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search.filters.applied.f.access: openAccess × Author: Rehberg, Joachim × End date: 2009 × Start date: 2000 × DDC: 510 × Type: Text × Version: publishedVersion × Subject: Partial differential equations ×

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    Optimal control for the thermistor problem
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Hömberg, Dietmar; Meyer, Christian; Rehberg, Joachim; Ring, Wolfgang
    This paper is concerned with the state-constrained optimal control of the two-dimensional thermistor problem, a quasi-linear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Existence, uniqueness and continuity for the state system are derived by employing maximal elliptic and parabolic regularity. By similar arguments the linearized state system is discussed, while the adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem.
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    Optimal regularity for elliptic transmission problems including C1 interfaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Elschner, Johannes; Rehberg, Joachim; Schmidt, Gunther
    We prove an optimal regularity result for elliptic operators $-nabla cdot mu nabla:W^1,q_0 rightarrow W^-1,q$ for a $q>3$ in the case when the coefficient function $mu$ has a jump across a $C^1$ interface and is continuous elsewhere. A counterexample shows that the $C^1$ condition cannot be relaxed in general. Finally, we draw some conclusions for corresponding parabolic operators.