Final Report Summary - ANOPTSETCON (Analysis of optimal sets and optimal constants: old questions and new results)
As suggested by the name of our Project, “Analysis of Optimal Set and optimal Constants: old questions and new results”, our original aim was quite ambitious. Namely, we decided to attack simultaneously a number of different problems, all of which presently well studied in the literature, but with some old open questions still not solved.
These problems did not have apparently so much in common, coming from different areas of Mathematics (but mainly Calculus of Variations). In fact, the reason why we put these questions together, was that we had a quite general strategy of work in mind, which we hoped to be able to use to find solutions for all these different problems. And in turn, trying to say it simple, this “strategy” was in the end to use some different classical and well-known tools at once. We have given our contribution to strengthen each of these tools, and we have used them often in an original way, but the true novelty of our approach was precisely to bring together these quite different techniques. We can now describe first our tools, and then the main results we got.
These tools were respectively symmetrization arguments, geometric constructions, and mass transportation (we have used this last one only at the very beginning of the project).
The symmetrization arguments are a number of classical results, which allow to pass from a generic set, or a generic function, to another similar set or function which has some additional symmetry, in the meantime controlling how its main properties have been modified by the process; the main difficulty to use these techniques is that one has often to perform several slight changes, in the right order, instead of making a single severe one, which modifies too much the object and loses all its characteristics.
By “geometric constructions” we mean many different elementary techniques to work directly on a set or on a function by using geometrical tools. An example can be suitable cut-and-paste constructions which allow to show important properties of optimal sets or optimal functions; another example can be the idea, classical for instance in the Numerical Analysis, to subdivide a set in several simple ones, typically squares or triangles, and to work separately on each of them, so to have only to deal with very specific situations. The main source of difficulties here, is to understand how to make the subdivision of the domain, since for our purposes this choice is essential; moreover, it is also difficult, once worked on each tile of the subdivision, to “glue” together all the results to get a unique one on the whole domain.
Finally, the mass transportation is a classical problem in the Mathematics, first proposed in the 18th century by G. Monge, and which has found dramatic improvements in the last 30 years. In particular, tools coming from mass transportation are now well-known and widely used. For our purposes, the most difficult part in using these tools was that they are in principle quite far from the problems we were dealing with.
Most of the results that we have got in this period can be divided in four big subgroups (there are also few ones which do not fit in any of these groups, as for instance the solution of the Auerbach-Santalò conjecture, and some results about the Cheeger problem and the mass transportation problem).
First of all, we found many result about quantitative estimates for geometrical and functional inequalities: this direction of research had started before the beginning of the project, and was somehow the original core of our strategy. In particular, we found quantitative estimates for the second eigenvalue of the Laplacian, for the isodiametric problem, for certain anisotropic Sobolev and log-Sobolev inequalities, and for some mixed Euclidean-Gaussian isoperimetric problems.
Concerning the spectral problems, we found some extremely general existence results for minimizers of functional depending on the eigenvalues, and we studied the Lipschitz regularity of the eigenfunctions, which is a first step toward the regularity of the optimal sets.
In the study of the isoperimetric problem in an Euclidean space with density (which is a general case that contains, in particular, the case of the manifolds), we also got a sharp existence result for the existence of isoperimetric regions, and we started the study of their regularity, finding the first regularity results for the case of densities with low regularity, namely, less than Lipschitz.
Finally, in our study of the approximation of homeomorphisms, we could completely treat the case of the Sobolev ones, showing that any homeomorphism in W^{1,p} can be approximated by diffeomorphisms, for every p equal to or bigger than one. The same result, but only for p bigger than one, had been found a couple of years before than ours, by using “standard” and very strong techniques. The main advantage of our method, which is rather original and somehow more “rough” than the standard one, relies not only in the fact that we also reach the case of p=1, but more importantly in the fact that it seems to allow only to deal with the fundamental case of bi-Sobolev homeomorphisms, namely, those which are Sobolev and whose inverse is also Sobolev. Up to now, we could only get some preliminary results in this bi-Sobolev setting, but they are the first ones which have been ever found, and they seem quite promising. In particular, we have been able to deal with the case of p=1 and with the BV case.
To conclude, it is important to mention that, mainly thanks to the grant of this project, we have been able to organize or co-organize a number of conferences and of summer schools, all of which have had a great number of participants and an important impact in the community. This have given a wide visibility to our results, and has allowed us to discuss and interact with most of the leading experts in the areas of our interest.
These problems did not have apparently so much in common, coming from different areas of Mathematics (but mainly Calculus of Variations). In fact, the reason why we put these questions together, was that we had a quite general strategy of work in mind, which we hoped to be able to use to find solutions for all these different problems. And in turn, trying to say it simple, this “strategy” was in the end to use some different classical and well-known tools at once. We have given our contribution to strengthen each of these tools, and we have used them often in an original way, but the true novelty of our approach was precisely to bring together these quite different techniques. We can now describe first our tools, and then the main results we got.
These tools were respectively symmetrization arguments, geometric constructions, and mass transportation (we have used this last one only at the very beginning of the project).
The symmetrization arguments are a number of classical results, which allow to pass from a generic set, or a generic function, to another similar set or function which has some additional symmetry, in the meantime controlling how its main properties have been modified by the process; the main difficulty to use these techniques is that one has often to perform several slight changes, in the right order, instead of making a single severe one, which modifies too much the object and loses all its characteristics.
By “geometric constructions” we mean many different elementary techniques to work directly on a set or on a function by using geometrical tools. An example can be suitable cut-and-paste constructions which allow to show important properties of optimal sets or optimal functions; another example can be the idea, classical for instance in the Numerical Analysis, to subdivide a set in several simple ones, typically squares or triangles, and to work separately on each of them, so to have only to deal with very specific situations. The main source of difficulties here, is to understand how to make the subdivision of the domain, since for our purposes this choice is essential; moreover, it is also difficult, once worked on each tile of the subdivision, to “glue” together all the results to get a unique one on the whole domain.
Finally, the mass transportation is a classical problem in the Mathematics, first proposed in the 18th century by G. Monge, and which has found dramatic improvements in the last 30 years. In particular, tools coming from mass transportation are now well-known and widely used. For our purposes, the most difficult part in using these tools was that they are in principle quite far from the problems we were dealing with.
Most of the results that we have got in this period can be divided in four big subgroups (there are also few ones which do not fit in any of these groups, as for instance the solution of the Auerbach-Santalò conjecture, and some results about the Cheeger problem and the mass transportation problem).
First of all, we found many result about quantitative estimates for geometrical and functional inequalities: this direction of research had started before the beginning of the project, and was somehow the original core of our strategy. In particular, we found quantitative estimates for the second eigenvalue of the Laplacian, for the isodiametric problem, for certain anisotropic Sobolev and log-Sobolev inequalities, and for some mixed Euclidean-Gaussian isoperimetric problems.
Concerning the spectral problems, we found some extremely general existence results for minimizers of functional depending on the eigenvalues, and we studied the Lipschitz regularity of the eigenfunctions, which is a first step toward the regularity of the optimal sets.
In the study of the isoperimetric problem in an Euclidean space with density (which is a general case that contains, in particular, the case of the manifolds), we also got a sharp existence result for the existence of isoperimetric regions, and we started the study of their regularity, finding the first regularity results for the case of densities with low regularity, namely, less than Lipschitz.
Finally, in our study of the approximation of homeomorphisms, we could completely treat the case of the Sobolev ones, showing that any homeomorphism in W^{1,p} can be approximated by diffeomorphisms, for every p equal to or bigger than one. The same result, but only for p bigger than one, had been found a couple of years before than ours, by using “standard” and very strong techniques. The main advantage of our method, which is rather original and somehow more “rough” than the standard one, relies not only in the fact that we also reach the case of p=1, but more importantly in the fact that it seems to allow only to deal with the fundamental case of bi-Sobolev homeomorphisms, namely, those which are Sobolev and whose inverse is also Sobolev. Up to now, we could only get some preliminary results in this bi-Sobolev setting, but they are the first ones which have been ever found, and they seem quite promising. In particular, we have been able to deal with the case of p=1 and with the BV case.
To conclude, it is important to mention that, mainly thanks to the grant of this project, we have been able to organize or co-organize a number of conferences and of summer schools, all of which have had a great number of participants and an important impact in the community. This have given a wide visibility to our results, and has allowed us to discuss and interact with most of the leading experts in the areas of our interest.