Periodic Reporting for period 3 - VAREG (Variational approach to the regularity of the free boundaries)
Reporting period: 2023-06-01 to 2024-11-30
This project is dedicated to the theoretical study of free boundary problems, which are PDE problems on domains whose boundary is not a priori known, but depends on the solution of the PDE itself.
A typical example is a block of melting ice; here the free boundary is the surface of the ice and the PDE is the heat equation. The focus of the project is on the regularity of the free boundaries arising in the context of variational minimization problems as, for instance, the one-phase, the two-phase and the vectorial Bernoulli problems; the obstacle and the thin-obstacle problems. The aim is to develop new techniques for the analysis of the fine structure of the free boundaries, both at regular points and around singularities.
A typical example is a block of melting ice; here the free boundary is the surface of the ice and the PDE is the heat equation. The focus of the project is on the regularity of the free boundaries arising in the context of variational minimization problems as, for instance, the one-phase, the two-phase and the vectorial Bernoulli problems; the obstacle and the thin-obstacle problems. The aim is to develop new techniques for the analysis of the fine structure of the free boundaries, both at regular points and around singularities.
1) Regularity of two-phase free boundaries. In [De Philippis-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, with my PhD student Lorenzo Ferreri, and with Luca Spolaor, in a series of papers we developed a theory based on the approach of Almgren introduced in 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 (this result is optimal). 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: Preprint 09/2024] 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) Classical solutions to the soap film capillarity problem. In [Bevilacqua-Stuvard-Velichkov: Preprint 07/2024] we prove the so-called "non-collapsing conjecture" of King-Maggi-Stuvard for soap films 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-Caarelli-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. Furthermlore, 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.
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, with my PhD student Lorenzo Ferreri, and with Luca Spolaor, in a series of papers we developed a theory based on the approach of Almgren introduced in 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 (this result is optimal). 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: Preprint 09/2024] 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) Classical solutions to the soap film capillarity problem. In [Bevilacqua-Stuvard-Velichkov: Preprint 07/2024] we prove the so-called "non-collapsing conjecture" of King-Maggi-Stuvard for soap films 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-Caarelli-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. Furthermlore, 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.
The regularity of the Bernoulli two-phase free boundaries (1) and the application to the optimal sets for the second eigenvalue (2) were two of the main objectives of the project.
The theory developed in [Ferreri-Spolaor-Velichkov: Preprint 07/2024] allows to analyze the fine structure of the boundary branching set in free boundary problems.
In the paper "Regularity of the free boundary of the optimal sets for integral functionals" (https://arxiv.org/abs/2212.09118(opens in new window)) we proved for the first time a regularity theorem for one of the main classes of shape optimization problems.
In the paper [Bevilacqua-Stuvard-Velichkov: Preprint 07/2024] we develop a new method that allows to rule out the existence of branching points in geometric variational problems.
In the paper [Ognibene-Velichkov: Preprint 12/2024] we prove a structure theorem for free interfaces with triple junctions in the context of free boundary problems.
The theory developed in [Ferreri-Spolaor-Velichkov: Preprint 07/2024] allows to analyze the fine structure of the boundary branching set in free boundary problems.
In the paper "Regularity of the free boundary of the optimal sets for integral functionals" (https://arxiv.org/abs/2212.09118(opens in new window)) we proved for the first time a regularity theorem for one of the main classes of shape optimization problems.
In the paper [Bevilacqua-Stuvard-Velichkov: Preprint 07/2024] we develop a new method that allows to rule out the existence of branching points in geometric variational problems.
In the paper [Ognibene-Velichkov: Preprint 12/2024] we prove a structure theorem for free interfaces with triple junctions in the context of free boundary problems.