 2015Stefanie Heyden, Sergio Conti and Michael Ortiz
A nonlocal model of fracture by crazing in polymers
Mech. Materials, 90: 131139 2015
http://dx.doi.org/10.1016/j.mechmat.2015.02.006
Abstract: We derive and numerically verify scaling laws for the macroscopic fracture energy of poly mers undergoing crazing from a micromechanical model of damage. The model posits a local energy density that generalizes the classical network theory of polymers so as to account for chain failure and a nonlocal regularization based on straingradient elasticity. We specifically consider periodic deformations of a slab subject to prescribed opening dis placements on its surfaces. Based on the growth properties of the energy densities, scaling relations for the local and nonlocal energies and for the specific fracture energy are derived. We present finiteelement calculations that bear out the heuristic scaling relations. 
 
 Aicke Hinrichs, Lev Markhasin, Jens Oettershagen and Tino Ullrich
Optimal quasiMonte Carlo rules on higher order digital nets for the numerical integration of multivariate periodic functions
2015
http://arxiv.org/pdf/1501.01800v1

 
 Martin Huesmann
Transport estimates for random measures in dimension one
ArXiv eprint 2015
http://arxiv.org/abs/1510.03601

 
 Juhi Jang, Juan J. L. Velázquez and Hyung Ju Hwang
On the structure of the singular set for the kinetic FokkerPlanck equations in domains with boundaries
2015
http://arxiv.org/abs/1509.03366

 
 Christian Ketterer
Evolution variational inequality and Wasserstein control in variable curvature context
ArXiv eprints 2015
http://arxiv.org/abs/1509.02178

 
 Christian Ketterer
On the geometry of metric measure spaces with variable curvature bounds
ArXiv eprints 2015
http://arxiv.org/abs/1506.03279

 
 Christian Ketterer
Cones over metric measure spaces and the maximal diameter theorem
J. Math. Pures Appl. (9), 103(5): 12281275 2015
http://dx.doi.org/10.1016/j.matpur.2014.10.011

 
 Christian Ketterer
Obata's Rigidity Theorem for Metric Measure Spaces
Anal. Geom. Metr. Spaces, 3(Art. 16): 278295 2015
http://dx.doi.org/10.1515/agms20150016

 
 Arthur H. M. Kierkels and Juan J. L. Velázquez
On the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear Schrödinger equation
J. Stat. Phys., 159(3): 668712 2015
http://dx.doi.org/10.1007/s1095501511940
Abstract: We study the mathematical properties of a kinetic equation which describes the long time behaviour of solutions to the weak turbulence equation associated to the cubic nonlinear Schrödinger equation.In particular, we give a precise definition of weak solutions and prove global existence of solutions for all initial data with finite mass. We also prove that any nontrivial initial datum yields the instantaneous onset of a condensate, i.e.~a Dirac mass at the origin for any positive time. Furthermore we show that the only stationary solutions with finite total measure are Dirac masses at the origin. We finally construct solutions with finite energy, which is transferred to infinity in a selfsimilar manner. 
 
 Herbert Koch and Nikolai Nadirashvili
Partial analyticity and nodal sets for nonlinear elliptic systems
2015
http://arxiv.org/abs/1506.06224

 
 Herbert Koch and Stefan Steinerberger
Convolution Estimates for Singular Measures and Some Global Nonlinear BrascampLieb Inequalities
Proceedings of the Royal Society of Edinburgh, Section: A Mathematics, 145(6): 12231237 2015
http://arxiv.org/abs/1404.4536
Abstract: We give an L2 x L2 → L2 convolution estimate for singular measures supported on transversal hypersurfaces in ℝn, which improves earlier results of Bejenaru et al. as well as Bejenaru and Herr. The quantities arising are relevant to the study of the validity of bilinear estimates for dispersive partial differential equations. We also prove a class of global, nonlinear Brascamp–Lieb inequalities with explicit constants in the same spirit. 
 
 Herbert Koch, Angkana Rüland and Wenhui Shi
The Variable Coefficient Thin Obstacle Problem: Optimal Regularity and Regularity of the Regular Free Boundary
2015
http://arXiv.org/abs/1504.03525

 
 Herbert Koch
Selfsimilar solutions to supercritical gKdV
Nonlinearity, 28(3): 545575 2015
http://dx.doi.org/10.1088/09517715/28/3/545

 
 Jan Maas and Daniel Matthes
Longtime behavior of a finite volume discretization for a fourth order diffusion equation
ArXiv eprints 2015
http://arxiv.org/abs/1505.03178

 
 Barbara Niethammer, Sebastian Throm and Juan J. L. Velázquez
A revised proof of uniqueness of selfsimilar profiles to Smoluchowski's coagulation equation for kernels close to constant
2015
http://arxiv.org/abs/1510.03361

 
 Barbara Niethammer, Juan J. L. Velázquez and Michael Helmers
Mathematical analysis of a coarsening model with local interactions
2015
http://arxiv.org/abs/1509.04917

 
 Barbara Niethammer, Sebastian Throm and Juan J. L. Velázquez
Selfsimilar solutions with fat tails for Smoluchowski's coagulation equation with singular kernels
Ann. I. H. Poincaré  AN 2015
http://dx.doi.org/10.1016/j.anihpc.2015.04.002

 
 Celia Reina, Anja Schlömerkemper and Sergio Conti
Derivation of F=FeFp as the continuum limit of crystalline slip
preprint 2015
http://arxiv.org/abs/1504.06775
Abstract: In this paper we provide a proof of the multiplicative kinematic description of crystal elastoplasticity in the setting of large deformations, i.e.~$\F=\Fe\Fp$ for a two dimensional single crystal. The proof starts by considering a general configuration at the mesoscopic scale, where the dislocations are discrete line defects (points in the twodimensional description used here) and the displacement field can be considered continuous everywhere in the domain except at the slip surfaces, over which there is a displacement jump. At such scale, as previously shown by two of the authors, there exists unique physicallybased definitions of the total deformation tensor $\F$ and the elastic and plastic tensors $\Fe$ and $\Fp$ that do not require the consideration of any nonrealizable intermediate configuration and do not assume any a priori relation between them of the form $\F=\Fe\Fp$. This mesoscopic description is then passed to the continuum limit via homogenization i.e., by increasing the number of slip surfaces to infinity and reducing the lattice parameter to zero. We show for twodimensional deformations of initially perfect single crystals that the classical continuum formulation is recovered in the limit with $\F=\Fe\Fp$, $\det \Fp= 1$ and $\mathbfG=\textCurl\ \Fp$ the dislocation density tensor. 
 
 Angkana Rüland
Unique continuation for fractional Schrödinger equations with rough potentials
Comm. Partial Differential Equations, 40(1): 77114 2015
http://dx.doi.org/10.1080/03605302.2014.905594

 
 Martin Rumpf and Benedikt Wirth
Variational time discretization of geodesic calculus
IMA J. Numer. Anal., 35(3): 10111046 2015
http://dx.doi.org/10.1093/imanum/dru027
Abstract: We analyze a variational time discretization of geodesic calculus on finite and certain classes of infinitedimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete logarithm, discrete exponential maps, and discrete parallel transport, and we prove convergence to their continuous counterparts. The presented analysis is based on the direct methods in the calculus of variation, on convergence, and on weighted finite element error estimation. The convergence results of the discrete geodesic calculus are experimentally confirmed for a basic model on a twodimensional Riemannian manifold. This provides a theoretical basis for the application to shape spaces in computer vision, for which we present one specific example. 
 