We study measure-valued solutions of the inhomogeneous continuity equation
\( \partial_t \rho_t + div (v \rho_t) = g \rho_t \,, \)where the coefficients \(v \,\)and \(g \,\)are of low regularity. A new superposition principle is proven for positive measure solutions and coefficients for which the recently-introduced dynamic Hellinge–Kantorovich energy is finite. This principle gives a decomposition of the solution into curves \(t \mapsto h(t) δ\)\(γ(t) \) that satisfy the characteristic system \( γ’ (t)=v(t,γ(t)) \), \(h’ (t)=g(t,γ(t))h(t) \)in an appropriate sense. In particular, it provides a generalization of existing superposition principles to the low-regularity case of \(g \,\)where characteristics are not unique with respect to \(h \). Two applications of this principle are presented. First, uniqueness of minimal total-variation solutions for the inhomogeneous continuity equation is obtained if characteristics are unique up to their possible vanishing time. Second, the extremal points of dynamic Hellinger–Kantorovich-type regularizers are characterized. Such regularizers arise, for example, in the context of dynamic inverse problems and dynamic optimal transport.
@article{2022-Bre-Car-Fan,author={Bredies, Kristian and Carioni, Marcello and Fanzon, Silvio},doi={10.1080/03605302.2022.2109172},journal={Communications in Partial Differential Equations},number={10},pages={2023--2069},title={A superposition principle for the inhomogeneous continuity equation with Hellinger--Kantorovich-regular coefficients},url={https://doi.org/10.1080/03605302.2022.2109172},volume={47},year={2022},}
In this paper, we characterize the extremal points of the unit ball of the Benamou–Brenier energy and of a coercive generalization of it, both subjected to the homogeneous continuity equation constraint. We prove that extremal points consist of pairs of measures concentrated on absolutely continuous curves which are characteristics of the continuity equation. Then, we apply this result to provide a representation formula for sparse solutions of dynamic inverse problems with finite-dimensional data and optimal-transport based regularization.
@article{2021-Bre-Car-Fan-Rom,author={Bredies, Kristian and Carioni, Marcello and Fanzon, Silvio and Romero, Francisco},doi={10.1112/blms.12509},journal={Bulletin of the London Mathematical Society},number={5},pages={1436--1452},title={On the extremal points of the ball of the Benamou--Brenier energy},url={https://doi.org/10.1112/blms.12509},volume={53},year={2021},}
In this paper we introduce Peierls–Nabarro type models for edge dislocations at semi-coherent interfaces between two heterogeneous crystals, and prove the optimality of uniformly distributed edge dislocations. Specifically, we show that the elastic energy \(Γ\)-converges to a limit functional comprised of two contributions: one is given by a constant \(c_∞> 0 \,\)gauging the minimal energy induced by dislocations at the interface, and corresponding to a uniform distribution of edge dislocations; the other one accounts for the far field elastic energy induced by the presence of further, possibly not uniformly distributed, dislocations. After assuming periodic boundary conditions and formally considering the limit from semi-coherent to coherent interfaces, we show that \(c_∞\,\)is reached when dislocations are evenly-spaced on the one dimensional circle.
@article{2020-Fan-Pon-Sca,author={Fanzon, Silvio and Ponsiglione, Marcello and Scala, Riccardo},doi={10.1007/s00526-020-01787-5},journal={Calculus of Variations and Partial Differential Equations},number={4},pages={141},title={Uniform distribution of dislocations in Peierls--Nabarro models for semi-coherent interfaces},url={https://doi.org/10.1007/s00526-020-01787-5},volume={59},year={2020},}
In this paper we propose and study a novel optimal transport based regularization of linear dynamic inverse problems. The considered inverse problems aim at recovering a measure valued curve and are dynamic in the sense that (i) the measured data takes values in a time dependent family of Hilbert spaces, and (ii) the forward operators are time dependent and map, for each time, Radon measures into the corresponding data space. The variational regularization we propose is based on dynamic (un-)balanced optimal transport which means that the measure valued curves to recover (i) satisfy the continuity equation, i.e., the Radon measure at time \(t \)is advected by a velocity field \(v \)and varies with a growth rate \(g \), and (ii) are penalized with the kinetic energy induced by \(v \)and a growth energy induced by \(g \). We establish a functional-analytic framework for these regularized inverse problems, prove that minimizers exist and are unique in some cases, and study regularization properties. This framework is applied to dynamic image reconstruction in undersampled magnetic resonance imaging (MRI), modelling relevant examples of time varying acquisition strategies, as well as patient motion and presence of contrast agents.
@article{2020-Bre-Fan,author={Bredies, Kristian and Fanzon, Silvio},doi={10.1051/m2an/2020056},journal={ESAIM: Mathematical Modelling and Numerical Analysis},number={6},pages={2351--2382},title={An optimal transport approach for solving dynamic inverse problems in spaces of measures},url={https://doi.org/10.1051/m2an/2020056},volume={54},year={2020},}
In this paper we show the emergence of polycrystalline structures as a result of elastic energy minimization. For this purpose, we consider a well-known variational model for two-dimensional systems of edge dislocations, within the so-called core radius approach, and we derive the \(Γ\)-limit of the elastic energy functional as the lattice space tends to zero. In the energy regime under investigation, the symmetric and skew part of the strain become decoupled in the limit, the dislocation measure being the curl of the skew part of the strain. The limit energy is given by the sum of a plastic term, acting on the dislocation density, and an elastic term, which depends on the symmetric strains. Minimizers under suitable boundary conditions are piecewise constant antisymmetric strain fields, representing in our model a polycrystal whose grains are mutually rotated by infinitesimal angles. In the energy regime under investigation, the symmetric and skew part of the strain become decoupled in the limit, the dislocation measure being the curl of the skew part of the strain. The limit energy is given by the sum of a plastic term, acting on the dislocation density, and an elastic term, which depends on the symmetric strains. Minimizers under suitable boundary conditions are piecewise constant antisymmetric strain fields, representing in our model a polycrystal whose grains are mutually rotated by infinitesimal angles.
@article{2019-Fan-Pal-Pon,author={Fanzon, Silvio and Palombaro, Mariapia and Ponsiglione, Marcello},doi={10.1137/18M118726X},journal={SIAM Journal on Mathematical Analysis},number={5},pages={3956--3981},title={Derivation of Linearized Polycrystals from a Two-Dimensional System of Edge Dislocations},url={https://doi.org/10.1137/18M118726X},volume={51},year={2019},}
We study the higher gradient integrability of distributional solutions \(u \)to the equation \(div (σ∇u)=0 \)in dimension two, in the case when the essential range of \(σ\,\)consists of
only two elliptic matrices, i.e., \(σ∈\) { \( \sigma_1, \sigma_2 \) } a.e. in \(Ω\). In Nesi et al. (Ann Inst H Poincaré Anal Non Linéaire 31(3):615–638, 2014), for every pair of elliptic matrices \(\sigma_1 \)and \(\sigma_2 \),
exponents
\(p = p\)\(\sigma_1, \sigma_2 \) \( ∈(0,+∞) \)and
\(q = q\)\(\sigma_1, \sigma_2 \) \(∈(1,2) \)have been found so that if
\(u ∈W\)\(1,q\) \( (Ω) \)is a solution to the elliptic equation then
\(∇u ∈L^p(Ω,weak) \)and the optimality of the upper exponent \(p \)has been proved. In this paper we complement the above result by proving the optimality of the lower exponent \(q \). Precisely, we show that for every arbitrarily small \(δ\), one can find a particular microgeometry, i.e., an arrangement of the sets
\(σ\)\(-1 \) \( (\sigma_1) \)and
\(σ\)\(-1 \) \( (\sigma_2) \), for which there exists a solution \(u \)to the corresponding elliptic equation such that
\( ∇u ∈L\)\(q-δ\)
but \( ∇u ∉L^q \). The existence of such optimal microgeometries is achieved by convex integration methods, adapting to the present setting the geometric constructions provided in Astala et al. (Ann Scuola Norm Sup Pisa Cl Sci 5(7):1–50, 2008) for the isotropic case.
@article{2017-Fan-Pal,author={Fanzon, Silvio and Palombaro, Mariapia},doi={10.1007/s00526-017-1222-9},journal={Calculus of Variations and Partial Differential Equations},number={5},pages={137},title={Optimal lower exponent for the higher gradient integrability of solutions to two-phase elliptic equations in two dimensions},url={https://doi.org/10.1007/s00526-017-1222-9},volume={56},year={2017},}
We propose and analyze a simple variational model for dislocations at semi-coherent interfaces. The energy functional describes the competition between two terms: a surface energy induced by dislocations and a bulk elastic energy, spent to decrease the amount of dislocations needed to compensate the lattice misfit. We prove that, for minimizers, the former scales like the surface area of the interface, the latter like its diameter. The proposed continuum model is built on some explicit computations done in the framework of the semi-discrete theory of dislocations. Even if we deal with finite elasticity, linearized elasticity naturally emerges in our analysis since the far-field strain vanishes as the interface size increases.
@article{2017-Fan-Pal-Pon,author={Fanzon, Silvio and Palombaro, Mariapia and Ponsiglione, Marcello},doi={10.1007/s00332-017-9366-5},journal={Journal of Nonlinear Science},number={5},pages={1435--1461},title={A Variational Model for Dislocations at Semi-coherent Interfaces},url={https://doi.org/10.1007/s00332-017-9366-5},volume={27},year={2017},}
