\epsilon-regularity for systems involving non-local, antisymmetric operators

We prove an epsilon-regularity theorem for critical and super-critical systems with a non-local antisymmetric operator on the right-hand side. These systems contain as special cases, Euler-Lagrange equations of conformally invariant variational functionals as Rivi\`ere treated them, and also Euler-Lagrange equations of fractional harmonic maps introduced by Da Lio-Rivi\`ere. In particular, the arguments presented here give new and uniform proofs of the regularity results by Rivi\`ere, Rivi\`ere-Struwe, Da-Lio-Rivi\`ere, and also the integrability results by Sharp-Topping and Sharp, not discriminating between the classical local, and the non-local situations

Fractional De Giorgi classes and applications to nonlocal regularity theory

We present some recent results obtained by the author on the regularity of solutions to nonlocal variational problems. In particular, we review the notion of fractional De Giorgi class, explain its role in nonlocal regularity theory, and propose some open questions in the subject.Comment: Short note based on a talk given by the author at a conference held in Bari on May 29-30, 2017, as part of the INdAM intensive period "Contemporary research in elliptic PDEs and related topics

A regularity theory for intrinsic minimising fractional harmonic maps

We define and develop an interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps are critical points of an energy whose first variation is a Dirichlet to Neumann map for the harmonic map problem on a half-space with a Riemannian metric which can degenerate/become singular along the boundary, depending on the fractional power. Similarly to the approach used to prove regularity for stationary intrinsic semi-harmonic maps, we take advantage of the connection between fractional harmonic maps and free boundary problems for harmonic maps in order to develop a partial regularity theory for the fractional harmonic maps we consider. In particular, we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces with the aforementioned metrics up to the boundary; fractional harmonic maps then inherit this regularity. As a by-product of our methods we shed some new light on the monotonicity of the average energy of solutions of the degenerate linear elliptic equation related to fractional harmonic functions

n/p-harmonic maps: regularity for the sphere case

We introduce $n$/$p$-harmonic maps as critical points of the energy \mathcal{E}_{n,p}(v) = \intl_{\R^n} \abs{\laps{\alpha} v}^{p} where pointwise $v: D \subset \R^n \to \mathcal{S}^{N-1}$, for the $N$-sphere $\S^{N-1} \subset \R^N$ and $\alpha = \frac{n}{p}$. This energy combines the non-local behaviour of the fractional harmonic maps introduced by Rivi\`{e}re and first author with the degenerate arguments of the $n$-laplacian. In this setting, we will prove H\"older continuity

Embedding of Alumina Reinforcing Elements in the Composite Extrusion Process

Abstract. Extruded aluminum profiles are essential for lightweight constructions in contemporary transport and automotive applications. The reinforcement of such aluminum-based profiles with high-strength materials offers a high potential for weight reduction and an improvement of functional and mechanical properties. In comparison to conventional composite extrusion using fiber or particle reinforced billets, the alternatively developed process for the embedding of endless reinforcing elements provides enormous advantages regarding extrusion forces, load-adapted reinforcement, and tool abrasion. In this extrusion process with conventional billets, modified tools with portholes are used to position reinforcing elements from outside the pressing tool and to embed them into the material flow during the pressing operation. This composite extrusion process is part of the research work started in 2003 and carried out within the scope of the Collaborative Research Center SFB/TR10. To increase the potential of composite extrusion with endless reinforcing elements, the manufacture of composite extrusion profiles with high-strength non-metallic alumina wires is planned. Due to the wires' specific properties, e.g. high stiffness, their deflection behavior must be analyzed to guarantee a stable feeding-in process. In this paper the specific behavior of alumina reinforcing elements regarding the feeding-in process is analyzed by experimental investigations. The main influencing factors are determined and a process window is deduced

Microstructure analysis of aluminum extrusion: grain size distribution in AA6060, AA6082 and AA7075 alloys

Microstructure and material flow of aluminum alloys have a significant influence on the mechanical properties and surface quality. In extrusion of aluminum billets at high temperatures the microstructure is dependent on the alloy and the forming and temperature history. A prediction of grain size and precipitation is of increasing importance in order to design the process by adjustment of parameters such as punch speed, temperatures, and quenching. To give references for microstructure prediction based on material flow, and with it strain and strain rate history, this paper deals with the microstructure during the extrusion process of AA6060, AA6082, and AA7075 alloys. Billets have been partly extruded to axisymmetric round profiles and the microstructure of the press rests consisting of the billet rests in container and die has been considered. Furthermore, these rests have been analyzed to show the material flow, dynamic and static recrystallization based on macro etchings and visible microstructure under different conditions, e.g. as in the area of high strain rate near the container wall, or in dead zones [1]. To allow an accurate simulation of the extrusion process, punch force and temperature conditions during the tests have been measured and are presented in this paper, too. Keyw

Extrusion Benchmark 2007-Effect of pocket shape in the extrusion of aluminum profiles

The 2007 workshop and benchmark has been a great success in combining industrial needs and academic knowledge in the area of extrusion technology with a particular attention to FEM computation. The number of international participants and presenters exceeded by far the expectations of the organizers. The current state of the numerical simulation was exploited by the extrusion benchmark: the quality of simulation results generally increased a lot with respect to the previous benchmark edition, some of them evidencing a good correspondence to experimental results. However, the use of simulation still has to be treated with care in order to obtain reliable answers for industrial needs. In order to keep monitoring the simulation quality and improving the knowledge transfer between industry and universities, the next Ex-trusion Workshop and Benchmark on Finite Element Simulation of Extrusion Processes is currently planned to be held in 2009 at the Institute of Forming Technology and Lightweight Construction (IUL), Technical University of Dortmund in Germany

Grain size prediction in AA6060 profile extrusion

In the first part of the work an experimental procedure to investigate the evolution of recrystallization in aluminum alloys is presented and discussed. Several cups, obtained by means of inverse extrusion, were produced at different temperatures and process speeds. The specimens were analyzed in order to examine the grain size distribution. The coefficients for dynamic recrystallization models were obtained by regression analysis after thermo-mechanical FEM simulations of the experiments realized with the code Deform 3D. The specimens were then heated in a furnace and cooled in order to reproduce static recrystallization of the material. The grain distribution was examined and the coefficients for the equation for static recrystallization prediction were regressed, too. In the second part of the work the extrusion of a round-shaped profile is described and the grain size distribution on the profile and on the billet rest are analyzed. The developed equations were applied to the rod extrusion and the simulated grain size distributions were compared with the experimental ones

Extrusion Workshop and Benchmark on Finite Element Simulation of Extrusion Processes, Bologna, 2007

The demand for properties and quality in extruded profiles stretches the ability of extrud-ers and die makers to the limit: ever more complex sections, hard alloys, emerging tech-nologies, and microstructure control are just some of manufacturers’ every-day concerns. Key factors for innovation and competitiveness are skilled engineering analysis and reli-able software; however, no reference community exists for the extrusion analyst and there is no common base for evaluating commercial code capabilities. With the organization of the extrusion workshop and benchmark in September 2007 in Bologna extruders as well as users and developers of finite element codes have been asked to come together as a reference community in order to fill this lack of knowledge and to create an open discus-sion on this topic of technological frontier