Our research in Applied Mathematical Methods focuses on central themes of mathematical analysis with a strong orientation toward modeling complex phenomena in physical and engineering contexts. At the core of our activity are variational and functional tools applied to the study of nonlinear problems, often characterized by complex geometric structures, nonlocal effects, or significant physical constraints.

Functional analysis provides the theoretical framework for investigating nonlinear operators and functional spaces involved in complex differential equations. In this context, the calculus of variations represents a fundamental methodology for analyzing minimization problems and deriving limit models, particularly through Γ-convergence techniques.

The study of partial differential equations focuses on nonlinear formulations, including the presence of nonlocal effects, singularities, or non-standard asymptotic conditions. Special attention is given to evolution equations, used to describe dynamics emerging in physical systems, such as anomalous diffusion, interface propagation, or transport in heterogeneous media.

Optimal transport and its variational interpretation are key tools for analyzing gradient flows and modeling energy-driven evolutions, often in metric spaces or generalized settings.

A significant part of our activity is dedicated to the development of mathematical models for materials science and elasticity theory, with the goal of describing complex mechanical phenomena based on fundamental constitutive laws. The research addresses issues related to incompressibility, surface tension, nonlinear behaviors, and dimensional reduction in thin structures such as plates and membranes.

The variety of topics addressed is unified by a coherent approach based on rigorous analytical methods, enabling significant results both from a theoretical perspective and in relation to widely relevant applied models.

Involved Laboratories

Representative Publications

• Huang, Y., Mainini, E., Vázquez, J. L., & Volzone, B. (2024). Nonlinear aggregation-diffusion equations with Riesz potentials.  Journal of Functional Analysis, 287(2), 110465.
• Dolera, E., Favaro, S., & Mainini, E. (2024). Strong posterior contraction rates via Wasserstein dynamics. Probability Theory and Related Fields, 179(3-4), 1013–1045.
• Camerlenghi, F., Dolera, E., Favaro, S., & Mainini, E. (2024). Wasserstein posterior contraction rates in non-dominated Bayesian nonparametric models. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 60(2), 1027–1055.
• Mainini, E., & Percivale, D. (2023). Newton's second law as limit of variational problems. Advances in Continuous and Discrete Models, 2023(1), 1–12.
• Mainini, E., & Percivale, D. (2024). On the weighted inertia-energy approach to forced wave equations. Journal of Differential Equations, 336, 1–25.