I would describe my research as driven by questions arising in the calculus of variations, which have a rich geometric measure theory flavor and require a pinch of harmonic analysis ideas to be solved.
As a student, my formation was oriented toward problems arising in homogenization theory. Over time, I became more interested in the theoretical questions of the calculus of variations, particularly those with a strong geometric flavor. I am very familiar with its applied side, given that I have worked on variational models arising in materials science.
Currently, my research revolves around the following guiding question: I want to understand the geometric, functional, and continuity properties that a function « $u$ » and a measure « $\mu$ » gain by solving a linear PDE
$$\mathcal A u = \mu.$$
The motivation to study this general setting stems from the versatility of the differential operator « $\mathcal A$ » to describe diverse linearized physical models. For example, in the mathematical theory of linear elasticity, materials may undergo deformations that result in the formation of microscopic plane-like dislocations or fractures. To make sense of the gradient deformation tensor on the region where these ruptures occur, it is often necessary to consider $\mathrm{sym}(Du)=\mu$ in the space of measures.
Aluminum alloy under strain forces (image by Jover Carrasco et al., Metals 2021, licensed under CC BY 4.0.)
The mean-curvature does not sit on the tip of the cone (image by A. Arroyo Rabasa)