Periodic Reporting for period 1 - LNLFB-Problems (Local and Nonlocal Free Boundary Problems)
Reporting period: 2020-09-01 to 2022-08-31
Free Boundary (FB) problems arise in many applications such us Biology, Physics, Finance, Fluid Dynamics: they are usually described by some Partial Differential Equations (PDEs) that exhibit in addition some unknown interfaces. A classical example is the Stefan problem, which describes the melting of an ice block into water: the main idea is to model the space-time variation of the temperature through a parabolic problem, distinguishing between the region where it is zero (ice) and where it is positive (water). The most challenging problem is to study the evolution and regularity of both the temperature distribution and its FB, that is the region which separates the ice from the water. The peculiarity of the "double unknown" makes this class of problems significantly complex: the advancements in this field have been possible thanks to some innovative techniques which combine tools from different areas like PDEs, Calculus of Variations and Geometric Measure Theory.
In the meantime, nonlocal diffusion problems have been the subject of considerable research too and, as FB problems, they possess significant applications in Finance, Industry as well as in Probability and Statistics. The interest in nonlocal problems comes from a simple empiric observation: a large class of physical phenomena are characterized by long-range interactions between particles involved in the diffusion process, which cannot be investigated with models with local diffusion (that is, based on Brownian motion). Models with nonlocal diffusion (for example, the fractional heat equation) seem to be the right tools to fruitfully describe them, since they take into account that the moving particles can jump from a value to another in a time step: this generates probability densities exhibiting fat tails, in deep contrast with standard diffusion.
The MSCA 892017 (LNLFB-Problems) has been mainly developed around this two main areas, with two corresponding parts and four main objectives:
- The first part is devoted to elliptic problems and focuses on two objectives: the first one is to prove
some quantitative regularity estimates and classification results for solutions to a class of one-phase semilinear problems related to combustion theory, while the second is to investigate the singular sets of fractional harmonic maps and relate such singularities to nonlocal minimal sets.
- The second part is devoted to one-phase FB problems of parabolic type with other two objectives: the first one, is to construct solutions of some parabolic non-local one-phase problems while, the second, is to investigate the optimal regularity of such solutions and their FBs.
In the meantime, nonlocal diffusion problems have been the subject of considerable research too and, as FB problems, they possess significant applications in Finance, Industry as well as in Probability and Statistics. The interest in nonlocal problems comes from a simple empiric observation: a large class of physical phenomena are characterized by long-range interactions between particles involved in the diffusion process, which cannot be investigated with models with local diffusion (that is, based on Brownian motion). Models with nonlocal diffusion (for example, the fractional heat equation) seem to be the right tools to fruitfully describe them, since they take into account that the moving particles can jump from a value to another in a time step: this generates probability densities exhibiting fat tails, in deep contrast with standard diffusion.
The MSCA 892017 (LNLFB-Problems) has been mainly developed around this two main areas, with two corresponding parts and four main objectives:
- The first part is devoted to elliptic problems and focuses on two objectives: the first one is to prove
some quantitative regularity estimates and classification results for solutions to a class of one-phase semilinear problems related to combustion theory, while the second is to investigate the singular sets of fractional harmonic maps and relate such singularities to nonlocal minimal sets.
- The second part is devoted to one-phase FB problems of parabolic type with other two objectives: the first one, is to construct solutions of some parabolic non-local one-phase problems while, the second, is to investigate the optimal regularity of such solutions and their FBs.
The work performed during the 24 months has involved the ER, the supervisor and some researchers from foreign universities, following the Gantt chart presented in the project. The main work packages (WP) and corresponding results can be summarized as follows:
- The first WP has been completed in the first 12 months (local diffusion), while its nonlocal counterpart will be the subject of future research. In collaboration with the supervisor (J. Serra), the ER has developed an “improvement of flatness” technique for semilinear one-phase problems and exploited it to obtain 1D symmetry of global minimizers of the energy.
- The second objective (the most challenging part of the project) is the subject of current research: some partial results have already been obtained (linearised operator, main model of singularity and classification of homogeneous solutions), while the “improvement of flatness” part is under study (this is a joint work with M. Medina and J. Serra).
- The third and the fourth WPs are almost complete: I established some uniform Holder bounds for the elliptic perturbation of a nonlocal parabolic one-phase equation (when the diffusion process is driven by the “fractional power of the heat operator”) and, in a forthcoming paper, I will show the existence of solutions to the limit problem and the optimal regularity (joint work with T. Sanz).
- In addition to the results mentioned above, I was also involved in two research projects which were not planned in the proposal. In the first, I proved convergence of solutions to the Porous Medium equation posed in cylindrical domains to a special wave solutions for large times (joint work with A. Garriz and F. Quiros). In the second, I showed a structure theorem for the FB of solutions to the parabolic obstacle problem with fully nonlinear diffusion and proved full regularity of the regular part (joint work with T. Kukuljan).
The project’s results have been widely presented to the PDE community in workshops, conferences, seminars and research stays: 3 workshops/conferences, 9 talks, 3 research stays and 5 researchers hosted at ETHZ during the MSCA. In these occasions I received great feedback and undertake new scientific collaborations. Further, I was involved in many scientific popularization activities (Science is Wonderful, European Researcher’s Night, Welcome Home), tutoring activities for MSCA applicants (2 online workshops) and many training activities and courses such as Project and Time Management. All the work done was incredibly useful for my personal and professional growth.
- The first WP has been completed in the first 12 months (local diffusion), while its nonlocal counterpart will be the subject of future research. In collaboration with the supervisor (J. Serra), the ER has developed an “improvement of flatness” technique for semilinear one-phase problems and exploited it to obtain 1D symmetry of global minimizers of the energy.
- The second objective (the most challenging part of the project) is the subject of current research: some partial results have already been obtained (linearised operator, main model of singularity and classification of homogeneous solutions), while the “improvement of flatness” part is under study (this is a joint work with M. Medina and J. Serra).
- The third and the fourth WPs are almost complete: I established some uniform Holder bounds for the elliptic perturbation of a nonlocal parabolic one-phase equation (when the diffusion process is driven by the “fractional power of the heat operator”) and, in a forthcoming paper, I will show the existence of solutions to the limit problem and the optimal regularity (joint work with T. Sanz).
- In addition to the results mentioned above, I was also involved in two research projects which were not planned in the proposal. In the first, I proved convergence of solutions to the Porous Medium equation posed in cylindrical domains to a special wave solutions for large times (joint work with A. Garriz and F. Quiros). In the second, I showed a structure theorem for the FB of solutions to the parabolic obstacle problem with fully nonlinear diffusion and proved full regularity of the regular part (joint work with T. Kukuljan).
The project’s results have been widely presented to the PDE community in workshops, conferences, seminars and research stays: 3 workshops/conferences, 9 talks, 3 research stays and 5 researchers hosted at ETHZ during the MSCA. In these occasions I received great feedback and undertake new scientific collaborations. Further, I was involved in many scientific popularization activities (Science is Wonderful, European Researcher’s Night, Welcome Home), tutoring activities for MSCA applicants (2 online workshops) and many training activities and courses such as Project and Time Management. All the work done was incredibly useful for my personal and professional growth.
As mentioned in the above paragraphs, the project has progressed quite linearly despite the challenging goals and, in collaboration with my supervisor, I have obtained interesting results with a strong impact on the PDE community. More specifically:
- The classification of global minimizers of semilinear one-phase problems (up to space dimension 4) is very robust and the only other result in this direction was established by O. Savin in 2009 (De Giorgi conjecture). It is left to understand whether the classification holds true in dimension 5 and 6 (super challenging issue!), and disprove it in spaces of dimension higher that 7.
- The study of singularities for fractional harmonic maps is believed to have an incredible impact for two reasons: first, it connects singularities with nonlocal minimal surfaces (a totally new approach in the field) and, second, it will allow to approximate solutions to local problems with nonlocal ones (this methods will be extended to a wide range of problems in the future).
- Exploiting the elliptic regularization to construct solutions to parabolic FB problems has two significant aspects that I believe will play an important role in the future: first, it goes extremely beyond the current state of the art by exploiting the stability of the approximating sequence (which, different from minimality, could pass to the limit) while, secondly, it is extremely flexible and many other problems can be attacked with the same approach (Mean Curvature Flow, Harmonic Maps Flow and so on).
As already mentioned, the topics studied in the project and the problems that are still open will be the center of my future research.
- The classification of global minimizers of semilinear one-phase problems (up to space dimension 4) is very robust and the only other result in this direction was established by O. Savin in 2009 (De Giorgi conjecture). It is left to understand whether the classification holds true in dimension 5 and 6 (super challenging issue!), and disprove it in spaces of dimension higher that 7.
- The study of singularities for fractional harmonic maps is believed to have an incredible impact for two reasons: first, it connects singularities with nonlocal minimal surfaces (a totally new approach in the field) and, second, it will allow to approximate solutions to local problems with nonlocal ones (this methods will be extended to a wide range of problems in the future).
- Exploiting the elliptic regularization to construct solutions to parabolic FB problems has two significant aspects that I believe will play an important role in the future: first, it goes extremely beyond the current state of the art by exploiting the stability of the approximating sequence (which, different from minimality, could pass to the limit) while, secondly, it is extremely flexible and many other problems can be attacked with the same approach (Mean Curvature Flow, Harmonic Maps Flow and so on).
As already mentioned, the topics studied in the project and the problems that are still open will be the center of my future research.