Variational theories for multiphase problems in materials science and imaging

Abstract

Among the questions stemming from materials science and imaging that can be efficiently tackled with similar Calculus of Variations and PDEs techniques there are the multiphase problems.In particular these are problems whose associated energy density, considered a domain representing a material sample or a grey-scale image, is either discontinuous, modeling possible abrupt changes along interfaces between different subsets of the considered domain, or periodically oscillating, due to the periodic small-scale structure of the considered sample.In the first part of this PhD thesis we focus on materials science problems with this structure, considering samples with heterogeneities or microstructure formations, respectively made of different media in a fixed configuration or having multiple phases of the same material in the microstructure.Composite magnetoelastic materials are prime examples, as the combined elastic and magnetostrictive effects have different outcomes on the material sample depending on the underlying microstructure and small-scale structure. In particular, we perform a simultaneous homogenization and linearization analysis for magnetoelastic energy functionals featuring a mixed Eulerian-Lagrangian structure, and characterize their asymptotic behaviour in the sense of Γconvergence.While the homogenization procedure is exploited in order to translate the effect of small-scale structure of the composite to the macroscopic material response of the whole sample, the relaxation procedure follows a different approach, with the focus on characterizing the effective behaviour of a material accounting for its internal microstructure. Among the samples benefiting from this analysis are the two-well elastoplastic materials, namely characterized by two different elastic phases which can interact with each other under suitable rank-one connectedness assumptions. In order to study them, we characterize the quasiconvex hull of the domain of a chosen energy density, and provide upper and lower bounds on, respectively, its rank-one convex and polyconvex envelopes. In some specific modeling cases, we also construct an explicittrial function through a lamination procedure, which has the prescribed properties on most of the domain. This is a first step towards the characterization of the quasiconvex envelope of the energy density, for which an in-approximation result is needed.In the context of image denoising and reconstruction procedures, as different intensity blocks in a gray-scale image can be modeled as different phases of the same material, or different materials in a composite sample, adjacent to each other, similar energy densities are to be considered. An example of such densities characterizing the anisotropy arising in such multiphase scenarios arethe generalized Φ-functions, a class of maps that includes, for example, variable exponent maps or double-phase integrands, and are the basis for the theory of generalized Orlicz (or MusielakOrlicz) spaces.A natural question in this setting is to understand under what assumptions on the domain it is possible to prove global higher integrability of minimizers of such related functionals. Considering double-phase-type functionals under a uniform local capacity density condition on the complement of the considered domain, we prove global higher integrability of minimizers, an integral Hardy inequality and the equivalence of this capacity density condition to a boundary Poincaré inequality.These generalized Φ-functions are not only useful as energy densities in order to model the behaviour of grey-scale images or multiphase composites, but they can also have an active role in algorithms for image denoising acting as regularizers, as they may have different effects on different phases, again thanks to their ability to model and select different phases all at once.After introducing the space of bounded deformation fields with generalized Orlicz growth, we define a notion of Musielak-Orlicz anisotropic Total Generalized Variation, establish a duality representation and show well-posedness of the corresponding image reconstruction problem.