Periodic Reporting for period 4 - HiCoS (Higher Co-dimension Singularities: Minimal Surfaces and the Thin Obstacle Problem)
Período documentado: 2022-08-01 hasta 2023-10-31
The main objective of the project is the study of singular solutions arising in two families of variational problems, that are minimal surfaces and free boundary problems.
Singular solutions are naturally ubiquitous in many contexts and often have a prominent analytical and physical interest. A solution is said singular if it cannot be described by a regular functions or, more generally, by a regular geometrical object.
The analysis of singularities is by now a classical topic in mathematics and it has required, and still requires, the development of new analytical tools beyond the present state of the art.
As it often happens, there is no general theory of singularities and every problem, or family of problems, needs to develop a set of techniques which are specifically tailored. In this project the problems we consider are two classical questions in the geometric calculus of variations: the analysis of minimal surfaces and the thin obstacle problem. There are deep connections between these two classes of problems and the project is aimed to develop analytical techniques which can be applied to both of them.
We are looking at geometric surfaces minimizing suitable energies, that is the area in the first case, and the elastic energy in the second case, under suitable conditions: fixed boundaries and, in the second case, the constraint of lying above a a lower dimensional obstacle (like a membrane pushed by the blade of a knife). These problems arise in several analytical, geometric and physical instances and have been studied extensively in the last century. Nevertheless, many questions remains still open from the mathematical point of view and their solutions would represent a concrete progress in mathematics. The main objectives of the project is the theoretical analytical characterization of the singular solutions to these problems, not the aspects related to the applications.
A particular issue which is present in both questions, yet not well-understood from a mathematical point of view, is related to the higher co-dimension of the objects under consideration, for example two-dimensional minimal surfaces in a space with four dimensions, or free boundaries of dimension one in a three dimensional space. More generally, we look at surfaces of dimension K in a space of dimension N with N-K>1: the terms higher co-dimension refer to the difference of dimensions N-K being bigger than one. This is a crucial aspect of the project, because many of the analytical and geometric principles which are valid in co-dimension one (i.e. for minimal hypersurfaces or for the classical obstacle problems) do not apply to higher co-dimensions and there are compelling questions about the singularities of these two classical problems: for example, it is not known whether the singularities of higher co-dimension minimal surfaces are finite in measure or infinite, or what the free boundary look like in the thin obstacle problem.
The main unifying theme of the project is the central role played by geometric measure theory, which underlines various common aspects of these two classes of questions and makes them suited to be treated in a unified framework. Geometric measure theory is a classical field in mathematics studying the structure of measures from a geometrical point of view. The final goal of the project is to develop suitable analytical techniques that provides valuable insights on the mathematics at the basis of higher co-dimension singularities.
Singular solutions are naturally ubiquitous in many contexts and often have a prominent analytical and physical interest. A solution is said singular if it cannot be described by a regular functions or, more generally, by a regular geometrical object.
The analysis of singularities is by now a classical topic in mathematics and it has required, and still requires, the development of new analytical tools beyond the present state of the art.
As it often happens, there is no general theory of singularities and every problem, or family of problems, needs to develop a set of techniques which are specifically tailored. In this project the problems we consider are two classical questions in the geometric calculus of variations: the analysis of minimal surfaces and the thin obstacle problem. There are deep connections between these two classes of problems and the project is aimed to develop analytical techniques which can be applied to both of them.
We are looking at geometric surfaces minimizing suitable energies, that is the area in the first case, and the elastic energy in the second case, under suitable conditions: fixed boundaries and, in the second case, the constraint of lying above a a lower dimensional obstacle (like a membrane pushed by the blade of a knife). These problems arise in several analytical, geometric and physical instances and have been studied extensively in the last century. Nevertheless, many questions remains still open from the mathematical point of view and their solutions would represent a concrete progress in mathematics. The main objectives of the project is the theoretical analytical characterization of the singular solutions to these problems, not the aspects related to the applications.
A particular issue which is present in both questions, yet not well-understood from a mathematical point of view, is related to the higher co-dimension of the objects under consideration, for example two-dimensional minimal surfaces in a space with four dimensions, or free boundaries of dimension one in a three dimensional space. More generally, we look at surfaces of dimension K in a space of dimension N with N-K>1: the terms higher co-dimension refer to the difference of dimensions N-K being bigger than one. This is a crucial aspect of the project, because many of the analytical and geometric principles which are valid in co-dimension one (i.e. for minimal hypersurfaces or for the classical obstacle problems) do not apply to higher co-dimensions and there are compelling questions about the singularities of these two classical problems: for example, it is not known whether the singularities of higher co-dimension minimal surfaces are finite in measure or infinite, or what the free boundary look like in the thin obstacle problem.
The main unifying theme of the project is the central role played by geometric measure theory, which underlines various common aspects of these two classes of questions and makes them suited to be treated in a unified framework. Geometric measure theory is a classical field in mathematics studying the structure of measures from a geometrical point of view. The final goal of the project is to develop suitable analytical techniques that provides valuable insights on the mathematics at the basis of higher co-dimension singularities.
During the project we have made some progress on the understanding the regularity of free boundary for the thin obstacle problems and for minimal surfaces.
In particular, we have done the following achievements.
1. In collaboration with M. Focardi (Uni. Florence), we have investigated the global geometric properties of the free boundary for the thin obstacle problem. In a publication we have answered the following long-standing open question affirmatively, whether the free boundary of the thin obstacle problem has finite measure.
Until our contribution there were only partial results in this direction, characterizing special parts of the free boundary, but there was no global estimate. We have exploited some recent progress in geometric measure theory and a novel use of the frequency function to prove a global result on the structure of the free boundary.
2. In collaboration with C. De Lellis (IAS), A. Marchese (Uni. Trento) and D. Valtorta (Uni. Zurich), we have analyzed the structure of the higher multiplicity points of minimizing harmonic multiple valued functions. This concept has been introduced in the study of higher co-dimension minimal surfaces as a suitable linearization of the problem: in particular, multiple valued harmonic functions can be seen as the solutions to a linear partial differential equation geometrically approximating higher co-dimension minimal surfaces. In a publication we have given the first result in the literature on the structure and the measure of singular points of surfaces of higher co-dimension.
3. In collaboration with M. Focardi (Uni. Florence) we have studied the problem of a minimal surface with a thin obstacle. The regularity for such problem was open since many years (the problem was posed by J. Nitsche in 1969) and we have shown the optimal regularity of the solutions and the global structure of the free boundary points for flat obstacles. The general case of Nitsche's problem is still an open question.
Further works performed within the project includes the study of nonlinear free boundary problems, the regularity of stationary surfaces, the evolution of free boundaries according to curvature flows and the characterization of grain boundaries in material sciences, to mention a few.
In particular, we have done the following achievements.
1. In collaboration with M. Focardi (Uni. Florence), we have investigated the global geometric properties of the free boundary for the thin obstacle problem. In a publication we have answered the following long-standing open question affirmatively, whether the free boundary of the thin obstacle problem has finite measure.
Until our contribution there were only partial results in this direction, characterizing special parts of the free boundary, but there was no global estimate. We have exploited some recent progress in geometric measure theory and a novel use of the frequency function to prove a global result on the structure of the free boundary.
2. In collaboration with C. De Lellis (IAS), A. Marchese (Uni. Trento) and D. Valtorta (Uni. Zurich), we have analyzed the structure of the higher multiplicity points of minimizing harmonic multiple valued functions. This concept has been introduced in the study of higher co-dimension minimal surfaces as a suitable linearization of the problem: in particular, multiple valued harmonic functions can be seen as the solutions to a linear partial differential equation geometrically approximating higher co-dimension minimal surfaces. In a publication we have given the first result in the literature on the structure and the measure of singular points of surfaces of higher co-dimension.
3. In collaboration with M. Focardi (Uni. Florence) we have studied the problem of a minimal surface with a thin obstacle. The regularity for such problem was open since many years (the problem was posed by J. Nitsche in 1969) and we have shown the optimal regularity of the solutions and the global structure of the free boundary points for flat obstacles. The general case of Nitsche's problem is still an open question.
Further works performed within the project includes the study of nonlinear free boundary problems, the regularity of stationary surfaces, the evolution of free boundaries according to curvature flows and the characterization of grain boundaries in material sciences, to mention a few.
In the work performed in the project, we have made some significant progress about the structure of the singularities for higher co-dimension minimal surfaces and free boundary problems with thin obstacles.
In particular, the global structure of the free boundary for the thin obstacle problem and for the Nitsche problem of minimal surfaces with thin obstacles was not within reach using the techniques and the knowledge previously developed, and similarly there were completely no previous results on the structure of the singular set for multiple valued harmonic functions, or more general higher co-dimension minimal surfaces.
In our work we have introduced a new approach combining some techniques in geometric measure theory and novel estimates on the monotonic quantity named frequency, which allowed to solve long-standing regularity questions and have the potentiality to be used in several other problems in the geometric calculus of variations.
In particular, the global structure of the free boundary for the thin obstacle problem and for the Nitsche problem of minimal surfaces with thin obstacles was not within reach using the techniques and the knowledge previously developed, and similarly there were completely no previous results on the structure of the singular set for multiple valued harmonic functions, or more general higher co-dimension minimal surfaces.
In our work we have introduced a new approach combining some techniques in geometric measure theory and novel estimates on the monotonic quantity named frequency, which allowed to solve long-standing regularity questions and have the potentiality to be used in several other problems in the geometric calculus of variations.