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,
especially 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.
Keywords: free boundary regularity, the one-phase Bernoulli problem, Alt-Caffarelli,
the two-phase Bernoulli problem, Alt-Caffarelli-Friedman, the vectorial Bernoulli problem,
the obstacle problem, the Signorini problem, epiperimetric inequality,
logarithmic epiperimetric inequality, monotonicity formulas.
References:
• Lecture notes on the regularity of the one-phase free boundaries: pdf .
• The vectorial Bernoulli problem: MTV, CSY, KL, KL2, MTV2.
Starting from 2022 the seminars and the meetings of the workgroup of the project are held in the Deparment of Mathematics (University of Pisa)
with the participation of the members of the workgroup and the visitors invited on the project.
The main topics are regularity theory, free boundary problems, calculus of variations, elliptic PDEs, geometric analysis.
The dates of the meetings and other notifications are diffused on the mailing list of the workgroup.
18/05/2023. Luca Spolaor (UC San Diego).
Title: Interior regularity for stationary two-dimensional multivalued maps.
Abstract. Q-valued maps minimizing a suitably defined Dirichlet energy were introduced
by Almgren in his proof of the optimal regularity of area minimizing currents in any
dimension and codimension. In this talk I will discuss the extension of Almgren's
results to stationary Q-valued maps in dimension 2.
This is joint work with Jonas Hirsch (Leipzig).
05/04/2023. Roberto Ognibene (Università di Pisa).
Title: Capacity and torsional rigidity: two measures of spectral stability.
Abstract. In this talk, I will discuss the behavior of the spectrum of the Laplacian
on bounded domains, subject to varying mixed boundary conditions.
More precisely, let us assume the boundary of the domain to be split into two parts,
on which homogeneous Neumann and Dirichlet boundary conditions are respectively prescribed;
let us then assume that, alternately, one of these regions “disappears” and the
other one tends to cover the whole boundary. In this framework, I will first describe
under which conditions the eigenvalues of the mixed problem converge to the ones
of the limit problem (where a single kind of boundary condition is imposed);
then, I will sharply quantify the rate of this convergence by providing an
explicit first-order asymptotic expansion of the “perturbed” eigenvalues.
These results have been obtained in collaboration with L. Abatangelo, V. Felli and B. Noris.
14/03/2023 (cancelled). Carlos Kenig (University of Chicago).
New channels of energy for wave
equations, new non-radiative solutions
and soliton resolution.
Abstract. We will discuss the role of nonradiative solutions to nonlinear
wave equations, in connection with
soliton resolution. Using new
channels of energy estimates we
characterize all radial nonradiative solutions for a general
class of nonlinear wave equations. This is joint work with C.Collot, T.
Duyckaerts, and F. Merle.
08/03/2023. Damià Torres-Latorre (Universitat de Barcelona).
Title: Optimal regularity for supercritical parabolic obstacle problems.
Abstract. The parabolic nonlocal obstacle problem is said to be in the
supercritical regime (s < 1/2) when the time derivative is of higher order
than the diffusion operator. We will discuss the optimal C1,1 regularity
of solutions and the C1,α regularity of the free boundary.
The arguments rely on comparison principles and the scaling of the equation
to circumvent the fact that blow-ups, the usual technique for free
boundary problems, are not useful in this context.
This is a joint work with X. Ros-Oton.
01/03/2023. Seongmin Jeon (KTH).
Almost minimizers for the parabolic thin obstacle problem.
Abstract. We consider almost minimizers for the parabolic thin
obstacle (or Signorini) problem with zero obstacle.
We establish their Hσ, σ/2 - regularity for every
σ, strictly between zero and one, as well as Hβ,β/2 - regularity of their
spatial gradients on the either side of the thin space for some β again in (0,1).
We then extend these regularity results to the variable Hölder continuous coefficient setting.
We also discuss the regularity of the "regular" part of the free boundary.
This is based on joint work with Arshak Petrosyan.
06/12/2022. Carlo Gasparetto (SISSA).
A short proof of Allard's theorem.
Abstract. Allard's theorem roughly states that a minimal surface,
that is close enough to a plane, coincides with the graph of a smooth function
which enjoys suitable a priori estimates. In this talk we will show
how one can prove this result by exploiting viscosity technique and a
weighted monotonicity formula.
This talk is based on a joint work with Guido De Philippis and Felix Schulze.
11/10/2022 (cancelled). Hui Yu (National University of Singapore).
Rate of blow-up in the thin obstacle problem.
Abstract. The thin obstacle problem is a classical free boundary problem
arising from the study of an elastic membrane resting on a lower-dimensional
obstacle. Concerning the behavior of the solution near a contact point between
the membrane and the obstacle, many important questions remain open.
In this talk, we discuss a unified method that leads to a rate of convergence
to `tangent cones' at contact points with integer frequencies in general
dimensions as well as 7/2-frequency points in 3d.
This talk is based on recent joint works with Ovidiu Savin (Columbia).
25/05/2022. Salvatore Stuvard (Università degli Studi di Milano).
Existence of canonical multi-phase Brakke flows.
Abstract (pdf).
18/05/2022. Edoardo Mainini (Università di Genova).
Linearization of finite elasticity.
Abstract. We discuss the linearization of finite elasticity by means of Gamma-convergence for the case of pure traction problems.
For hyperelastic bodies subject to equilibrated force fields, we show that the limit energy can be different from the global energy
of linear elasticity subject to the same force field, unless suitable conditions are fulfilled.
We also discuss linearization under incompressibility constraint.
12/04/2022. Cristiana De Filippis (University of Parma).
Nonuniform ellipticity and nonlinear potentials.
Abstract. Nonuniform Ellipticity and Nonlinear Potential Theory are two classical topics in the analysis of Partial Differential Equations.
In this talk I show how those themes merge to solve the longstanding open problem of deriving Schauder estimates
for minima of functionals (resp. solutions to elliptic equations) featuring polynomial nonuniform ellipticity.
This is joint work with Giuseppe Mingione (University of Parma).
06/04/2022. Dario Mazzoleni (University of Pavia).
L2-Gradient Flows of Spectral Functionals.
Abstract (pdf).
09/03/2022. Nicola Soave (Politecnico di Milano).
Free boundary problems in the spatial segregation of competing systems.
Abstract. In this talk we present some results concerning the spatial segregation in systems with strong competition.
In particular, we focus on two different (but strongly related) issues: long-range segregation models and systems characterized by asymmetric diffusion.
The content of the talk is part of ongoing project with H. Tavares, S. Terracini and A. Zilio.
21/02/2022. Giulia Bevilacqua (Politecnico di Torino).
The Kirchhoff-Plateau problem.
Abstract (pdf).
Invited speakers and seminars before 2022
25/09/2023 - 28/09/2023. Luigi De Masi (Università di Padova)
25/04/2023 - 27/05/2023. Luca Spolaor (UC San Diego)
25/02/2023 - 04/03/2023. Seongmin Jeon (KTH)
04/12/2022 - 08/12/2022. Carlo Gasparetto (SISSA)
30/11/2022 - 02/11/2022. Gianmaria Verzini (Politecnico di Milano)
7/11/2022 - 18/11/2022. Carlo Gasparetto (SISSA)
11/10/2022 - 15/10/2022. Hui Yu (National University of Singapore)
22/9/2022 - 25/9/2022. Max Engelstein (University of Minnesota)
22/5/2022 - 27/5/2022. Salvatore Stuvard (Università di Milano)
16/5/2022 - 19/5/2022. Edoardo Mainini (Università di Genova)
10/4/2022 - 13/4/2022. Cristiana De Filippis (Università di Parma)
3/4/2022 - 8/4/2022. Dario Mazzoleni (Università di Pavia)
8/3/2022 - 10/3/2022. Nicola Soave (Politecnico di Milano)
20/2/2022 - 23/2/2022. Giulia Bevilacqua (Politecnico di Torino)
16/1/2022 - 29/1/2022. Mickaël Nahon (Université de Savoie, Chambery)
5/12/2021 - 9/12/2021. Luca Spolaor (UC San Diego)
25/10/2021 - 28/10/2021. Alessandro Audrito (ETH, Zürich)
10/9/2021 - 24/9/2021. Ekaterina Mukoseeva (University of Helsinki)
4/7/2021 - 10/7/2021. Mickaël Nahon (Université de Savoie, Chambery)
