7 research outputs found
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 div(Ļāu) = 0 in dimension two, in the case when the essential range of Ļ consists of only two elliptic matrices, i.e., Ļ ā {Ļ1,Ļ2} a.e. in Ī©. In [9], for every pair of elliptic matrices Ļ1 and Ļ2 exponents pĻ1,Ļ2 ā (2,+ā) and qĻ1,Ļ2 ā (1,2) have been found so that if u ā W1,qĻ1,Ļ2(Ī©) is solution to the elliptic equation then āu ā LpĻ1,Ļ2(Ī©) and the optimality of the upper exponent pĻ1,Ļ2 has been proved. In this paper we complement the above result by proving the optimality of the lower exponent qĻ1,Ļ2. Precisely, we show that for every arbitrarily small Ī“, one can find a particular microgeometry, i.e. an arrangement of the sets Ļ-1(Ļ1) and Ļ-1(Ļ2), for which there exists a solution u to the corresponding elliptic equation such that āu ā LqĻ1,Ļ2-Ī“, but āu É LqĻ1,Ļ2-Ī“. The existence of such optimal microgeometries is achieved by convex integration methods, adapting to the present setting the geometric constructions provided in [2] for the isotropic case
Uniform distribution of dislocations in PeierlsāNabarro models for semi-coherent interfaces
In this paper we introduce PeierlsāNabarro 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 Ī -converges to a limit functional comprised of two contributions: one is given by a constant cā> 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 cā is reached when dislocations are evenly-spaced on the one dimensional circle
On the extremal points of the ball of the BenamouāBrenier energy
Funder: Christian Doppler Research Association; Id: http://dx.doi.org/10.13039/501100006012Funder: Royal Society; Id: http://dx.doi.org/10.13039/501100000288Abstract: In this paper, we characterize the extremal points of the unit ball of the BenamouāBrenier 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 finiteādimensional data and optimalātransport based regularization
A Generalized Conditional Gradient Method for Dynamic Inverse Problems with Optimal Transport Regularization
Funder: University of GrazAbstractWe 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 54:2351ā2382, 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 BenamouāBrenier energy constrained via the homogeneous continuity equation. Employing the characterization of the extremal points of the BenamouāBrenier energy (Bredies et al. in Bull Lond Math Soc 53(5):1436ā1452, 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.</jats:p