Periodic Reporting for period 2 - VAREG (Variational approach to the regularity of the free boundaries)
Période du rapport: 2021-12-01 au 2023-05-31
A free boundary problem is a boundary value problem that involves partial differential equation on a domain whose boundary is free, that is, it is not a priori known and depends on the solution of the PDE itself. These problems naturally arise in many different models in Physics, Engineering and Economy. A typical example is a block of melting ice; in this case, the free boundary is the surface of the ice, the PDE is the heat equation and its solution (the state function) is the temperature distribution.
In this project, we study free boundary problems from a purely theoretical point of view. The focus 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. Many tools and methods developed in this context can find application to other problems and domains, including shape optimization problems, area-minimizing surfaces, harmonic maps, free discontinuity problems, parabolic and non-local free boundary problems.
In this project, we study free boundary problems from a purely theoretical point of view. The focus 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. Many tools and methods developed in this context can find application to other problems and domains, including shape optimization problems, area-minimizing surfaces, harmonic maps, free discontinuity problems, parabolic and non-local free boundary problems.
1) Regularity of the two-phase free boundaries. With Luca Spolaor and Guido De Philippis, 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.
Ref: https://link.springer.com/article/10.1007/s00222-021-01031-7
2) Regularity of the optimal sets for the second Dirichlet eigenvalue. With Dario Mazzoleni and Baptiste Trey, we prove that the optimal sets of the second eigenvalue of the Dirichlet Laplacian are the disjoint union of two regular domains (up to a small set of one-phase singular points).
Ref: https://ems.press/journals/aihpc/articles/5300752
3) Higher regularity of one-phase and two-phase free boundaries in dimension two. Using tools from Complex Analysis, with Guido De Philippis and Luca Spolaor, we consider one and two-phase problems in dimension two. In particular, for the one-phase problem with geometric constraint (the free boundaryh is constrained in the upper half-plane) we prove that the contact points of the free boundary with the boundary of the constraint is a discrete set. In order to prove this, we introduce new tools as the conformal hodograph transform.
Ref: https://arxiv.org/abs/2110.14075
4) Optimal transmission problems. With Serena Guarino Lo Bianco and Domenico La Manna, we prove an existence and regularity result for an optimal transmission problem. This is a two-phase free boundary problem for which the condition of the free interface is of Robin type (not a Dirichlet one, as in the classical two-phase problem). This is the first result for two-phase Robin problems that opens the way to the study of free boundary clusters.
Ref: https://jep.centre-mersenne.org/item/JEP_2021__8__1_0/
5) Free boundary systems. With Giorgio Tortone and Francesco Paolo Maiale we proved an epsilon-regularity theorem for a free boundary system. Free boundary systems are a general type of vectorial free boundary problems. The specific type of free boundary system, together with the new Boundary Harnack Principle that we proved opened the way to the regularity of the free boundary of the optimal sets for integral functionals
Ref: Regularity of the optimal sets for a class of integral shape functionals (https://arxiv.org/abs/2212.09118); Epsilon-regularity theorem for free boundary systems (https://arxiv.org/abs/2108.03606) Boundary Harnack Principle on optimal domains (https://arxiv.org/abs/2112.01217) Shape optimization problems in control form (https://arxiv.org/abs/2105.03711)
6) Vectorial Bernoulli free boundaries. With Max Engelstein, Luca Spolaor and Guido De Philippis, we proved a rectifiability result for the vectorial Bernoulli free boundaries. Using the Naber-Valtorta quantitative rectifiability result, we proved that the blow-up at almost-every free boundary point is unique.
Ref: https://arxiv.org/abs/2107.12485
7) Log-epiperimetric inequality and obstacle problems. With Luca Spolaor, we investigated a new approach towards the log-epiperimetric inequality for the obstacle and the thin-obstacle problems. We plan to use this approach to prove a reverse log-epiperimetric inequality that can be used in the analysis of the asymptotic behavior of global solutions.
Ref: https://arxiv.org/abs/2007.02346
8) An elastic-plastic torsion problem on manifolds. The elastic-plastic torsion problem is an obstacle problem in which the obstacle is given by the distance function to a fixed point on a manifold. A second formulation consists in minimizing a functional among all 1-Lipschitz functions that vanish in the given point. The first formulation allows to study the local structure of the contact set, while the second one can be used to numerically approximate the cut locus. Using some free boundary techniques, with François Generau and Edouard Oudet, we proved that the two problems are equivalent.
Ref: https://link.springer.com/article/10.1007/s00205-022-01821-0
Ref: https://link.springer.com/article/10.1007/s00222-021-01031-7
2) Regularity of the optimal sets for the second Dirichlet eigenvalue. With Dario Mazzoleni and Baptiste Trey, we prove that the optimal sets of the second eigenvalue of the Dirichlet Laplacian are the disjoint union of two regular domains (up to a small set of one-phase singular points).
Ref: https://ems.press/journals/aihpc/articles/5300752
3) Higher regularity of one-phase and two-phase free boundaries in dimension two. Using tools from Complex Analysis, with Guido De Philippis and Luca Spolaor, we consider one and two-phase problems in dimension two. In particular, for the one-phase problem with geometric constraint (the free boundaryh is constrained in the upper half-plane) we prove that the contact points of the free boundary with the boundary of the constraint is a discrete set. In order to prove this, we introduce new tools as the conformal hodograph transform.
Ref: https://arxiv.org/abs/2110.14075
4) Optimal transmission problems. With Serena Guarino Lo Bianco and Domenico La Manna, we prove an existence and regularity result for an optimal transmission problem. This is a two-phase free boundary problem for which the condition of the free interface is of Robin type (not a Dirichlet one, as in the classical two-phase problem). This is the first result for two-phase Robin problems that opens the way to the study of free boundary clusters.
Ref: https://jep.centre-mersenne.org/item/JEP_2021__8__1_0/
5) Free boundary systems. With Giorgio Tortone and Francesco Paolo Maiale we proved an epsilon-regularity theorem for a free boundary system. Free boundary systems are a general type of vectorial free boundary problems. The specific type of free boundary system, together with the new Boundary Harnack Principle that we proved opened the way to the regularity of the free boundary of the optimal sets for integral functionals
Ref: Regularity of the optimal sets for a class of integral shape functionals (https://arxiv.org/abs/2212.09118); Epsilon-regularity theorem for free boundary systems (https://arxiv.org/abs/2108.03606) Boundary Harnack Principle on optimal domains (https://arxiv.org/abs/2112.01217) Shape optimization problems in control form (https://arxiv.org/abs/2105.03711)
6) Vectorial Bernoulli free boundaries. With Max Engelstein, Luca Spolaor and Guido De Philippis, we proved a rectifiability result for the vectorial Bernoulli free boundaries. Using the Naber-Valtorta quantitative rectifiability result, we proved that the blow-up at almost-every free boundary point is unique.
Ref: https://arxiv.org/abs/2107.12485
7) Log-epiperimetric inequality and obstacle problems. With Luca Spolaor, we investigated a new approach towards the log-epiperimetric inequality for the obstacle and the thin-obstacle problems. We plan to use this approach to prove a reverse log-epiperimetric inequality that can be used in the analysis of the asymptotic behavior of global solutions.
Ref: https://arxiv.org/abs/2007.02346
8) An elastic-plastic torsion problem on manifolds. The elastic-plastic torsion problem is an obstacle problem in which the obstacle is given by the distance function to a fixed point on a manifold. A second formulation consists in minimizing a functional among all 1-Lipschitz functions that vanish in the given point. The first formulation allows to study the local structure of the contact set, while the second one can be used to numerically approximate the cut locus. Using some free boundary techniques, with François Generau and Edouard Oudet, we proved that the two problems are equivalent.
Ref: https://link.springer.com/article/10.1007/s00205-022-01821-0
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 achieved results allow to attack other major objectives of the project, in particular, the description of the structure of the vectorial free boundaries around singularities, but they also open new research directions.
Optimal transmission problems. This is a new class of two-phase free boundary problems in which the function is not identically vanishing on the free boundary but only satisfies a transmission condition. In (4), we gave a variational formulation of the problem and study the free interface in the zones where the state function is positive. A next objective is to describe the free boundary around branching points at which the transmission (two-phase) interface splits into two one-phase free boundaries on which the functions is vanishing. This behaviour is similar the one of the classical two-phase Bernoulli free boundaries and its understanding will open the way to the study of free boundary clusters.
Higher order expansion of the free boundaries around singularities. In (1) we prove that around a branching point of the two-phase problem, the free boundary is the union of the boundaries of two disjoint smooth sets, but this result doesn't provide information on the contact set between these two boundaries. For instance, they can touch infinitely many times forming a Cantor-like set. The study of this contact set requires techniques from the obstacle and the thin-obstacle problem. In (3) we explored the applications of complex analytic tools as quasi-conformal maps and a conformal hodograph transform to analyse the structure of these contact sets in dimension two. The complete description of the fine structure of the contact set is now a major objective, both in dimension two and higher than two.
In the paper "Regularity of the free boundary of the optimal sets for integral functionals" (https://arxiv.org/abs/2212.09118) we proved for the first time a regularity theorem for on of the main classes of shape optimization problems. In particular, we developed a new theory for stable solutions of the one-phase Bernoulli problem.
The achieved results allow to attack other major objectives of the project, in particular, the description of the structure of the vectorial free boundaries around singularities, but they also open new research directions.
Optimal transmission problems. This is a new class of two-phase free boundary problems in which the function is not identically vanishing on the free boundary but only satisfies a transmission condition. In (4), we gave a variational formulation of the problem and study the free interface in the zones where the state function is positive. A next objective is to describe the free boundary around branching points at which the transmission (two-phase) interface splits into two one-phase free boundaries on which the functions is vanishing. This behaviour is similar the one of the classical two-phase Bernoulli free boundaries and its understanding will open the way to the study of free boundary clusters.
Higher order expansion of the free boundaries around singularities. In (1) we prove that around a branching point of the two-phase problem, the free boundary is the union of the boundaries of two disjoint smooth sets, but this result doesn't provide information on the contact set between these two boundaries. For instance, they can touch infinitely many times forming a Cantor-like set. The study of this contact set requires techniques from the obstacle and the thin-obstacle problem. In (3) we explored the applications of complex analytic tools as quasi-conformal maps and a conformal hodograph transform to analyse the structure of these contact sets in dimension two. The complete description of the fine structure of the contact set is now a major objective, both in dimension two and higher than two.
In the paper "Regularity of the free boundary of the optimal sets for integral functionals" (https://arxiv.org/abs/2212.09118) we proved for the first time a regularity theorem for on of the main classes of shape optimization problems. In particular, we developed a new theory for stable solutions of the one-phase Bernoulli problem.