My research in the calculus of variations develops analytical and geometric methods for nonlinear partial differential equations arising in continuum mechanics, defect theory, image processing and shape optimization. Combining functional analysis, geometric measure theory and variational function spaces for continuum mechanics, I investigate the structure of some variational models arising in material science, their asymptotic behaviour and the singular phenomena they describe.
A common theme is the passage from local or microscopic structures to effective continuum descriptions. This includes the analysis of singular measures, differential constraints, free boundaries, topological defects and heterogeneous materials through tools such as rigidity, blow-up methods, integral representation, relaxation, Γ-convergence and homogenization.
A major part of my research concerns function spaces that arise naturally in the calculus of variations and continuum mechanics, including BV, BD, BDdev and, more generally, spaces of bounded 𝒜-variation. These spaces provide a natural framework for describing discontinuities, singular measures and irreversible deformations while retaining the differential structure of the underlying physical models.
In recent work with M. Caroccia, I have investigated BDdev(Ω), where only the deviatoric part of the symmetric gradient is controlled. This space is closely connected with perfect plasticity and incompressible deformation. Our work establishes rigidity results and functional properties that clarify the global structure of fields satisfying this weaker differential constraint.
Rigidity is also central to my work on blow-up analysis. Refined iterative blow-up methods for maps with bounded 𝒜-variation provide local descriptions of singularities and extend classical arguments from BV and BD to a wider class of differential operators. These techniques help identify the geometric form of singular measures and the local behaviour of variational energies near concentration sets.
Another important direction concerns the integral representation of variational functionals. In many continuum models, the energy is initially defined only as an abstract functional on an admissible class. An integral representation theorem identifies the local densities that generate this energy and shows how it decomposes into bulk and singular contributions. Such results are essential for relaxation, lower semicontinuity and homogenization because they reveal the precise local structure of the effective energy. In collaboration with M. Caroccia and M. Focardi, I established integral representation results for local energy functionals on BD.
These developments are closely connected with the theory of 𝒜-free measures, namely measures satisfying a linear differential constraint of the form 𝒜μ = 0. This viewpoint encompasses familiar conditions such as divergence-free and curl-free fields and provides a unified framework for studying oscillation, concentration and singularity formation in variational problems.
A significant part of my research is devoted to the variational analysis of nonlinear problems through Γ-convergence, relaxation, lower semicontinuity and homogenization. These tools provide the mathematical foundation for deriving effective continuum models, studying singular limits and understanding how microscopic structures influence macroscopic behaviour.
Lower semicontinuity provides the basic existence criterion for minimizers, while relaxation constructs the appropriate generalized energy when minimizing sequences develop oscillations, concentrations or singularities. Homogenization derives effective macroscopic laws for heterogeneous media, and Γ-convergence supplies the natural notion of convergence for the associated minimization problems.
These methods recur throughout my work in elasticity, plasticity, topology optimization and fracture, where they provide rigorous justification for reduced, relaxed and effective theories.
Another direction of my research concerns the variational description of topological singularities through geometric measure theory. The objective is to represent singular defects in a form that preserves both their geometric structure and their compatibility with continuum and variational models.
This line of research is closely linked to my earlier distributional approach to dislocations, in which defects are described through distributions rather than smooth classical fields. That formulation made it possible to represent dislocations directly at the continuum scale and provided one of the conceptual foundations for my later work on incompatibility-based elasticity and elasto-plasticity.
Building on these ideas, I have used currents, Jacobians and functions of bounded variation to study singular maps and defect structures. In collaboration with L. De Luca and R. Scala, the paper A new approach to Topological Singularities via a Weak Notion of Jacobian for Functions of Bounded Variation extends the Jacobian to singular BV maps and provides a variational framework for describing topological defects. This opens a route toward the analysis and homogenization of dislocation clusters and other singular structures into effective continuum theories.
A further direction concerns gradient-free variational models for free-boundary problems in topology optimization and image processing. In collaboration with S. Amstutz and A. A. Novotny, I developed perimeter approximations that avoid explicit gradient regularization while preserving the essential variational structure of the problem.
These models were first introduced for topology optimization, where they provide an efficient framework for optimizing material distributions without explicitly tracking interfaces. The same ideas were later extended to image segmentation and classification, where variational principles determine the partition of an image into regions rather than relying solely on local edge detection.
This work illustrates how methods from the calculus of variations can be transferred between mechanics, optimization and computer vision, combining rigorous analysis with computationally effective models.
Current research combines variational analysis, geometric measure theory, function spaces arising in the calculus of variations and continuum mechanics, and modern computational methods. Particular emphasis is placed on higher-order continuum models, differential constraints, weak Jacobians, homogenization of defect structures, and structure-preserving finite element discretizations derived from differential complexes. The long-term objective is to bridge microscopic singular structures and mathematically rigorous, computationally predictive continuum theories.