Mathematical Analysis

The free boundary problems are PDE problems (often elliptic or parabolic) in which the domain of the solution is not a priori prescribed but is part of the variables of the problem. The boundary conditions are overdetermined and might involve the trace of the function, its gradient (or just the normal derivative), and the curvature of the boundary. Solutions to free boundary problems are obtained for instance from the minimization of variational functionals that take into account both the state function (the solution of the PDE) and its domain of definition. Well-known examples of free boundary problems are the one-phase and the two-phase Bernoulli problems, the obstacle problem, the thin-obstacle problem, the optimal partitions problems, Mumford-Shah, free transmission problems, and their parabolic counterparts, for instance, the Stefan problem.

The focus is on the free boundary regularity theory, which is the study of the fine local structure of the free boundary and relates to the analysis of the interplay between the geometry of the domain and the behavior of the state function (the solution of the PDE). Of particular interest are theorems that guarantee (under some geometric conditions) that the boundary is a smooth manifold (the so-called epsilon-regularity theorems) and the analysis of the free boundaries around singularities. The methods include blow-up analysis, monotonicity formulas, density estimates, estimates on the dimension of the singular set, viscosity solutions, analysis of the stable cones, improvement-of-flatness methods, epiperimetric inequalities.

The free boundary problems are strongly related to problems in Geometric Analysis as the regularity of area-minimizing surfaces and currents; in Elliptic PDEs, for instance to the analysis of the nodal sets to the solutions of PDEs or the behavior at the boundary of the solutions (Boundary Harnack inequalities); and also to some Shape Optimization problems.

The idea of deforming a geometric object by means of a parabolic PDEs to simplify it (e.g. to reduce its topological complexity or to make it more symmetric) goes back to Eells and Sampson who introduced the first geometric evolution equation: the harmonic map flow. Since 1964, geometric flows have been applied to a variety of topological, analytical, and physical problems, giving in some cases very fruitful results.

Among other geometric equations, we mention the mean curvature flow. This flow can be regarded as the gradient flow of the area functional: an n-dimensional time-dependent surface evolves with normal velocity its mean curvature, at any point and time. In place of the area functional, one may consider generalized perimeters, such as the fractional perimeter, and the corresponding gradient flows. Concerning the study of these flows, we are interested in the long-time existence of (weak and strong) solutions and in the classification of possible singularities. Besides the urge to introduce a non-local version of mean curvature flow, one wants also to extend it to the more singular geometric objects.

Motived by the study of grain boundaries, Brakke introduces the first weak versions of the flow and in the years a plethora of weak definitions (level set approach, minimizing movements, thresholding scheme, canonical Brakke flow) has been developed, but the one-dimensional version of this problem is also studied in the framework of classical PDEs. In such a situation the flow is called network flow.

We are interested in several aspects of evolution: well-posedness, long-time behavior, analysis of singularities, flow past singularities, and stability. Our main goal is to exploit the known results to a higher dimension.

The development of Geometric Measure Theory (GMT) began about one hundred years ago, with the first intrinsic notions of dimension for general sets, which could be possibly non-integer, and the corresponding intrinsic notions of measure, where “intrinsic” means that it does not rely on parametrizations.
In the second half of the last century, one of the driving problems was the existence of d-dimensional surfaces with minimal area and prescribed boundary (Plateau problem), which led to the development of several “weak” notions of surface, such as finite perimeter set (Caccioppoli, De Giorgi), integral currents (Federer and Fleming), and varifolds (Almgren).

In the past fifty years these notions played a fundamental role in the treatment of many problems of geometric and variational nature besides minimal surfaces, from the asymptotic behavior of singular perturbations problems in continuum mechanics and physics (e.g, scalar and vector models of Ginzburg-Landau type) to the notions of weak solutions of geometric evolution problems (e.g., the mean curvature flow), and even the notion of generalized surfaces (Young measures) as a means to solve all types of variational and non variational problems.

At the same time, many attempts have been made to extend these theories to various non-Euclidean settings (sub-Riemannian manifolds and metric spaces with bounds on curvature). These attempts have led to several fundamental questions that have not yet been fully answered–indeed in many cases our understanding is still quite limited.

The Italian mathematical community in Calculus of Variations and Geometric Measure Theory, whose origins can be traced back to the work of Ennio De Giorgi here in Pisa, gave essential contributions to the developments mentioned above. Among the topics currently studied in our department, we mention the following:

• GMT in sub-Riemannian manifolds: area formulas, contact sets, currents;
• Minimal partitions;
• Gauss-Green formulas, sets of finite perimeter and rectifiability in stratified Lie groups;
• Weak solutions to geometric evolution problems;
• Description of the regularity, variational, and fine properties associated with a PDE constraint (e.g., differential inclusions);
• Structure of PDE-constrained measures and differentiability of Lipschitz functions.

The field of shape optimization had very strong growth in the last decades, mainly due to the several applications in engineering, but also for the mathematical interest related to several problems in spectral geometry. The goal is to find a domain $\Omega$ that minimizes a given cost functional, that may involve geometrical quantities as perimeter or measure, as well as other quantities related to PDEs like for instance the related energies or eigenvalues. Showing the existence of optimal shapes, and their regularity is often a very challenging issue that needs the development of new and deep tools. 

Our research focuses on the statistical properties of deterministic and random dynamical systems.

These studies have a wide range of applications, from a number of theoretical and geometrical questions to the understanding of the behavior of models of real-world phenomena like the dynamics of fluids, the evolution of climate and its extreme events, the dynamics of extended systems in a mean-field coupling and other complex systems. The methods used in this research are often related to probability, functional analysis, transport, and geometric measure theory.

Our research also focused on reliable computational tools for the study of the statistical properties of dynamics, computer-aided proofs, and their relation to theoretical computer science.

Our research focuses on the pure and applied aspects of mathematical fluid dynamics of Newtonian and non-Newtonian fluids.

The research interests span from the most theoretical questions of existence, uniqueness, and regularity of weak solutions towards applications to the numerical analysis of such flows (with finite element/differences methods) and to the study of turbulent flows, with applications to geophysical problems. The methods we use stem from those of classical analysis of PDEs in Sobolev spaces, but for the problems we consider we are using also: convex integration, analysis in Orlicz, weighted spaces, and approximate methods like reduced-order ones. More specifically for the Equations of classical fluid mechanics: incompressible Euler and Navier-Stokes equations. Existence, uniqueness, properties, and regularity of weak and suitable solutions.

Concerning non-Newtonian fluids, we are interested in the existence, regularity, numerical approximation, and long-time behavior of classes of p-fluids or electrorheological ones.

For the turbulent flows, we study the mathematical properties and the mathematical modeling of such methods, having in mind the application to the study of boundary layers and air-sea interaction, together with other multi-phase problems arising in the study of volcanic eruptions.

The nonlinear analysis group focuses on partial differential equations of elliptic and hyperbolic types in which the presence of the nonlinear term is crucial for the existence and qualitative properties of solutions.

The research interests span from nonlinear Schroedinger equations and Gross-Pitaevskij in a Euclidean setting to Yamabe problems on compact manifolds. The main topics of the research concerning elliptic equations are the existence and multiplicity of concentrated solutions, qualitative properties, blow-up analysis, and compactness of solutions. On the time-dependent side, we focus on the existence of solitons, stability and instability issues, and long-time dynamics for equations of physical interest. Another field of research is related to functional inequalities in presence of nonlocal terms: the existence of maximizers and the role of symmetry.