# Lehrstuhl I Analysis

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Item Non-local approximation of free-discontinuity problems in linear elasticity and application to stochastic homogenisation(2023-06-08) Marziani, Roberta; Solombrino, FrancescoShow more We analyse the Γ-convergence of general non-local convolution type functionals with varying densities depending on the space variable and on the symmetrized gradient. The limit is a local free-discontinuity functional, where the bulk term can be completely characterized in terms of an asymptotic cell formula. From that, we can deduce an homogenisation result in the stochastic setting.Show more Item A data-driven framework for evolutionary problems in solid mechanics(2022-11-17) Poelstra, Klaas; Bartel, Thorsten; Schweizer, BenShow more Data-driven schemes introduced a new perspective in elasticity: While certain physical principles are regarded as invariable, material models for the relation between strain and stress are replaced by data clouds of admissible pairs of these variables. A data-driven approach is of particular interest for plasticity problems, since the material modeling is even more unclear in this field. Unfortunately, so far, data-driven approaches to evolutionary problems are much less understood. We try to contribute in this area and propose an evolutionary data-driven scheme. We present a first analysis of the scheme regarding existence and data convergence. Encouraging numerical tests are also included.Show more Item Homogenization and low Mach number limit of compressible Navier-Stokes equations in critically perforated domains(2022-07-02) Bella, Peter; Oschmann, FlorianShow more In this note, we consider the homogenization of the compressible Navier-Stokes equations in a periodically perforated domain in R3. Assuming that the particle size scales like ε3, where ε>0 is their mutual distance, and that the Mach number decreases fast enough, we show that in the limit ε→0, the velocity and density converge to a solution of the incompressible Navier-Stokes equations with Brinkman term. We strongly follow the methods of Höfer, Kowalczyk and Schwarzacher [https://doi.org/10.1142/S0218202521500391], where they proved convergence to Darcy’s law for the particle size scaling like εα with α∈(1,3).Show more Item Homogenization of the full compressible Navier-Stokes-Fourier system in randomly perforated domains(2022-03-26) Oschmann, FlorianShow more We consider the homogenization of the compressible Navier-Stokes-Fourier equations in a randomly perforated domain in R3. Assuming that the particle size scales like ε^α, where ε>0 is their mutual distance and α>3, we show that in the limit ε→0, the velocity, density, and temperature converge to a solution of the same system. We follow the methods of Lu and Pokorný [https://doi.org/10.1016/j.jde.2020.10.032] and Pokorný and Skříšovský [https://doi.org/10.1007/s41808-021-00124-x] where they considered the full system in periodically perforated domains.Show more Item Inhomogeneous Helmholtz equations in wave guides – existence and uniqueness results with energy methods(2022-03-30) Schweizer, BenShow more The Helmholtz equation −∇⋅(a∇u)−ω2u=f is considered in an unbounded wave guide Ω:=R×S⊂Rd , S⊂Rd−1 a bounded domain. The coefficient a is strictly elliptic and either periodic in the unbounded direction x1∈R or periodic outside a compact subset; in the latter case, two different periodic media can be used in the two unbounded directions. For non-singular frequencies ω , we show the existence of a solution u. While previous proofs of such results were based on analyticity arguments within operator theory, here, only energy methods are used.Show more Item Homogenization of compressible fluids in perforated domains(2022) Oschmann, Florian; Bella, Peter; Feireisl, EduardShow more The present thesis deals with the homogenization of Navier-Stokes (NSE) and Navier-Stokes-Fourier (NSF) equations in perforated domains, describing the motion of a compressible and heat conducting fluid. We start the thesis with the description of the flow of compressible fluids governed by the Navier-Stokes equations, which we derive from several physical principles. In the second chapter, we show how to construct a right inverse to the divergence operator in different domains, which will be crucial in order to get the homogenization results mentioned below. The third chapter is devoted to the homogenization of different types of equations in different perforated domains. We consider two different types of perforations: well-separated and randomly distributed. The homogenization procedure deals with the behavior of NSE and NSF as the perforation becomes denser and takes place first in randomly perforated domains for the case of tiny holes. We start with NSE and assume a certain growth rate for the fluid's pressure, which we later relax in the direction of physical relevance. Heat-conducting fluids are considered in the following section. In all the aforementioned, the limiting equations are the same as in the perforated domain. The last section, however, deals with the case of critically sized holes in periodically perforated domains. We will show that, under an additional scaling assumption on the pressure, the limiting equations have an additional friction term occurring in the momentum balance, thus providing a first step towards the homogenization of compressible NSE in the critical regime. Finally, in the last chapter, we give an outlook on possible future work and open problems.Show more Item Uniform convergence to equilibrium for a family of drift-diffusion models with trap-assisted recombination and self-consistent potential(2021-07-06) Fellner, Klemens; Kniely, MichaelShow more We investigate a recombination–drift–diffusion model coupled to Poisson's equation modelling the transport of charge within certain types of semiconductors. In more detail, we study a two-level system for electrons and holes endowed with an intermediate energy level for electrons occupying trapped states. As our main result, we establish an explicit functional inequality between relative entropy and entropy production, which leads to exponential convergence to equilibrium. We stress that our approach is applied uniformly in the lifetime of electrons on the trap level assuming that this lifetime is sufficiently small.Show more Item Dimension reduction for elastoplastic rods and homogenization of elastoplastic lattices(2021) Poelstra, Klaas Hendrik; Schweizer, Ben; Mielke, AlexanderShow more We derive effective equations for periodic lattices of linearly elastoplastic rods in the limit of both infinitesimal periodicity and infinitesimal relative width of the rods. For this derivation we use the method of evolutionary Γ-convergence for quadratic rate-independent systems. As a first step towards this goal we derive effective equations for a single rod. After introducing appropriate scalings, the main difficulty lies in the proof of Γ-convergence for the stored energy. For the study of periodic lattices we then introduce the notion of periodic graph frameworks, discuss infinitesimal rigidity properties of such frameworks and define a notion of two-scale convergence. The stored energy of a lattice of rods ist just the sum of the energies of the individual rods, coupled by boundary conditions at the nodes. For this energy we again prove Γ-convergence. In the presence of volume loads we observe qualitatively different behaviour depending on the relative rate of convergence of the periodicity parameter and the thickness parameter.Show more Item Higher integrability for variational integrals with non-standard growth(2021-04-09) Schäffner, MathiasShow more We consider autonomous integral functionals of the form F[u]:=∫Ωf(Du)dx where u:Ω→RN,N≥1, where the convex integrand f satisfies controlled (p, q)-growth conditions. We establish higher gradient integrability and partial regularity for minimizers of F assuming qp<1+2n−1, n≥3. This improves earlier results valid under the more restrictive assumption qp<1+2n.Show more Item Derivation of a homogenized bending–torsion theory for rods with micro-heterogeneous prestrain(2020-05-29) Bauer, Robert; Neukamm, Stefan; Schäffner, MathiasShow more In this paper we investigate rods made of nonlinearly elastic, composite–materials that feature a micro-heterogeneous prestrain that oscillates (locally periodic) on a scale that is small compared to the length of the rod. As a main result we derive a homogenized bending–torsion theory for rods as Γ-limit from 3D nonlinear elasticity by simultaneous homogenization and dimension reduction under the assumption that the prestrain is of the order of the diameter of the rod. The limit model features a spontaneous curvature–torsion tensor that captures the macroscopic effect of the micro-heterogeneous prestrain. We devise a formula that allows to compute the spontaneous curvature–torsion tensor by means of a weighted average of the given prestrain. The weight in the average depends on the geometry of the composite and invokes correctors that are defined with help of boundary value problems for the system of linear elasticity. The definition of the correctors depends on a relative scaling parameter γ, which monitors the ratio between the diameter of the rod and the period of the composite’s microstructure. We observe an interesting size-effect: For the same prestrain a transition from flat minimizers to curved minimizers occurs by just changing the value of γ. Moreover, in the paper we analytically investigate the microstructure-properties relation in the case of isotropic, layered composites, and consider applications to nematic liquid–crystal–elastomer rods and shape programming.Show more Item On time-harmonic Maxwell’s equations in periodic media(2019) Urban, Maik; Schweizer, Ben; Bonnet-Ben Dhia, Anne-SophieShow more In this thesis we study the propagation of time-harmonic electromagnetic waves through periodic media for two different regimes. In the first part, we consider a periodic medium in a bounded domain with a period that is much smaller than the wavelength of the electromagnetic wave hitting the medium. The medium is a periodic assembly of conducting microstructures and void space. In order to describe the effective behaviour of the field propagating through this medium, we homogenise the time-harmonic Maxwell equations. There is a vast literature on homogenising Maxwell's equations under rather restrictive assumptions on the (conducting) microstructure. Using a new averaging method---the so-called geometric average---allows us to consider a large class of microstructures that have not been treated before in the literature. We derive the effective Maxwell equations for two cases: perfectly conducting microstructures and highly conductive microstructures. The second part of this thesis is concerned with the propagation of electromagnetic waves in a closed, unbounded and periodic waveguide. In this part we assume that the wavelength of the fields and the period of the medium are of the same order. Imposing suitable assumptions on the geometry and the fields, Maxwell's equations reduce to a scalar Helmholtz equation. We truncate the waveguide to obtain a bounded domain and replace the radiation condition at infinity appropriately. In order to establish the existence of a solution to the Helmholtz equation in this bounded periodic waveguide, we derive a limiting absorption principle for sesquilinear forms. Using this principle, we show that up to an at most countable set of singular frequencies there exists a unique solution to the Helmholtz equation in the bounded waveguide that satisfies the replacement of the radiation condition at infinity.Show more