Abstract: Michael Ortiz and Gustavo Gioia showed in the 90s that the complex patterns arising in compressed elastic films can be analyzed within the context of the calculus of variations. Their initial work focused on films partially debonded from the substrate, subject to isotropic compression arising from the difference in thermal expansion coefficients between film and substrate. In the following two decades different geometries have been studied, as for example anisotropic compression. We review recent mathematical progress in this area, focusing on the rich phase diagram of partially debonded films with a lateral boundary condition.

Abstract: We prove the convergence of the extremal processes for variable speed branching Brownian motions where the ”speed functions”, that describe the timeinhomogeneous variance, lie strictly below their concave hull and satisfy a certain weak regularity condition. These limiting objects are universal in the sense that they only depend on the slope of the speed function at 0 and the final time t. The proof is based on previous results for two-speed BBM obtained in Bovier and Hartung (2014) and uses Gaussian comparison arguments to extend these to the general case.

Abstract: In this paper we will prove the existence of weak solutions to the Korteweg-de Vries initial value problem on the real line with H^-1 initial data; moreover, we will study the problem of orbital and asymptotic H^s stability of solitons for integers s≥ -1; finally, we will also prove new a priori H^-1 bounds for solutions to the Korteweg-de Vries equation. The paper will utilise the Miura transformation to link the Korteweg-de Vries equation to the modified Korteweg-de Vries equation.

Abstract: The speed of growth for a particular stochastic growth model introduced by Borodin and Ferrari in [Comm. Math. Phys. 325 (2014), 603-684], which belongs to the KPZ anisotropic universality class, was computed using multi-time correlations. The model was recently generalized by Toninelli in [arXiv:1503.05339] and for this generalization the stationary measure is known but the time correlations are unknown. In this note, we obtain algebraic and combinatorial proofs for the expression of the speed of growth from the prescribed dynamics.

Abstract: Motivated by simulations of carbon nanocones (see Jordan and Crespi, Phys. Rev. Lett., 2004), we consider a variational plate model for an elastic cone under compression in the direction of the cone symmetry axis. Assuming radial symmetry, and modeling the compression by suitable Dirichlet boundary conditions at the center and the boundary of the sheet, we identify the energy scaling law in the von-Kármán plate model. Specifically, we find that three different regimes arise with increasing indentation $δ$: initially the energetic cost of the logarithmic singularity dominates, then there is a linear response corresponding to a moderate deformation close to the boundary of the cone, and for larger $δ$ a localized inversion takes place in the central region. Then we show that for large enough indentations minimizers of the elastic energy cannot be radially symmetric. We do so by an explicit construction that achieves lower elastic energy than the minimum amount possible for radially symmetric deformations.

Abstract: We prove that the classical line-tension approximation for dislocations in crystals, i.e., the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length, follows as the $\Gamma$-limit of regularized linear-elasticity as the lattice parameter becomes increasingly small or, equivalently, as the dislocation measure becomes increasingly dilute. We consider two regularizations of the theory of linear-elastic dislocations: a core-cutoff and a mollification of the dislocation measure. We show that both regularizations give the same energy in the limit, namely, an energy defined on matrix-valued divergence-free measures concentrated on lines. The corresponding self-energy per unit length $\psi(b,t)$, which depends on the local Burgers vector and orientation of the dislocation, does not, however, necessarily coincide with the self-energy per unit length $\psi_0(b,t)$ obtained from the classical theory of the prelogarithmic factor of linear-elastic straight dislocations. Indeed, microstructure can occur at small scales resulting in a further relaxation the classical energy down to its $\calH^1$-elliptic envelope.

Abstract: Functions of bounded deformation (BD) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation (BV), but are less well understood. We discuss here the relation to BV under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that BD functions which are piecewise affine on a Caccioppoli partition are in GSBV, and we prove that $SBD^p$ functions are approximately continuous $H^n-1$-a.e. away from the jump set. On the negative side, we construct a function which is $BD$ but not in BV and has distributional strain consisting only of a jump part, and one which has a distributional strain consisting of only a Cantor part.

Abstract: We consider vectorial variational problems in nonlinear elasticity of the form I[u]=∫W(Du)dx, where W is continuous on matrices with a positive determinant and diverges to infinity along sequences of matrices whose determinant is positive and tends to zero. We show that, under suitable growth assumptions, the functional ∫Wqc(Du)dx is an upper bound on the relaxation of I, and coincides with the relaxation if the quasiconvex envelope W qc of W is polyconvex and has p-growth from below with p≧n. This includes several physically relevant examples. We also show how a constraint of incompressibility can be incorporated in our results.