have the same structure away from the limiting vortices.

AbstractIn this paper we consider the branched transportation problem in two dimensions associated with a cost per unit length of the form1+β⁢θ{1+\beta\,\theta}, where θ denotes the amount of transported mass andβ>0{\beta>0}is a fixed parameter (notice that the limit caseβ=0{\beta=0}corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ({ℱε}ε>0{\{\mathcal{F}_{\varepsilon}\}_{\varepsilon>0}}) which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the Γ-convergence of{ℱε}{\{\mathcal{F}_{\varepsilon}\}}asε↓0{\varepsilon\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.

In this paper we produce a Γ-convergence result for a class of energies F ε,a k modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that F ε,a 1 Γ-converges to a branched transportation energy whose cost per unit length is a function f a n −1 depending on a parameter a > 0 and on the codimension n − 1. The limit cost f a ( m ) is bounded from below by 1 + m so that the limit functional controls the mass and the length of the limit object. In the limit a ↓ 0 we recover the Steiner energy. We then generalize the approach to any dimension and codimension. The limit objects are now k -currents with prescribed boundary, the limit functional controls both their masses and sizes. In the limit a ↓ 0, we recover the Plateau energy defined on k -currents, k < n . The energies F ε,a k then could be used for the numerical treatment of the k -Plateau problem.

The aim of this note is to review some recent results on a family of functionals penalizing oblique oscillations. These functionals naturally appeared in some varia-tional problem related to pattern formation and are somewhat reminiscent of those introduced by Bourgain, Brezis and Mironescu to characterize Sobolev functions. We obtain both qualitative and quantitative results for functions of finite energy. It turns out that this problem naturally leads to the study of various differential inclusions and has connections with branched transportation models. We review in this paper some recent results obtained in [GM19a, GM19b, GM19c] on non-convex functionals penalizing oblique oscillations. We are mainly interested in both qualitative and quantitative rigidity results for functions with finite energy. We also obtain concentration and rectifiability properties of the corresponding 'defect' measures. We will focus here on the most important results and sacrifice generality for clarity. In particular, we will restrict ourselves to a periodic setting to avoid boundary effects. 1 The energy For n 1 , n 2 ≥ 1 and n = n 1 + n 2 , we decompose R n = X 1 ⊕ X 2 with X 1 ⊥ X 2 and n l = dim X l and consider the n dimensional torus T n := (R/Z) n = T n 1 ⊕ T n 2. For x ∈ R n we write x = x 1 + x 2 its decomposition in X 1 ⊕ X 2. For (x, z) ∈ T n × R n , we introduce the notation Du(x, z) := u(x + z) − u(x) for the discrete derivative. We also fix a radial non-negative kernel 1 ρ ∈ L 1 (R n , R +) with R n ρdx = 1, and supp ρ ⊂ B 1 .

Following the ideas of V. V. Zhikov and A. L. Pyatnitski, and more precisely the stochastic two-scale convergence, this paper establishes a homogenization theorem in a stochastic setting for two nonlinear equations : the equation of harmonic maps into the sphere and the Landau-Lifschitz equation. These equations have strong nonlinear features, in particular, in general their solutions are not unique.

Let h : R → R+ be a lower semi-continuous subbadditive and even function such that h(0) = 0 and h(θ) ≥ α|θ| for some α > 0. The h-mass of a k-polyhedral chain P =∑j θjσj in R n (0 ≤ k ≤ n) is defined as M h (P) := j h(θj) H k (σj). If T = τ (M, θ, ξ) is a k-rectifiable chain, the definition extends to M h (T) := M h(θ) dH k. Given such a rectifiable flat chain T with M h (T) < ∞ and ∂T polyhedral, we prove that for every η > 0, it decomposes as T = P + ∂V with P polyhedral, V rectifiable, M h (V) < η and M h (P) < M h (T) + η. In short, we have a polyhedral chain P which strongly approximates T in h-mass and preserves the homological constraint ∂P = ∂T. These results are motivated by the study of approximations of M h by smoother functionals but they also provide explicit formulas for the lower semicontinuous envelope of T → M h (T) + I ∂S (∂T) with respect to the topology of the flat norm.

In this paper we define and study a finite volume scheme for a concrete carbonation model proposed by Aiki and Muntean in [Adv. Math. Sci. Appl.19(2009) 109–129]. The model consists in a system of two weakly coupled parabolic equations in a varying domain whose length is governed by an ordinary differential equation. The numerical sheme is obtained by a Euler discretisation in time and a Scharfetter-Gummel discretisation in space. We establish the convergence of the scheme. As a by-product, we obtain existence of a solution to the model. Finally, some numerical experiments show the efficiency of the scheme.

We study the strong segregation limit for mixtures of Bose-Einstein condensates modelled by a Gross-Pitaievskii functional. Our first main result is that in presence of a trapping potential, for different intracomponent strengths, the Thomas-Fermi limit is sufficient to determine the shape of the minimizers. Our second main result is that for asymptotically equal intracomponent strengths, one needs to go to the next order. The relevant limit is a weighted isoperimetric problem. We then study the minimizers of this limit problem, proving radial symmetry or symmetry breaking for different values of the parameters. We finally 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.