Jürgen Sprekels

On the identification of coefficients of elliptic problems by asymptotic regularization

The problem of recovering coefficients of elliptic problems from measured data is considered. An algorithm is developed to identify the unknown coefficients without a minimization technique. The method is based on the construction of certain time-dependent problems which contain the original equation as asymptotic steady state. A Liapunovtype a-priori estimate is fundamental to prove that the solution of the time-dependent regularized equations approach a solution of the original problem as t →∞. A related behavior is proved for the solution of corresponding finite-dimensional Galerkin approximations. A stability result is proved for the Galerkin approximations.

Real-time control of the free boundary in a two-phase stefan problem

A two-phase Stefan-problem is considered where the progress of the free boundary is observed by fully automatic real-time controls (thermostats or photo-electric cells). The heat flux at both fixed boundaries can be determined by heaters which respond to the signals of the controls, possibly with a certain time lag. The corresponding mathematical model leads to a two-phase Stefan problem with nonlinear and discontinuous boundary conditions with delays at the fixed boundaries. The problem is transformed into a set-valued fixed point equation, and the existence of a solution is shown with the aid of a theorem due to Bohnenblust-Karlin. The consequence of this result is that a free boundary with the well-known smoothness properties develops under the impact of a fully automatic real-time control via thermostats or photo-electric cells. Some numerical experiments conclude the paper. They indicate that the model is realistic.

Global Solutions to a Penrose–Fife Phase‐field Model Under Flux Boundary Conditions for the Inverse Temperature

In this paper, we study an initial-boundary value problem for a system of phase-field equations arising from the Penrose-Fife approach to model the kinetics of phase transitions. In contrast to other recent works in the field, the correct form of the boundary condition for the temperature field is assumed which leads to additional difficulties in the mathematical treatment. It is demonstrated that global existence of strong solutions can be shown under essentially the same assumptions on the data as in the previous papers where a simplified boundary condition for the heat exchange with the surrounding medium has been used.

Optimal boundary control problems for shape memory alloys under state constraints for stress and temperature

We consider two optimal control problems for first order martensitic phase transitions in a deformation-driven experiment on shape memory alloys including state constraints for the total stress and the temperature. We control by the elongation of a thin rod and by the outside temperature. The control problems are stated, and the necessary conditions of optimality are derived.

Analysis and Optimization of Nonsmooth Arches

It is our aim to present a new treatment for some classical models of arches and for their optimization. In particular, our approach allows us to study nonsmooth arches, while the standard assumptions from the literature require

Sur les arches lipschitziennes

W^{3, infty}

Optimization Problems for Thin Elastic Structures

-regularity for the parametric representation. Moreover, by a duality-type argument, the deformation of the arches may be explicitly expressed by integral formulas. %This also provides a complete solution of the so-called nAs examples for the shape optimization problems under study, we mention the design of the middle curve of a clamped arch such that, under a prescribed load, the obtained deflection satisfies certain desired properties. In all cases, no smoothness is required for the design parameters.

Optimization of clamped plates with discontinuous thickness

Resume On etudie le modele de Kirchhoff–Love dans le cas quand la courbe moyenne de larche a des coins. Notre approche nutilise pas le principe de Dirichlet ou linegalite de Korn. La formulation variationnelle que nous proposons est basee sur la theorie du controle optimal et on obtient des formules explicites pour la deformation.

Propriétés de bang-bang généralisées dans l'optimisation des plaques

We discuss shape optimization problems and variational methods for fundamental mechanical structures like beams, plates, arches, curved rods, and shells.

Optimization of Differential Systems with Hysteresis

We consider a new variational method for a clamped plate model and related shape optimization problems. Our approach allows the study of plates with discontinuous thickness.