Abstract: In 2007 Choe, Ghomi and Ritoré proved the relative isoperimetric inequality outside a convex set $C$. More precisely they showed that if $E\subset \mathbb{R}^N \setminus C$ has finite mass $m$, then the perimeter of $E$ outside $C$ is equal to the surface measure of the half sphere enclosing the half ball with mass $m$. Moreover, they showed that if $C$ is smooth and the equality holds in this isoperimetric inequality, then $E$ is a half ball sitting on $C$. In this talk I will discuss the equality case when $C$ is any convex set, not necessarily smooth.
This is a joint work with M. Morini.

Abstract:We provide the first general result for the asymptotics of the area preserving mean curvature flow in two dimensions showing that flat flow solutions starting from any bounded set of finite perimeter converge to a finite union of equally sized disjoint disks with exponential rate.

Abstract: Let $\Omega \subset R^d$ be a set with finite Lebesgue measure such that, for a fixed radius $r>0$, the Lebesgue measure of $\Omega \cap B _ r (x)$ is equal to a positive constant when $x$ varies in the essential boundary of $\Omega$. We prove that $\Omega$ is a ball (or a finite union of equal balls) provided it satisfies a nondegeneracy condition, which holds in particular for any set of diameter larger than $r$ which is either open and connected, or of finite perimeter and indecomposable.
This is a joint work with Ilaria Fragalà.

Abstract: Finding graphs of prescribed mean curvature is one of the classical problems of Calculus of Variations, strongly motivated by capillarity theory. We present a novel, weak formulation of the equation \[ \mathop{\rm div} \frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}} = \mu \] where the prescribed mean curvature $\mu$ can be a signed measure, and for which existence of solutions can be proved under minimal assumptions on $\mu$. The existence proof is obtained by minimizing a suitable functional on a class of $BV$ functions with Dirichlet boundary conditions. It combines various tools like the notion of $\lambda$-pairing, a refined form of Anzellotti-Giaquinta theorem in $BV$, and a convex duality argument.
This is a joint collaboration with Giovanni Comi.

Abstract: For an integral functional defined on functions $(u,v)$ in $W^{1,1}\times L^1$ featuring a strong interaction term between $u$ and $v$, we calculate its relaxation in the space of functions with bounded variations and Radon measures. Emerging free discontinuities can interact with the measure variable, which can lead to non-trivial behavior of minimizing sequences. As it turns out, these can exhibit interesting oscillatory and concentrating behavior even if the leading terms in the functional (in the sense of highest order derivatives) are convex and coercive.
Joint work: Martin Kružìk (Prague), Elvira Zappale (Rome)

Abstract: We consider a geometric problem consisting of an evolution equation for a closed hypersurface coupled to a parabolic equation on this evolving surface. More precisely, the evolution of the hypersurface is determined by a scaled mean curvature flow that depends on a quantity defined on the surface via a diffusion equation. This system arises as a gradient flow of a simple energy functional. Assuming suitable parabolicity conditions, we derive short-time existence for the system. The proof is based on linearization and a contraction argument. For this, we parameterize the hypersurface via a height function and thus the system, originally defined on an evolving surface, can be transformed onto a fixed reference surface. The result is formulated in a classical sense, holds for the case of embedded and immersed hypersurfaces alike and provides an existence time independent of small changes in the initial surface. Afterwards, several properties of the solution are analyzed. Emphasis is placed on to what extent the surface in our setting evolves the same as for the usual mean curvature flow. To this end, we show that the surface area is strictly decreasing but give an example of a surface that exists for infinite times nevertheless. Moreover, mean convexity is conserved whereas convexity is not. Finally, we construct an embedded hypersurface that develops a self-intersection in the course of time.
Finally, we discuss how solutions can be computed numerically with the help of an evolving surface finite element discretization. We will discuss optimal error bounds and present numerical experiments illustrating the above discussed qualitative properties of the flow as well as the convergence behaviour.
The analytical part is a joint work with Helmut Abels and Felicitas Burger (both University Regensburg) and the numerical part is joint work with Charlie Elliott (University of Warwick) and Balàzs Kovàcs (University Regensburg).

Abstract: The curve shortening flow is an evolution equation in which a curve moves with normal velocity equal to its curvature (at any point and time) and can be interpreted as the gradient flow of the length. We consider the same flow for networks (finite unions of sufficiently smooth curves whose end points meet at junctions). Because of the variational nature of the problem, one expects that for almost all the times the evolving network will possess only triple junctions where the unit tangent vectors forms angles of 120 degrees (regular junctions). However, even if the initial network has only regular junctions, this property is not preserved by the flow and junctions of four or more curves may appear during the evolution. The aim of this talk is first to describe the process of singularity formation and then to explain the resolution of such singularities and how to continue the flow in a classical PDE framework.
This is a research in collaboration with Jorge Lira (Universidade Federal do Cearà), Rafe Mazzeo (Stanford University) and Mariel Sàez (Pontificia Universidad Catòlica de Chile).

Abstract: Mean curvature flow is one of the most fundamental geometric evolution equations and appears in many surface-tension driven problems. Although the equation has an instantaneous smoothing effect, generically, singularities appear in finite time. One is led to consider weak solutions which persist through these singular events. Folklore says that mean curvature flow is a gradient flow with the caveat that the underlying metric is completely degenerate. In this talk, after discussing previous weak solution concepts, I will present a new notion which has its roots in the theory of gradient flows and relies on basic geometric measure theory. I will show that these solutions arise naturally in the sharp-interface limit of the Allen-Cahn equation and in addition satisfy a weak-strong uniqueness principle. The latter property is a fundamental difference to well-known Brakke solutions, which a priori may disappear at any given time and are therefore fatally non-unique.
This is joint work with Sebastian Hensel (U Bonn).

Abstract: We discuss a Plateau problem in non-parametric form with a partial free boundary. We show existence and regularity of solutions, and then we compare the obtained solutions with the solutions of a parametric plateau problem with corresponding boundary data.

Abstract: We discuss the existence of generalized clusters with a finite or infinite number of components, having minimal total perimeter and given masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving homeomorphisms. Such clusters are a natural relaxation of a cluster and can be thought of as ``albums'' with possibly infinite pages, the total perimeter being calculated by summation over all pages, and being minimal among all generalized clusters with the same masses.

Abstract: We consider the model equation arising in the theory of viscoelasticity $$ \partial_{tt} u(t)-\big[1+k_t(0)\big]\Delta u(t) -\int_0^\infty k'_t(s)\Delta u(t-s) d s + f(u(t)) = g $$ in the presence of a (convex, nonnegative and summable) memory kernel $k_t(\cdot)$ explicitly depending on time. Such a model is apt to describe, for instance, the dynamics of aging viscoelastic materials. From the mathematical viewpoint, this translates into the study of dynamical systems acting on time-dependent spaces, according to the recently established theory of Di Plinio et al. [3]. After giving the proper notion of solution, along with a global well-posedness result, we focus on the asymptotic properties of the solutions, proving the existence and the regularity of the time-dependent global attractor for the dynamical process generated by the equation. In addition, when $k_t$ approaches a multiple $m\delta_0$ of the Dirac mass at zero as $t\to\infty$, we show that the asymptotic dynamics of our problem is close to the one of its formal limit $$\partial_{tt}u(t)-\Delta u(t) -m\Delta\partial_{t} u(t)+ f(u(t)) = g$$ describing viscoelastic solids of Kelvin-Voigt type. This talk is based on the papers [1,2].
[1] M. Conti, V. Danese, C. Giorgi, V. Pata. A model of viscoelasticity with time-dependent memory kernels. Amer. J. Math., 140 (2018), 349--389.
[2] M. Conti, V. Danese, V. Pata. Viscoelasticity with time-dependent memory kernels, II: Asymptotic behavior of solutions. Amer. J. Math., 140 (2018), 1687--1729.
[3] Di Plinio, G.S. Duane, R. Temam. Time dependent attractor for the oscillon equation. Discrete Contin. Dyn. Syst., 29 (2011), 141--167.

Abstract: We prove Taylor scaling (in some contexts also known as Friedel scaling) for dislocation lines characterized by line-tension and moving by curvature under the action of an applied shear stress in a plane containing a random array of obstacles. Specifically, we show-in the sense of optimal scaling-that the critical applied shear stress for yielding, or percolation-like unbounded motion of the dislocation, scales in proportion to the square root of the obstacle density. For sufficiently small obstacle densities, Taylor scaling dominates the linear-scaling that results from purely energetic considerations as well as the two-thirds Labusch scaling, and, therefore, characterizes the dominant rate-limiting mechanism in that regime.
Joint work with L.Courte (Luxembourg) and M.Ortiz (Bonn, Pasadena).

Abstract: In a two-dimensional Hele-Shaw cell, provided that the viscous forces dominate the inertial ones, the well-known Navier-Stokes-Cahn-Hilliard system for an incompressible binary flow can be approximated by the so-called Cahn-Hilliard-Hele-Shaw (CHHS) (or Cahn-Hilliard-Darcy) system (see [9]). In three dimensions, CHHS systems are used to describe fluid flow in a porous medium and in solid tumor growth modeling through diffuse interfaces (see, for instance, [2,3,5,6] and references therein). I intend to present and compare a number of recent results (see [1,4,7,8]), focusing, in particular, on uniqueness and regularity issues. All of them are characterized by a mixing entropy term which is not approximated, as usual, by a polynomial.
[1] C. Cavaterra, S. Frigeri, M. Grasselli. Nonlocal Cahn-Hilliard-Hele-Shaw systems with singular potential and degenerate mobility. J. Math. Fluid Mech., 24, 13 (2022).
[2] M. Dai, E. Feireisl, E. Rocca, G. Schimperna, M. E. Schonbek. Analysis of a diffuse interface model of multispecies tumor growth, Nonlinearity, 30 (2017), 1639--1658.
[3] L. Ded\'{e}, H. Garcke, K.F. Lam. A Hele-Shaw-Cahn-Hilliard model for incompressible two-phase flows with different densities. J. Math. Fluid Mech., 20 (2018), 531--567.
[4] F. Della Porta, A. Giorgini, M. Grasselli. The nonlocal Cahn-Hilliard-Hele-Shaw system with logarithmic potential. Nonlinearity, 31 (2018), 4851--4881.
[5] S. Frigeri, K. F. Lam, E. Rocca, G. Schimperna. On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials. Commun. Math. Sci., 16 (2018), 821--856.
[6] H. Garcke, K. F. Lam. Global weak solutions and asymptotic limits of a Cahn-Hilliard-Darcy system modelling tumour growth. AIMS Math., 1 (2016), 318--360.
[7] A. Giorgini. Well-posedness for a diffuse interface model for two-phase Hele-Shaw flows. J. Math. Fluid Mech., 22, 5 (2020).
[8] A. Giorgini, M. Grasselli, H. Wu. The Cahn-Hilliard-Hele-Shaw system with singular potential. Ann. Inst. H. Poincarè Anal. Non Linèaire, 35 (2018), 1079--1118.
[9] H.-G. Lee, J. S. Lowengrub, J. Goodman. Modeling pinch-off and reconnection in a Hele-Shaw cell. I. The models and their calibration. Phys. Fluids, 14 (2002), 492--512.

Abstract: We describe a general phase-field model for hyperelastic multiphase materials. We can consider shape memory alloys as an example of such materials. The model features an elastic energy functional that depends on the phase-field variable and a surface energy term that depends in turn on the elastic deformation, as it measures inter- faces in the deformed configuration. We prove existence of energy minimizing equilibrium states and $\Gamma$-convergence of diffuse-interface approximations to the sharp-interface limit.
It is a joint work with D. Grandi (Ferrara), E. Mainini (Genoa), and U. Stefanelli (Vienna).

Abstract: We study the homogenisation problem for elliptic high-contrast operators $A_{\varepsilon}$ whose coefficients degenerate as $\varepsilon \to 0$ on a set of randomly distributed inclusions. We discuss the limit operator (in the sense of the resolvent convergence) and the convergence of spectrum. On the bounded domain the limiting spectrum is equal to the spectrum of the limit operator, while in the whole space setting the spectrum of the limit operator is the subset of the limiting spectrum. Additionally we characterize the limiting spectrum in the case of finite correlation.
This is a joint work with Mikhail Cherdantsev (University of Cardiff) and Kirill Cherednichenko (University of Bath).
[1] M. Cherdantsev, K. Cherednichenko, I. Velčíć. Stochastic homogenisation of high-contrast media. Applicable Analysis, 98 (2019), 91--117.
[2] M. Cherdantsev, K. Cherednichenko, I. Velčíć. High-contrast random composites: homogenisation framework and new spectral phenomena. Preprint

Abstract: In this talk we consider the dynamics of binary media that can be modeled by the Cahn-Hilliard equation. We investigate the situation that the phase separation of the binary media depends on temperature. Firstly, we derive models by combining ideas from non-equilibrium thermodynamics with an energetic variational approach. Secondly, we discuss selection criteria for these models and provide some analytical results regarding well-posedness of one of the models.
This is joint work with Francesco De Anna, Chun Liu and Jan-Eric Sulzbach.

Inspired by a recent result of Lauteri and Luckhaus, with derive, via Gamma convergence, a surface tension model for polycrystals in dimension two. The starting point is a semi-discrete model accounting for the possibility of having crystal defects. The presence of defects is modelled by incompatible strain field with quantised curl. In the limit as the lattice spacing tends to zero we obtain an energy for grain boundaries that depends on the relative angle of the orientations of the two neighbouring grains. The energy density is defined through an asymptotic cell problem formula. By means of the bounds obtained by Lauteri and Luckhaus we also show that the energy density exhibits a logarithmic behaviour for small angle grain boundaries in agreement with the classical Shockley Read formula.
The talk is based on a paper in preparation in collaboration with Emanuele Spadaro.

Abstract: We introduce a notion of uniform convergence for local and nonlocal curvatures and we propose an abstract method to prove the convergence of the corresponding geometric flows, within the level set formulation. We apply such a general theory to characterize the limits of several nonlocal geometric evolutions. We study the limit of the $s$-fractional mean curvature flows as $s\to 0^+$ and $s\to 1^-$. Moreover, in analogy with $s$-fractional mean curvature flows, we introduce the notion of s-Riesz curvature flows and characterize its limit as $s\to 0^-$ . Furthermore, using a suitable core-radius regularization, we define s-fractional perimeters and s-fractional curvatures also for $s\ge 1$ and we show that - as the core-radius tends to $0$ - the corresponding geometric flows converge to the classical mean curvature flow. Finally, we discuss also the stability of the s-fractional heat flows as $s\to 0^+$ and $s\to 1^-$ .
The results discussed here are obtained in collaboration with A. Cesaroni (Padova), V. Crismale (Sapienza, University of Rome), L. De Luca (CNR, Rome), A. Kubin (TU Munich), A. Ninno (Sapienza, University of Rome), M. Novaga (Pisa).
[1] A. Cesaroni, L. De Luca, M. Novaga, M. Ponsiglione: Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows. Comm. Partial Differ. Equ. 46 (2021).
[2] L. De Luca, A. Kubin, M. Ponsiglione: The core-radius approach to supercritical fractional perimeters, curvatures and geometric flows. Preprint (2021).
[3] L. De Luca, M. Novaga, M. Ponsiglione: The 0-fractional perimeter between fractional perimeters and Riesz potentials. Ann. SNS Pisa Cl. Sci. XXII (2021).
[4] V. Crismale, L. De Luca, A. Kubin, A. Ninno, M. Ponsiglione: The variational approach to $s$-fractional heat flows and the limit cases $s\to 0^+$ and $s\to 1^-$. Preprint (2021).

Abstract: In this joint work with Alessandro Giacomini (Brescia) and Oscar Lopez-Pamiés (Urbana Champaign) we propose a simple model of healing for brittle fracture as a first step towards cavitation. The idea is that healing will also be brittle and that there is a competition between add-cracks and subtract-cracks. We present an existence theorem in a 2d topological setting, the only available framework as of yet.

Abstract: The numerical approximation of the Mumford-Shah functional, $u \mapsto MS(u)$, is a difficult question related to the fact that the jump set $J_u$ of the unknown $u$ is a free discontinuity. A possible approach is the use of a phase field method which consists in replacing $J_u$ by a smooth phase field variable $v$ equal to $1$ everywhere in the domain, but on a small $\varepsilon$-neighborhood of $J_u$, where it vanishes. The variational approximation of the Mumford-Shah functional by such $\varepsilon$-regularized functionals has been obtained by Ambrosio & Tortorelli in [1]. As a byproduct, the convergence of global minimizers becomes an immediate consequence of the fundamental theorem on $\Gamma$-convergence. Unfortunately, the Ambrosio-Tortorelli functional $(u,v) \mapsto AT_\varepsilon(u,v)$ fails to be convex with respect to the pair $(u,v)$ (while it is separately convex) and it leads to serious issues regarding the numerical approximation of its global minimizers (which however do exist). A possible remedy consists in performing an alternate minimization algorithm [3]. Limit points of such an algorithm only turn out to be critical points of the Ambrosio-Tortorelli functional, which have no reasons to be global minimizers. The question of convergence of critical points becomes therefore natural. In the one-dimensional case, this problem has been addressed in [4,5]. We extend here these results to any arbitrary dimension by proving that any critical points $(u_\varepsilon,v_\varepsilon)$ of $AT_\varepsilon$ converge to a critical point $u$ of $MS$ (in the sense of inner variations) upon the additional assumption of the convergence of the energies, i.e. $AT_\varepsilon(u_\varepsilon,v_\varepsilon) \to MS(u)$. As a byproduct, we also obtain the convergence of the second inner variation, which implies that stable critical points of $AT_\varepsilon$ converge to stable critical points of $MS$. The proof rests on elliptic PDEs and geometric measure theoretic arguments. We first establish a regularity theory in H\"older spaces for the system of elliptic partial differential equations satisfied by critical points of the Ambrosio-Tortorelli functional. It allows us to derive the first inner variations of $AT_\varepsilon$ which shows a varifold structure. Then, we characterize the limit varifold as the rectifiable varifold associated to $J_u$.
This is a joint work with Vincent Millot (Université Paris-Est) and Rémy Rodiac (Université Paris Saclay).
[1] L. Ambrosio, V. M. Tortorelli, On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7) 6 (1992) 105--123.
[2] J.-F. Babadjian, V. Millot, R. Rodiac: On the convergence of critical points of the Ambrosio-Tortorelli functional, In preparation.
[3] B. Bourdin, G. A. Francfort, J.-J. Marigo, The variational approach to fracture, J. Elasticity 9 (2008) 5--148.
[4] G. A. Francfort, N. Q. Le, S. Serfaty, Critical Points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case, ESAIM Control Optim. Calc. Var., 15 (2009), 576--598.
[5] N. Q. Le, Convergence results for critical points of the one-dimensional Ambrosio--Tortorelli functional with fidelity term, Adv. Differential Equations 15 (2010) 255--282.

A variational model in the context of the gradient theory for fluid-fluid phase transitions with small scale heterogeneities is studied. In the case where the scale of the small homogeneities is of the same order of the scale governing the phase transition, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy. Bounds on the homogenized surface tension are established. In addition, a characterization of the large-scale limiting behavior of viscosity solutions to non-degenerate and periodic Eikonal equations in half-spaces is given.
This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands), Adrian Hagerty, Cristina Popovici, Rustum Choksi (McGill), Jessica Lin (McGill), and Raghavendra Venkatraman (CMU).