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The homogenization of elliptic divergence-type fourth-order operators with periodic coefficients is studied in a (periodic) domain. The aim is to find an operator with constant coefficients and represent the equation through a perturbation around this operator. The resolvent is found as $L^2 \to L^2$ operator using the Neumann series for the periodic fundamental solution of biharmonic operator. Results are based on some auxiliary Lemmas suggested by Bensoussan in 1986, Zhikov in 1991, Yu. Grabovsky and G. Milton in 1998, Pastukhova in 2016. Operators of the type considered in the paper appear in the study of the elastic properties of thin plates. The choice of the operator with constant coefficients is discussed separately and chosen in an optimal way w.r.t. the spectral radius and convergence of the Neumann series and uses the known bounds for ''homogenized'' coefficients. Similar ideas are usually applied for the construction of preconditioners for iterative solvers for finite dimensional problems resulting from discretized PDEs. The method presented is similar to Cholesky factorization transferred to elliptic operators (as in references mentioned above). Furthermore, the method can be applied to non-linear problems., Comment: it was publishe in conference proceedings in 2019, see the reference below

Three-dimensional metallic microstructures find applications as stents in medicine, as ultrabroadband antennas in communications, in micromechanical parts or as structures of more fundamental interest in photonics like metamaterials. Direct metal printing of such structures using three-dimensional laser lithography is a promising approach, which is not extensively applied yet, as fabrication speed, surface quality, and stability of the resulting structures are limited so far. In order to identify the limiting factors, we investigate the influence of light-particle interactions and varying scan speed on heat generation and particle deposition in direct laser writing of silver. We introduce a theoretical model which captures diffusion of particles and heat as well as the fluid dynamics of the photo-resist. Chemical reactions are excluded from the model but particle production is calibrated using experimental data. We find that optical forces generally surmount those due to convection of the photo-resist. Simulations predict overheating of the photo-resist at laser powers similar to those found in experiments. The thermal sensitivity of the system is essentially determined by the largest particles present in the laser focus. Our results suggest that to improve particle deposition and to achieve higher writing speeds in metal direct laser writing, strong optical trapping of the emerging particles is desirable. Furthermore, precise control of the particle size reduces the risk of spontaneous overheating., Comment: Corrected formatting of metadata