# Modeling of Crystalline Interfaces and Thin Film Structures: A Joint Mathematics-Physics Symposium

## Schedule

 Tuesday, November 12th, 2019 09:00 - 09:35 Guus Rijnders (University of Twente) Novel Functionalities in Atomically Controlled Oxide Heterostructures by Pulsed Laser Deposition Abstract: In recent years, it has been shown that novel functionalities can be achieved in oxide heterostructures in which the interfaces are atomically controlled, in terms of atomic stacking as well as in terms of the local symmetry. I will highlight the recent developments in atomic controlled growth of epitaxial oxides by pulsed laser deposition, with a focus on heterostructures showing manipulated magnetic and electronic properties. Emergent phenomena in oxide heterostructures such as interface charge transfer [1], two dimensional electron gas and ferromagnetism between two non-magnetic materials, are induced by the dedicated coupling between spin, orbital, charge and lattice degrees of freedom. Developing strategies to engineer these intimate couplings in oxide heterostructures is crucial to achieve new phenomena and to pave the path towards novel functionalities with atomic scale dimensions. Strong oxygen octahedral coupling has recently been demonstrated, which transfers the octahedral rotation from one oxide into the other at the interface region. As a result, we possess control of the lateral magnetic and electronic anisotropies by atomic scale design of the oxygen octahedral rotation [2]. References [1] . Geessinck, G. Araizi-Kanoutas, N. Gauquelin, M.S. Golden, G. Koster, G. Rijnders. Charge transfer at the LaCoO3-LaTiO3 interface. To be published . [2] Z. Liao, G. Rijnders et al, Nat. Mater. 15 (2016), 425. 09:35 - 10:10 Marco Cicalese (TU Muenchen) Does the $N$-clock model approximate the $XY$-model? Abstract: The $N$-clock model is a two-dimensional ferromagnetic spin model on the square lattice in which the spin field is constrained to take values in a set of $N$ equi-spaced points of the unit circle. It is usually considered as an approximation of the $XY$ model, for which instead the spin field is allowed to attain all the values of the unit circle. In the theory of superconductivity the latter models phase transitions mediated by the formation and the interaction of co-dimension 2 topological defects as in the well-known Ginzburg-Landau functional. A breakthrough result by Fr\"ohlich and Spencer (CMP 1981) shows that the same kind of phase transitions appear in the $N$-clock model for $N$ large enough. By a variational analysis we find the explicit rate of divergence of $N$ (with respect to the number of interacting lattice points) for which the $N$-clock model asymptotically behaves like the $XY$ model at zero temperature. We moreover exhaustively discuss all the other regimes of $N$ and we show how Cartesian Currents can detect the energy concentration on sets of co-dimenion smaller or equal than 2. The results presented are contained in a recent paper in collaboration with G. Orlando (TUM) and M. Ruf (EPFL). 10:10 - 10:40 Coffee/Tea break 10:40 - 11:15 Antonin Chambolle (Ecole Polytechnique, CNRS) Crystalline evolution of mean-convex sets Abstract: In a recent paper [4], T. Laux and G. De Philippis have studied the mean curvature flow of (strictly) mean convex sets and characterized its major properties. In this talk we essentially address the same questions however in the nonsmooth case. We show that given an appropriate definition of strict "mean convexity" relative to an arbitrary surface tension (including crystalline), we can recover the same results, using mosty variational proofs based on [1,5] and the recent theory for crystalline flows [2,3]. In particular, this flow is (as expected) unique and decreasing. This is a joint work with Matteo Novaga (U. Pisa). References [1] F. Almgren, J.E. Taylor, and L. Wang, Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31 (1993), 387--438. [2] A. Chambolle, M. Morini, M. Novaga, and M. Ponsiglione, Existence and uniqueness for anisotropic and crystalline mean curvature flows. J. Amer. Math. Soc., to appear. [3] A. Chambolle, M. Morini, and M. Ponsiglione, Existence and uniqueness for a crystalline mean curvature flow. Comm. Pure Appl. Math. 70 (2017), 1023--1220. [4] G. De Philippis, T. Laux, Implicit time discretization for the mean curvature flow of outward minimizing sets. Ann. Sc. Norm. Super. Pisa Cl. Sci., to appear. [5] S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation. Calc. Var. Partial Differential Equations 3 (1995), 253--271. 11:15 - 11:50 Barbara Zwicknagl (TU Berlin) Low-volume fraction martensitic microstructures close to interfaces Abstract: In this talk, I shall discuss recent analytical results on variational models for martensitic microstructures. We consider (singularly perturbed) multiwell elastic energy functionals and the associated nonconvex vectorial minimization problems. We shall discuss in particular needle-like microstructures and geometrically linearized models in the limit of low volume fraction. This talk is based on joint works with S. Conti, J. Diermeier, M. Lenz, N. Lüthen, D. Melching, and M. Rumpf. 11:50 - 13:55 Lunch break 13:55 - 14:30 Cyrill Muratov (New Jersey Institute of Technology) Chiral domain walls and domain wall tilt in ferromagnetic nanostrips Abstract: Recent advances in nanofabrication make it possible to produce multilayer nanostructures composed of ultrathin film materials with thickness down to a few monolayers of atoms and lateral extent of several tens of nanometers. At these scales, ferromagnetic materials begin to exhibit unusual properties, such as perpendicular magnetocrystalline anisotropy and antisymmetric exchange, also referred to as Dzyaloshinskii-Moriya interaction (DMI), due of the increased importance of interfacial effects. The presence of surface DMI has been demonstrated to fundamentally alter the structure of domain walls. Here we use the micromagnetic modeling framework to analyse the existence and structure of chiral domain walls, viewed as minimizers of a suitable micromagnetic energy functional. We explicitly construct the minimizers in the one-dimensional setting, both for the interior and edge walls, for a broad range of parameters. Using varitional methods we analyze the asymptotics of the two-dimensional magnetization patterns in samples of large spatial extent in the presence of weak applied magnetic fields and present an analytical theory of domain wall tilt. We show that under an applied field the domain wall remains straight, but tilts at an angle to the direction of the magnetic field that is proportional to the field strength for moderate fields and sufficiently strong DMI. This is joint work with V.V. Slastikov, A.G. Kolesnikov, and O.A. Tretiakov. Support by NSF via grants DMS-1313687 and DMS-1614948 is gratefully acknowledged. References [1] C.B. Muratov and V.V. Slastikov. Domain structure of ultrathin ferromagnetic elements in the presence of Dzyaloshinskii-Moriya interaction. Proc. R. Soc. Lond. Ser. A 473 (2016), 20160666. [2] C.B. Muratov, V.V. Slastikov, A.G. Kolesnikov, and O.A. Tretiakov. Theory of Dzyaloshinskii domain wall tilt in ferromagnetic nanostrips. Phys. Rev. B 96 (2017), 134417. 14:30 - 15:05 Jian-Guo Liu (Duke University) Dynamics of a degenerate PDE model of epitaxial crystal growth Abstract: Epitaxial growth is an important physical process for forming solid films or other nano-structures. It occurs as atoms, deposited from above, adsorb and diffuse on a crystal surface. Modeling the rates that atoms hop and break bonds leads in the continuum limit to degenerate 4th-order PDE that involve exponential nonlinearity and the $p$-Laplacian with $p=1$, for example. We discuss a number of analytical results for such models, some of which involve subgradient dynamics for Radon measure solutions and a new notion of weak solutions. 15:05 - 15:35 Coffee/Tea break 15:35 - 16:10 Aldo Pratelli (University of Pisa) On the optimization of Riesz-like potentials Abstract: In this talk, we will discuss the extrema of functionals of the type $$P(E)+\iint_{E\times E} g(|y−x|)dxdy,$$ where $E\subseteq R^N$ is a set of finite perimeter with an assigned volume, $P(E)$ is its perimeter, and $g:R^+\to R^+$ is a given concave function. This problem is currently deeply studied by several people, with a particular emphasis on the case when $g(t)=t^{\alpha − N}$ for some $0 < \alpha < N$ .We will describe the general question and some of the main known facts, and we will discuss in particular the case of the small volume, and the stability of the extrema (this last, central issue is studied in several papers, see for instance [1-3]). References [1] A. Figalli, N. Fusco, F. Maggi, V. Millot, M. Morini, Isoperimetry and stability properties of balls with respect to nonlocal energies. Comm. Math. Phys. 336 (2015), 441--507. [2] R. Frank, E. Lieb, Proof of spherical flocking based on quantitative rearrangement inequalities. Preprint, 2019 [3] N. Fusco, A. Pratelli, Sharp stability for the Riesz potential. Preprint, 2019 . 16:10 - 16:45 Michael Goldman (University of Paris 7) Connectedness of drops in convex potentials Abstract: An old conjecture of Almgren states that for every convex and coercive potential $g: \mathbb{R}^d\to \mathbb{R}$, every convex and one-homogeneous anisotropy $\Phi : \mathbb{R}^d\to \mathbb{R}^+$ and every volume $V>0$, the minimizers of $\min_{|E|=V} \int_{\partial E} \Phi(\nu) d\mathcal{H}^{d-1} + \int_{E} g dx$ are convex. I will review the known results on this problem and present recent progress obtained with G. De Philippis on the connectedness of the minimizers for smooth potentials and anisotropies. Our proof is based on the introduction of a new two-point function'' which measures the lack of convexity and which gives rise to a negative second variation of the energy.
 Wednesday, November 13th, 2019 09:00 - 09:35 Jim Evans (Iowa State University and Ames Laboratory - USDOE) Assembly and stability of nanoclusters during thin film deposition Abstract: We consider epitaxial nanoclusters (NCs) formed by deposition on crystalline surfaces where NCs can be either two- or three-dimensional (depending on the relative strength of adhesion to surface energy). NC assembly is described by appropriate theories for nucleation and growth during deposition. Extensive theoretical efforts attempted to develop a beyond-mean-field for homogeneous nucleation of homoepitaxial 2D NCs to predict NC size distributions, and the stochastic geometry of the spatial NC distribution [1]. Recent efforts have also provided a precise description with ab-initio kinetics of far-from-equilibrium growth shapes [2]. For heteroepitaxial 3D NCs, nucleation is often heterogeneous, and modeling of growth shapes is more challenging as it must account for interlayer transport. In this presentation, we focus on post-deposition evolution which can involve either Ostwald Ripening - OR (dissolution of smaller NCs and growth of larger NCs), or Smoluchowski Ripening - SR (NC diffusion and coalescence) [3]. OR for 2D NCs can provide perfect 2D realizations of classic LSW theory in some systems. However, ?anomalous? behavior has been observed in others [4], and the presence of even trace additives can accelerate OR due to mass transport by complex formation (where analysis involves appropriate reaction-diffusion equations) [5]. For SR of 2D NCs, the size-dependence of NC diffusion, and the dynamics of coalescence or sintering are of central interest. For 3D NCs, a long-standing debate about whether OR or SR dominates is being answered by in-situ experimental imaging. For both 2D and 3D epitaxial NCs, we present recent modeling related to SR revealing a complex oscillatory size dependence of NC diffusivity, and also analyzing evolution during NC coalescence. References [1] J.W. Evans, P.A. Thiel, and M.C. Bartelt, Morphological evolution during epitaxial thin film growth: Formation of 2D islands and 3D mounds. Surface Science Reports 61 (2006) 1--128; Y. Han et al. Scaling of capture zone area distributions? J. Chem. Phys. 145 (2016), 211911. [2] Y. Han, D.-J. Liu and J.W. Evans, Real-time ab-initio KMC simulation of self-assembly and sintering of bimetallic nanoclusters: Au+Ag on Ag(100). Nano Letters 14 (2014), 4646. [3] K.C. Lai, et al., Reshaping, intermixing and coarsening for metallic nanocrystals: Non-equilibrium stat mech and coarse-grained modeling. Chemical Reviews 119 (2019), 6670--6768. [4] Y. Han, et al., Anisotropic coarsening: 1D decay of Ag on Ag(110). PRB 87 (2013), 155420. [5] H. Walen, et al., Mass transport enhancement via Cu2S3 on Cu(111). PRB 91 (2015), 045426. 09:35 - 10:10 Mariapia Palombaro (University of L'Aquila) Derivation of linearised polycrystals from a 2D system of edge dislocations Abstract: Many solids in nature exhibit a polycrystalline structure. A single phase polycrystal is formed by many individual crystal grains, having the same underlying periodic atomic structure, but rotated with respect to each other. The region separating two grains with different orientation is called grain boundary. Since the grains are mutually rotated, the periodic crystalline structure is disrupted at grain boundaries. As a consequence, grain boundaries are regions where dislocations occur, inducing high energy concentration. We will discuss a variational model that describes the emergence of polycrystalline structures as a result of elastic energy minimisation. The setting is that of linearised planar elasticity. Starting from the variational semi-discrete model for edge dislocations introduced in [1] within the so-called core radius approach, we derive by $\Gamma$-convergence as the lattice spacing tends to zero, a limit energy given by the sum of a plastic term, acting on the dislocation density, and an elastic term, which depends on the symmetric strains. Minimisers under suitable boundary conditions are piece-wise constant antisymmetric strain fields, representing in our model a polycrystal whose grains are mutually rotated by infinitesimal angles. In this respect our result can be regarded as a linearised version of the Read-Shockley formula [2]. This is joint work with S. Fanzon and M. Ponsiglione [3]. References [1] W. T. Read, W. Shockley. Dislocation models of crystal grain boundaries. Phys. Rev. 98 (1950), 275--289. [2] A. Garroni, G. Leoni, M. Ponsiglione. Gradient theory for plasticity via homogenization of discrete dislocations. J. Eur. Math. Soc. (JEMS) 12 (2010), 1231--1266. [3] J. S. Fanzon, M. Palombaro, M. Ponsiglione. Derivation of linearised polycrystals from a two-dimensional system of edge dislocations. SIAM Journal on Mathematical Analalysis, to appear. 10:10 - 10:40 Coffee/Tea break 10:40 - 11:15 Andrea Braides (University of Rome Tor Vergata) Homogenization of oscillating networks Abstract: We consider the asymptotic behaviour of integral energies with convex integrands defined on one-dimensional networks contained in a region of the three-dimensional space with a fast-oscillating boundary as the period of the oscillation tends to zero, keeping the oscillation themselves of fixed size. The limit energy, obtained as a $\Gamma$-limit with respect to an appropriate convergence, is defined in a `stratified' Sobolev space and is written as an integral functional depending on all, two or just one derivative, depending on the connectedness properties of the sublevels of the function describing the profile of the oscillations. In the three cases, the energy function is characterized through an usual homogenization formula for $p$-connected networks, a homogenization formula for thin-film networks and a homogenization formula for thin-rod networks, respectively. This is work in collaboration with Valeria Chiado Piat [2], and is related to previous work on fast oscillating boundaries [1] and on the homogenization of singular structures (e.g. in [3-5]). References [1] Ansini and A. Braides, Homogenization of oscillating boundaries and applications to thin films. J. Anal. Math. 83 (2000), 151--181. [2] A. Braides, V. Chiado Piat, Homogenization of networks in domains with oscillating boundaries. Appl. Anal. 98 (2019), 45--63. [3] A. Braides, V. Chiado Piat, Non convex homogenization problems for singular structures. Netw. Heterog. Media 3 (2008), 489--508. [4] V. V. Zhikov, Homogenization of elasticity problems on singular structures. Dokl. Ross. Akad. Nauk. 380 (2001), 741--745; translation in Dokl. Math. 64 (2001). [5] V. V. Zhikov and S. E. Pastukhova, Homogenization on periodic lattices. Dokl. Ross. Akad. Nauk. 391 (2003), 443--447; translation in Dokl. Math. 68 (2003). 11:15 - 11:50 Igor Velcic (University of Zagreb) Microscopical justification of the winterbottom shape Abstract: In this talk we will discuss the microscopical derivation of a continuum model for the Winterbottom problem, i.e., the problem of determining the equilibrium shapes for droplets attached to a wall. Our strategy consists in showing that properly defined atomistic energies of crystalline configurations $\Gamma$-converge, as the number of atoms grows, converge to a surface energy which is minimized by the Winterbottom shape. The work generalizes the result [1] which deals with the equilibrium shape of particles in a free crystalline configuration (without a substrate) where it was that the limit minimizing configuration is the Wulff shape. This problem finds applications in the framework of growth of thin films over substrates. This is a joint work with Paolo Piovano (University of Vienna). References [1] Y. A. Yeung, G. Friesecke and B. Schmidt. Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape. Calc. Var. Partial Differential Equations, 44 (2012), 81--110. [2] P. Piovano and I.Velcic. Microscopical justification of the Winterbottom shape. Preprint. 11:50 - 12:20 Group picture 12:20 - 19:00 Free afternoon. A visit to the lab of the Surface Physics group of the Institute of Applied Physics at TU Vienna is planned for interested workshop attendees from around 14:30 to 16:30 19:00 - 23:00 Social dinner at Heuriger Feuerwehr Wagner
 Thursday, November 14th, 2019 09:45 - 10:20 Christian Teichert (Montanuniversitaet Leoben) Interfaces between crystalline organic semiconductor nanostructures and 2D materials Abstract: Crystalline films of small conjugated molecules offer attractive potential for fabricating organic solar cells, organic light emitting diodes (LEDs), and organic field effect transistors (OFETs) on flexible substrates. Here, the novel two-dimensional (2D) van der Waals materials like conducting graphene (Gr), insulating ultra-thin hexagonal boron nitride (hBN) or semiconducting transition metal dichalcogenides come into play. Gr for instance offers potential application as a transparent conductive electrode in organic solar cells and LEDs replacing indium tin oxide, whereas hBN can be used as ultra-thin flexible dielectric in OFETs. Since small conjugated molecules like the rod-like molecules para-hexaphenyl (6P) or pentacene fit well to the honeycomb structure of 2D materials, their growth can be expected in a lying configuration. This has indeed been observed for growth of 6P on Pt(111) supported Gr by low-energy electron microscopy (LEEM) and by micro-beam low-energy electron diffraction ($\mu$-LEED) revealing the epitaxial relation between substrate and the molecular overlayer [1]. A similar interface can be assumed for the self-assembly of crystalline 6P needles on exfoliated, wrinkle-free Gr, where atomic-force microscopy (AFM) reveals several 10nm wide, a few nm high, and tens of $\mu$m long needles along discrete substrate directions [2]. For 6P on ultrathin hBN, such needles grow almost along the armchair direction of the substrates which could be supported by density functional theory (DFT) calculations of the energetically favorable molecular adsorption site [3]. AFM based manipulation in conjunction with molecular dynamics simulations revealed friction anisotropy and preferential sliding directions between the 6P nanocrystals and both 2D substrates [4]. For needle-like nanocrystals of the polar molecule dihydrotetraazaheptacene (DHTA7) on hBN, electrostatic force microscopy (EFM) revealed light-induced charge spreading depending on the polarization direction of light [5]. Here, DFT was not only employed to reveal the molecular adsorption site but also the most probable crystal structure as well as the optical properties of the molecules. Research has been performed together with G. Hlawacek, M. Kratzer, A. Matkovic, J. Genser, K.-P. Gradwohl, A. Cicek, B. Kaufmann, J. Liu, S. Klima (Leoben), R. van Gastl, F. Khokhar, H. Zandvliet, B. Poelsema (Univ. of Twente), B. Kollmann, D. Lueftner, P. Puschnig (University of Graz), Z. Shen, O. Siri, C. Becker (CINAM-CNRS, Aix Marseille University), B. Vasic, I. Stankovic, R. Gajicc (University of Belgrade) and was supported by the Austrian Science Fund (FWF) under project I 1788-N20. References [1] G. Hlawacek, et al., Nano Lett. 11 (2011), 333. [2] M. Kratzer, C. Teichert, Nanotechnology 27 (2016), 292001. [3] A. Matkovic, et al., Sci. Rep. 6 (2016), 38519. [4] B. Vasic, et al., Nanoscale 10 (2018), 18835. [5] A. Matkovic, et al., under review 10:20 - 10:40 Coffee/Tea break 10:40 - 11:15 Giovanni Bellettini (University of Siena) Some results on the relaxation of the area functional for graphs in dimension two and codimension two Abstract: Let us consider the functional % $$A(u, \Omega) = \int_\Omega \sqrt{1+\vert \nabla u_1\vert^2+ \vert \nabla u_2\vert^2 + \left( \frac{\partial u_1}{\partial x} \frac{\partial u_2}{\partial y} - \frac{\partial u_1}{\partial y} \frac{\partial u_2}{\partial x} \right)^2} ~dxdy,$$ % defined on smooth maps $u = (u_1,u_2) : \Omega \subset {\mathbf R}^2 \to \mathbf R^2$; $A(u,\Omega)$ gives the value of the two-dimensional area in $\mathbf R^4$, of the graph of $u$. We shall discuss some aspects of the relaxation of $A(\cdot,\Omega)$, in particular its values on nonsmooth maps, for instance on piecewise constant maps [1,2]. We shall show that the evaluation of the relaxation is related to the solution of certain Plateau-type problems, with various sort of boundary conditions. References [1] G. Bellettini, A. Elshorbagy, M. Paolini, R. Scala, On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions. Ann. Mat. Pura Appl. (2019), 1--33. [2] R. Scala, Optimal estimates for the triple junction function and other surprising aspects of the area functional. Ann. Sc. Norm. Super. Pisa Cl. Sci. 2 (2020), to appear. 11:15 - 11:50 Giuliano Lazzaroni (University of Firenze) Discrete energies with surface scaling: interactions beyond nearest neighbours versus non-interpenetration Abstract: We present some discrete models for crystals with surface scaling of the interaction energy. We assume that at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond nearest neighbours have the role of penalising changes of orientation and, to some extent, they may replace the positive-determinant constraint that is usually required when only nearest neighbours are accounted for. From joint works in collaboration with R. Alicandro and M. Palombaro [1, 2]. References [1] R. Alicandro, G. Lazzaroni, M. Palombaro, On the effect of interactions beyond nearest neighbours on non-convex lattice systems. Calc. Var. Partial Differential Equations 56 (2017). [2] R. Alicandro, G. Lazzaroni, M. Palombaro, Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours. Netw. Heterog. Media 13 (2018), 1--26. 11:50 - 13:55 Lunch break 13:55 - 14:30 Tevfik Onur Mentes (Elettra - Sincrotrone Trieste) Ultra-thin Co films: structure and magnetism Abstract: Ultra-thin magnetic films have been of interest as their low-dimensionality and the presence of interfaces sensitively modify the structural and magnetic properties [1]. From a technological view, such films present possibilities for applications in terms of recording density, fast domain wall motion and the presence of exotic magnetic domains. In this respect, among elemental metals cobalt is arguably the most studied one in ultra-thin film and multilayer configurations due to the magnetic anisotropy perpendicular to the film plane and the observation of chiral domain structures such as \emph{skyrmions} [2]. In the first part of the talk, the structural aspects of cobalt films grown on heavy metal substrates such as W(110) and Re(0001) will be treated. Along with the substrate interface and the hcp-fcc Co phase transition, we will also consider the subsequent growth of a graphene overlayer on the cobalt film [3]. In the second part, we will focus on the magnetic properties of cobalt films with perpendicular magnetization. In particular, we will show how x-ray speckles can be applied to studies of periodic magnetic domain patterns observed in such films. X-ray speckles, which had been developed for imaging aperiodic objects based on ideas from x-ray crystallography [4], provide interesting opportunities when combined with other x-ray imaging methods. References [1] C. A. F. Vaz, J. A. C. Bland, G. Lauhoff, Magnetism in ultrathin film structures. Rep. Prog. Phys. 71 (2008), 056501. [2] O. Boulle, et al., Room-temperature chiral magnetic skyrmions in ultrathin magnetic nanostructures. Nature Nanotech. 11 (2016), 449--454. [3] M. Jugovac, et al., Role of carbon dissolution and recondensation in graphene epitaxial alignment on cobalt. Carbon 152 (2019), 489--496. [4] D. Sayre, Some implications of a theorem due to Shannon. Acta Cryst. 5 (1952), 843. 14:30 - 15:05 Matteo Focardi (University of Firenze) How a minimal surface leaves a thin obstacle Abstract: In this talk I will present recent results on the optimal regularity of the solution to the thin obstacle problem for nonparametric minimal surfaces with zero obstacle. A detailed analysis of the global structure of the related free boundary, in particular its local finiteness in measure and its rectifiability, will be also considered. This is joint work with Emanuele Spadaro (U. Roma La Sapienza). References [1] M. Focardi, E. Spadaro, How a minimal surface leaves a thin obstacle. arXiv:1804.02890 15:05 - 15:35 Coffee/Tea break 15:35 - 16:10 Matteo Novaga (University of Pisa) The 0-fractional perimeter Abstract: I will present a unified point of view on fractional perimeters and Riesz potentials. De- noting by $H^s$ - for $s \in(0, 1)$ - the $s$-fractional perimeter and by $J^s$ - for $s \in(−d, 0)$ - the $s$-Riesz energies acting on characteristic functions, I will show that the functionals $H^s$ and $J^s$, up to a suitable additive renormalization diverging when $s\to 0,$ belong to a continuous one-parameter family of functionals, which for $s = 0$ gives back a new object we refer to as $0$-fractional perimeter. All the convergence results with respect to the parameter $s$ and to the renormalization procedures are obtained in the framework of $\Gamma$-convergence. As a byproduct of our analysis, we obtain the isoperimetric inequality for the $0$-fractional perimeter. These results are in collaboration with L. De Luca and M. Ponsiglione. 16:10 - 16:45 Marco Morandotti (Polytechnic of Torino) Analysis of a perturbed Cahn-Hilliard model for Langmuir-Blodgett films Abstract: A one-dimensional evolution equation including a transport term is considered; it models a process of thin films deposition. Existence and uniqueness of solutions, together with continuous dependence on the initial data and an energy equality are proved by combining a minimizing movement scheme with a fixed-point argument. Finally, it is shown that, when the contribution of the transport term is small, the equation possesses a global attractor and converges to a purely diffusive Cahn-Hilliard equation. This is joint work with Marco Bonacini and Elisa Davoli.
 Friday, November 15th, 2019 09:00 - 09:35 Michele Riva (TU Wien) Growth of $In_2O_3(111)$ Films with Optimized Surfaces 09:35 - 10:10 Roberto Alicandro (University of Cassino) Derivation of linear elasticity from atomistic energies with multiple wells Abstract: A rigorous derivation of linear elastic theories from non linear elasticity has been provided in terms of $\Gamma$-convergence for both continuum and atomistic models mainly in the case of single well potentials (see for example \cite{ref2,ref3,ref4}). On the other hand, energies with multiple wells naturally arise in many models, as for example in the gradient theory of solid-solid phase transitions. In the recent paper \cite{ref1}. it has been shown that linear elasticity can be rigorously derived from multi-well energies by adding a singular higher order term which penalizes the transitions between the wells and turns out to be necessary in order to recover good compactness properties of minimizing sequences of displacement fields. In this talk I will present a recent result in collaboration with G. Lazzaroni and M. Palombaro for the derivation of linear elasticity from a general class of atomistic energies with multiple wells for crystalline materials, showing that the role of the singular term in the continuum model in penalizing jumps from one well to another is played in this setting by interactions beyond nearest neighbours. References [1] R. Alicandro, G. Dal Maso, G. Lazzaroni, M.Palombaro, Derivation of a linearised elasticity model from singularly perturbed multiwell energy functionals. Arch. Rational Mech. Anal. 230 (2018), 1--45. [2] A. Braides, M. Solci, E. Vitali, A derivation of linear elastic energies from pair-interaction atomistic systems. Networks and Heterogeneous Media 2 (2007), 551--567. [3] G. Dal Maso, M. Negri, D. Percivale, Linearized elasticity as ?-limit of finite elasticity. Set-Valued Analysis 10 (2002), 165--183. [4] B. Schmidt, On the derivation of linear elasticity from atomistic models. Networks and Heterogeneous Media 4 (2009), 789--812. 10:10 - 10:40 Coffee/Tea break 10:40 - 11:15 Patrick Dondl (University of Freiburg) Pinning of interfaces by localized dry friction Abstract: We consider a differential inclusion to model the propagation of an interface, e.g., a phase boundary, in an environment with obstacles. The interaction of the interface with the obstacles is governed by a localized dry friction. The model implies that energy has to be expended to pass across an obstacle. Hence, the interface becomes arrested until enough curvature is accumulated such that it is energetically more favorable to pass across the obstacle. The treatment of our model in the context of pinning and depinning requires a comparison principle. We prove this property and hence the existence of viscosity solutions. Moreover, under reasonable assumptions, they are equivalent to weak solutions. Our main results asserts that for obstacles distributed according to a Poisson point process, interfaces become pinned, leading to the emergence of a rate-independent hysteresis. This is joint work with Luca Courte and Ulisse Stefanelli. 11:15 - 11:50 Gianni Dal Maso (SISSA, Trieste) On the jerky crack growth in elastoplastic materials Abstract: The purpose of the talk is to show that in elastoplastic materials cracks can grow only in an intermittent way. This result is rigorously proved in the framework of a simplified model.