1) Regularity of two-phase free boundaries. In [DePhilippis-Spolaor-Velichkov: Invent. Math. 2021] we prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. Precisely, we show that in a neighborhood of a contact point both phases are regular domains. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian. Using this result, in [Mazzoleni-Trey-Velichkov: Ann. Inst. H. Poincaré C 2022] we proved that the optimal sets of the second eigenvalue of the Dirichlet Laplacian are the disjoint union of two regular domains. Recently, in [Ferreri-Velichkov: J. Math. Pures Appl. 2025] we proved the smoothness of the free boundary for a two-phase Bernoulli free boundary problem in which the two-phases satisfy an inclusion constraint.
2) On the boundary branching set of the one-phase problem. Consider solutions of the one-phase problem with homogeneous Dirichlet boundary conditions on a portion of the fixed boundary. It is then known that the free boundary is a differentiable manifold that approaches the fixed boundary tangentially. Terminology: the "boundary contact set" is the intersection between the fixed and the free boundaries, while the "boundary branching set" is the set of points at which the fixed boundary enters in contact with the fixed boundary (in other words the branching set is the boundary of the contact set). The question is the following: what is the structure of the boundary branching set? In [De Philippis-Spolaor-Velichkov: J. Eur. Math. Soc.2024] we approached for the first time this question with tools from Complex Analysis; precisely, we showed that when the fixed boundary is a straight line in the plane, the boundary branching set is composed of isolated points. Recently, in a series of papers we developed a theory based on the approach of Almgren from his Big Regularity Paper. First, in [Ferreri-Spolaor-Velichkov: Preprint 05/2024] we showed that in the plane the branching set is always discrete when the fixed boundary is analytic. In [Ferreri-Spolaor-Velichkov: Preprint 07/2024] we proved that in dimension d>2 the boundary branching set has dimension d-2. Finally, in [Ferreri-Spolaor-Velichkov: Preprint 08/2024], we use this theory to provide a new method for proving Unique Continuation for nonlinear PDEs.
3) Free boundary systems. [Buttazzo-Maiale-Mazzoleni-Tortone-Velichkov: Arch. Rat. Mech. Anal. 2024] is the final of a series of papers in which we develop an existence and regularity theory for shape optimization problems with integral cost functionals. We proved weak existence results in [Buttazzo-Maiale-Velichkov: Rend.Acad.Lincei 2022]), while the regularity theory for the free boundary was developed in the following two papers: [Maiale-Tortone-Velichkov: Rev.Mat.Iberoam.2023] and [Maiale-Tortone-Velichkov: Ann. Sc. Norm. Sup. 2023], where we proved an Epsilon-Regularity Theorem and a Boundary Harnack Inequality
for Free Boundary Systems. In order to analyse the singularities on the free boundary, in [Buttazzo-Maiale-Mazzoleni-Tortone-Velichkov: Arch. Rat. Mech. Anal. 2024] we introduced a new notion of Stable One-Phase Solutions and we estimated on the dimension of the singular set via a new approach to the Stability Inequality of Caffarelli-Jerison-Kenig. We used this approach in [Mazzoleni-Tortone-Velichkov: J. Conv. Anal. 2024] to improve the free boundary regularity of solutions to the heat conduction problem of Aguilera-Caffarelli-Spruck.
4) Free boundary clusters. In [Guarino-LaManna-Velichkov: J. Ec. Polytechnique 2021] and [Guarino-LaManna-Velichkov: Calc. Var. PDE 2024] we developed existence and regularity theory for free boundary clusters.
5) Log-epiperimetric inequality and obstacle problems. In [Spolaor-Velichkov: Math. Eng. 2021], [Edelen-Spolaor-Velichkov: Calc. Var. PDE 2023], and [Carducci-Velichkov: J. Funct. Anal. 2025] we proved new (log-)epiperimetric inequalities for the obstacle and the thin-obstacle problems. In [Generau-Oudet-Velichkov: Arch. Rat. Mech. Anal. 2022] we studied the solutions of an obstacle-type problem in which the obstacle is given by the distance function to a fixed point on a manifold.
6) Regularity for soap films with free boundary. In [Bevilacqua-Stuvard-Velichkov: Math. Ann. 2025] we prove the non-collapsing conjecture of King-Maggi-Stuvard for solutions to the soap film capillarity problem spanning planar wires; key ingredients are a selection principle argument and uniform curvature estimates obtained via a new hodograph-type transformation.
7) Capillarity free boundary problems. In [Ferreri-Tortone-Velichkov: Preprint 10/2023] we develop new regularity theory for the one-phase free boundaries with capillarity, which arises in fluid dynamics and is not covered by the classical Alt-Caffarelli-Weiss theory.
8) Optimal partition problems. In [Ognibene-Velichkov: Preprint 12/2024] we prove a structure result for the domain walls arising in optimal partition problems. Precisely, we show that around any point of frequency 3/2, the free interface is composed of three smooth embedded manifolds with smooth common boundary, at which they meet forming 120 degree angles. This proves a result conjectured by Caffarelli and Lin in 2010. Furthermore, in [Ognibene-Velichkov: Preprint 4/2024] we study for the first time the structure of the free interfaces at the boundary of the fixed domain.
