@article{6762,
abstract = {We present and study novel optimal control problems motivated by the search for photovoltaic materials with high power-conversion efficiency. The material must perform the first step: convert light (photons) into electronic excitations. We formulate various desirable properties of the excitations as mathematical control goals at the Kohn-Sham-DFT level
of theory, with the control being given by the nuclear charge distribution. We prove that nuclear distributions exist which give rise to optimal HOMO-LUMO excitations, and present illustrative numerical simulations for 1D finite nanocrystals. We observe pronounced goal-dependent features such as large electron-hole separation, and a hierarchy of length scales: internal HOMO and LUMO wavelengths < atomic spacings < (irregular) fluctuations of the doping profiles < system size.},
author = {Friesecke, Gero and Kniely, Michael},
issn = {15403467},
journal = {Multiscale Modeling and Simulation},
number = {3},
pages = {926--947},
publisher = {SIAM},
title = {{New optimal control problems in density functional theory motivated by photovoltaics}},
doi = {10.1137/18M1207272},
volume = {17},
year = {2019},
}
@article{151,
abstract = {We construct planar bi-Sobolev mappings whose local volume distortion is bounded from below by a given function f∈Lp with p>1. More precisely, for any 1<q<(p+1)/2 we construct W1,q-bi-Sobolev maps with identity boundary conditions; for f∈L∞, we provide bi-Lipschitz maps. The basic building block of our construction are bi-Lipschitz maps which stretch a given compact subset of the unit square by a given factor while preserving the boundary. The construction of these stretching maps relies on a slight strengthening of the celebrated covering result of Alberti, Csörnyei, and Preiss for measurable planar sets in the case of compact sets. We apply our result to a model functional in nonlinear elasticity, the integrand of which features fast blowup as the Jacobian determinant of the deformation becomes small. For such functionals, the derivation of the equilibrium equations for minimizers requires an additional regularization of test functions, which our maps provide.},
author = {Fischer, Julian L and Kneuss, Olivier},
journal = {Journal of Differential Equations},
number = {1},
pages = {257 -- 311},
publisher = {Academic Press},
title = {{Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity}},
doi = {10.1016/j.jde.2018.07.045},
volume = {266},
year = {2019},
}
@article{6617,
abstract = {The effective large-scale properties of materials with random heterogeneities on a small scale are typically determined by the method of representative volumes: a sample of the random material is chosen—the representative volume—and its effective properties are computed by the cell formula. Intuitively, for a fixed sample size it should be possible to increase the accuracy of the method by choosing a material sample which captures the statistical properties of the material particularly well; for example, for a composite material consisting of two constituents, one would select a representative volume in which the volume fraction of the constituents matches closely with their volume fraction in the overall material. Inspired by similar attempts in materials science, Le Bris, Legoll and Minvielle have designed a selection approach for representative volumes which performs remarkably well in numerical examples of linear materials with moderate contrast. In the present work, we provide a rigorous analysis of this selection approach for representative volumes in the context of stochastic homogenization of linear elliptic equations. In particular, we prove that the method essentially never performs worse than a random selection of the material sample and may perform much better if the selection criterion for the material samples is chosen suitably.},
author = {Fischer, Julian L},
journal = {Archive for Rational Mechanics and Analysis},
publisher = {Springer},
title = {{The choice of representative volumes in the approximation of effective properties of random materials}},
doi = {10.1007/s00205-019-01400-w},
year = {2019},
}
@article{404,
abstract = {We construct martingale solutions to stochastic thin-film equations by introducing a (spatial) semidiscretization and establishing convergence. The discrete scheme allows for variants of the energy and entropy estimates in the continuous setting as long as the discrete energy does not exceed certain threshold values depending on the spatial grid size $h$. Using a stopping time argument to prolongate high-energy paths constant in time, arbitrary moments of coupled energy/entropy functionals can be controlled. Having established Hölder regularity of approximate solutions, the convergence proof is then based on compactness arguments---in particular on Jakubowski's generalization of Skorokhod's theorem---weak convergence methods, and recent tools on martingale convergence.
},
author = {Fischer, Julian L and Grün, Günther},
journal = {SIAM Journal on Mathematical Analysis},
number = {1},
pages = {411 -- 455},
publisher = {Society for Industrial and Applied Mathematics },
title = {{Existence of positive solutions to stochastic thin-film equations}},
doi = {10.1137/16M1098796},
volume = {50},
year = {2018},
}
@article{606,
abstract = {We establish the existence of a global solution for a new family of fluid-like equations, which are obtained in certain regimes in as the mean-field evolution of the supercurrent density in a (2D section of a) type-II superconductor with pinning and with imposed electric current. We also consider general vortex-sheet initial data, and investigate the uniqueness and regularity properties of the solution. For some choice of parameters, the equation under investigation coincides with the so-called lake equation from 2D shallow water fluid dynamics, and our analysis then leads to a new existence result for rough initial data.},
author = {Duerinckx, Mitia and Fischer, Julian L},
journal = {Annales de l'Institut Henri Poincare (C) Non Linear Analysis},
number = {5},
pages = {1267--1319},
publisher = {Elsevier},
title = {{Well-posedness for mean-field evolutions arising in superconductivity}},
doi = {10.1016/j.anihpc.2017.11.004},
volume = {35},
year = {2018},
}
@article{712,
abstract = {We establish a weak–strong uniqueness principle for solutions to entropy-dissipating reaction–diffusion equations: As long as a strong solution to the reaction–diffusion equation exists, any weak solution and even any renormalized solution must coincide with this strong solution. Our assumptions on the reaction rates are just the entropy condition and local Lipschitz continuity; in particular, we do not impose any growth restrictions on the reaction rates. Therefore, our result applies to any single reversible reaction with mass-action kinetics as well as to systems of reversible reactions with mass-action kinetics satisfying the detailed balance condition. Renormalized solutions are known to exist globally in time for reaction–diffusion equations with entropy-dissipating reaction rates; in contrast, the global-in-time existence of weak solutions is in general still an open problem–even for smooth data–, thereby motivating the study of renormalized solutions. The key ingredient of our result is a careful adjustment of the usual relative entropy functional, whose evolution cannot be controlled properly for weak solutions or renormalized solutions.},
author = {Fischer, Julian L},
issn = {0362546X},
journal = {Nonlinear Analysis: Theory, Methods and Applications},
pages = {181 -- 207},
publisher = {Elsevier},
title = {{Weak–strong uniqueness of solutions to entropy dissipating reaction–diffusion equations}},
doi = {10.1016/j.na.2017.03.001},
volume = {159},
year = {2017},
}