Optimal lower exponent for the higher gradient integrability of solutions to two-phase elliptic equations in two dimensions

We study the higher gradient integrability of distributional solutions u to the equation \({{\mathrm{div}}}(\sigma \nabla u) = 0\) in dimension two, in the case when the essential range of \(\sigma \) consists of only two elliptic matrices, i.e., \(\sigma \in \{\sigma _1, \sigma _2\}\) a.e. in \(\Omega \). 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 \(p_{\sigma _1,\sigma _2}\in (2,+\infty )\) and \(q_{\sigma _1,\sigma _2}\in (1,2)\) have been found so that if \(u\in W^{1,q_{\sigma _1,\sigma _2}}(\Omega )\) is solution to the elliptic equation then \(\nabla u\in L^{p_{\sigma _1,\sigma _2}}_{\mathrm{weak}}(\Omega )\) and the optimality of the upper exponent \(p_{\sigma _1,\sigma _2}\) has been proved. In this paper we complement the above result by proving the optimality of the lower exponent \(q_{\sigma _1,\sigma _2}\). Precisely, we show that for every arbitrarily small \(\delta \), one can find a particular microgeometry, i.e., an arrangement of the sets \(\sigma ^{-1}(\sigma _1)\) and \(\sigma ^{-1}(\sigma _2)\), for which there exists a solution u to the corresponding elliptic equation such that \(\nabla u \in L^{q_{\sigma _1,\sigma _2}-\delta }\), but \(\nabla u \notin L^{q_{\sigma _1,\sigma _2}}\). 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.     

A Variational Model for Dislocations at Semi-coherent Interfaces

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.     

A Generalized Conditional Gradient Method for Dynamic Inverse Problems with Optimal Transport Regularization

We develop a dynamic generalized conditional gradient method (DGCG) for dynamic inverse problems with optimal transport regularization. We consider the framework introduced in Bredies and Fanzon (ESAIM: M2AN 54:2351–2382, 2020), where the objective functional is comprised of a fidelity term, penalizing the pointwise in time discrepancy between the observation and the unknown in time-varying Hilbert spaces, and a regularizer keeping track of the dynamics, given by the Benamou–Brenier energy constrained via the homogeneous continuity equation. Employing the characterization of the extremal points of the Benamou–Brenier energy (Bredies et al. in Bull Lond Math Soc 53(5):1436–1452, 2021), we define the atoms of the problem as measures concentrated on absolutely continuous curves in the domain. We propose a dynamic generalization of a conditional gradient method that consists of iteratively adding suitably chosen atoms to the current sparse iterate, and subsequently optimizing the coefficients in the resulting linear combination. We prove that the method converges with a sublinear rate to a minimizer of the objective functional. Additionally, we propose heuristic strategies and acceleration steps that allow to implement the algorithm efficiently. Finally, we provide numerical examples that demonstrate the effectiveness of our algorithm and model in reconstructing heavily undersampled dynamic data, together with the presence of noise.