 2015Sergio Conti, Adriana Garroni and Stefan Müller
Dislocation microstructures and straingradient plasticity with one active slip plane
Preprint arXiv 1512.03076 2015
http://arxiv.org/abs/1512.03076

 
 Sergio Conti and Barbara Zwicknagl
Low volumefraction microstructures in martensites and crystal plasticity
preprint 2015
http://arxiv.org/abs/1507.04521
Abstract: We study microstructure formation in two nonconvex singularlyperturbed variational problems from materials science, one modeling austenitemartensite interfaces in shapememory alloys, the other one slip structures in the plastic deformation of crystals. For both functionals we determine the scaling of the optimal energy in terms of the parameters of the problem, leading to a characterization of the mesoscopic phase diagram. Our results identify the presence of a new phase, which is intermediate between the classical laminar microstructures and branching patterns. The new phase, characterized by partial branching, appears for both problems in the limit of small volume fraction, that is, if one of the variants (or of the slip systems) dominates the picture and the volume fraction of the other one is small. 
 
 Sergio Conti, Felix Otto and Sylvia Serfaty
Branched Microstructures in the GinzburgLandau Model of TypeI Superconductors
preprint 2015
http://arxiv.org/abs/1507.00836

 
 Sergio Conti, Janusz Ginster and Martin Rumpf
A $BV$ Functional and its Relaxation for Joint Motion Estimation and Image Sequence Recovery
ESAIM: Mathematical Modelling and Numerical Analysis, 49(5): 14631487 2015
http://dx.doi.org/10.1051/m2an/2015036
Abstract: The estimation of motion in an image sequence is a fundamental task in image processing. Frequently, the image sequence is corrupted by noise and one simultaneously asks for the underlying motion field and a restored sequence. In smoothly shaded regions of the restored image sequence the brightness constancy assumption along motion paths leads to a pointwise differential condition on the motion field. At object boundaries which are edge discontinuities both for the image intensity and for the motion field this condition is no longer well defined. In this paper a totalvariation type functional is discussed for joint image restoration and motion estimation. This functional turns out not to be lower semicontinuous, and in particular finescale oscillations may appear around edges. By the general theory of vector valued $BV$ functionals its relaxation leads to the appearance of a singular part of the energy density, which can be determined by the solution of a local minimization problem at edges. Based on bounds for the singular part of the energy and under appropriate assumptions on the local intensity variation one can exclude the existence of microstructures and obtain a model wellsuited for simultaneous image restoration and motion estimation. Indeed, the relaxed model incorporates a generalized variational formulation of the brightness constancy assumption. The analytical findings are related to ambiguity problems in motion estimation such as the proper distinction between foreground and background motion at object edges. 
 
 Sergio Conti, Adriana Garroni and Michael Ortiz
The linetension approximation as the dilute limit of linearelastic dislocations
Arch. Ration. Mech. Anal., 218(2): 699755 2015
http://dx.doi.org/10.1007/s0020501508697
Abstract: We prove that the classical linetension 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 linearelasticity as the lattice parameter becomes increasingly small or, equivalently, as the dislocation measure becomes increasingly dilute. We consider two regularizations of the theory of linearelastic dislocations: a corecutoff and a mollification of the dislocation measure. We show that both regularizations give the same energy in the limit, namely, an energy defined on matrixvalued divergencefree measures concentrated on lines. The corresponding selfenergy 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 selfenergy per unit length $\psi_0(b,t)$ obtained from the classical theory of the prelogarithmic factor of linearelastic straight dislocations. Indeed, microstructure can occur at small scales resulting in a further relaxation the classical energy down to its $\calH^1$elliptic envelope. 
 
 Sergio Conti and Peter Gladbach
A linetension model of dislocation networks on several slip planes
Mech. Materials, 90: 140147 2015
http://dx.doi.org/10.1016/j.mechmat.2015.01.013
Abstract: Dislocations in crystals can be studied by a PeierlsNabarro type model, which couples linear elasticity with a nonconvex term modeling plastic slip. In the limit of small lattice spacing, and for dislocations restricted to planes, we show that it reduces to a linetension model, with an energy depending on the local orientation and Burgers vector of the dislocation. This model predicts, for specific geometries, spontaneous formation of microstructure, in the sense that straight dislocations are unstable towards a zigzag pattern. Coupling between dislocations in different planes can lead to microstructures over several length scales. 
 
 Sergio Conti, Matteo Focardi and Flaviana Iurlano
Which special functions of bounded deformation have bounded variation
preprint 2015
http://arxiv.org/abs/1502.07464
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 $H^n1$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. 
 
 Sergio Conti, Adriana Garroni and Annalisa Massaccesi
Modeling of dislocations and relaxation of functionals on 1currents with discrete multiplicity
Calc. Var. PDE, 54(2): 18471874 2015
http://dx.doi.org/10.1007/s005260150846x
Abstract: In the modeling of dislocations one is lead naturally to energies concentrated on lines, where the integrand depends on the orientation and on the Burgers vector of the dislocation, which belongs to a discrete lattice. The dislocations may be identified with divergencefree matrixvalued measures supported on curves or with 1currents with multiplicity in a lattice. In this paper we develop the theory of relaxation for these energies and provide one physically motivated example in which the relaxation for some Burgers vectors is nontrivial and can be determined explicitly. From a technical viewpoint the key ingredients are an approximation and a structure theorem for 1currents with multiplicity in a lattice. 
 
 Sergio Conti and Georg Dolzmann
On the theory of relaxation in nonlinear elasticity with constraints on the determinant
Arch. Rat. Mech. Anal., 217(2): 413437 2015
http://dx.doi.org/10.1007/s0020501408359
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 pgrowth 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. 
 
 Carlota M. Cuesta, Maria Calle and Juan J. L. Velázquez
Interfaces determined by capillarity and gravity in a twodimensional porous medium
2015
http://arxiv.org/abs/1505.03676

 
 Roland Donninger
Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation
2015
http://arxiv.org/abs/1509.02041

 
 Roland Donninger and Birgit Schörkhuber
Stable blowup for wave equations in odd space dimensions
2015
http://arxiv.org/abs/1504.00808

 
 Dinh Dũng and Michael Griebel
Hyperbolic cross approximation in infinite dimensions
Journal of Complexity 2015
http://arxiv.org/pdf/1501.01119v1
Abstract: We give tight upper and lower bounds of the cardinality of
the index sets of certain hyperbolic crosses which reflect mixed
Sobolev–Korobovtype smoothness and mixed Sobolevanalytictype
smoothness in the infinitedimensional case where specific
summability properties of the smoothness indices are fulfilled.
These estimates are then applied to the linear approximation of
functions from the associated spaces in terms of the εdimension
of their unit balls. Here, the approximation is based on linear
information. Such function spaces appear for example for the
solution of parametric and stochastic PDEs. The obtained upper
and lower bounds of the approximation error as well as of the
associated εcomplexities are completely independent of any parametric
or stochastic dimension. Moreover, the rates are independent
of the parameters which define the smoothness properties
of the infinitevariate parametric or stochastic part of the solution.
These parameters are only contained in the order constants.
This way, linear approximation theory becomes possible in the
infinitedimensional case and corresponding infinitedimensional
problems get tractable. 
 
 Alexander Effland, Martin Rumpf, Stefan Simon, Kirsten Stahn and Benedikt Wirth
Bézier curves in the space of images
In Proceedings Scale Space and Variational Methods in Computer Vision, Volume 9087 of Lecture Notes in Computer Science
page 372384.
Publisher: Springer International
2015
http://dx.doi.org/10.1007/9783319184616_30
Abstract: Bézier curves are a widespread tool for the design of curves in Euclidian space. This paper generalizes the notion of Bézier curves to the infinitedimensional space of images. To this end the space of images is equipped with a Riemannian metric which measures the cost of image transport and intensity variation in the sense of the metamorphosis model by Miller and Younes. Bézier curves are then computed via the Riemannian version of de Casteljau's algorithm, which is based on a hierarchical scheme of convex combination along geodesic curves. Geodesics are approximated using a variational discretization of the Riemannian path energy. This leads to a generalized de Casteljau method to compute suitable discrete Bézier curves in image space. Selected test cases demonstrate qualitative properties of the approach. Furthermore, a Bézier approach for the modulation of face interpolation and shape animation via image sketches is presented. 
 
 Alberto Enciso, Daniel PeraltaSalas and Stefan Steinerberger
Prescribing the nodal set of the first eigenfunction in each conformal class
2015
http://arxiv.org/abs/1503.05105

 
 Matthias Erbar, Jan Maas and Prasad Tetali
Ricci curvature bounds for BernoulliLaplace and random transposition models
Ann. Fac. Sci. Toulouse Math., ArXiv eprints, 24(4): 781800 2015
http://arxiv.org/abs/1409.8605
Abstract: We calculate a Ricci curvature lower bound for some classical examples of random walks, namely, a chain on a slice of the $n$dimensional discrete cube (the socalled BernoulliLaplace model) and the random transposition shuffle of the symmetric group of permutations on $n$ letters. 
 
 Matthias Erbar, Kazumasa Kuwada and KarlTheodor Sturm
On the equivalence of the entropic curvaturedimension condition and Bochner's inequality on metric measure spaces
Invent. Math., 201(3): 9931071 2015
http://dx.doi.org/10.1007/s0022201405637

 
 Matthias Erbar and Martin Huesmann
Curvature bounds for configuration spaces
Calculus of Variations and Partial Differential Equations, 54(1): 397430 2015
http://dx.doi.org//10.1007/s0052601407901

 
 Max Fathi and Jan Maas
Entropic Ricci curvature bounds for discrete interacting systems
Ann. Appl. Prob. 2015
http://arxiv.org/abs/1501.00562

 
 Patrik L. Ferrari and Peter Nejjar
Anomalous shock fluctuations in TASEP and last passage percolation models
Probab. Theory Related Fields, 161(1): 61109 2015
http://dx.doi.org/10.1007/s0044001305446
Abstract: We consider the totally asymmetric simple exclusion process with initial conditions and/or jump rates such that shocks are generated. If the initial condition is deterministic, then the shock at time \(t\) will have a width of order \(t^{1/3}\). We determine the law of particle positions in the large time limit around the shock in a few models. In particular, we cover the case where at both sides of the shock the process of the particle positions is asymptotically described by the Airy\(_1\) process. The limiting distribution is a product of two distribution functions, which is a consequence of the fact that at the shock two characteristics merge and of the slow decorrelation along the characteristics. We show that the result generalizes to generic last passage percolation models. 
 