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- ItemMaximal parabolic regularity for divergence operators on distribution spaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Haller-Dintelmann, Robert; Rehberg, JoachimWe show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented
- ItemA Kohn-Sham system at zero temperature(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Cornean, Horia; Hoke, Kurt; Neidhardt, Hagen; Racec, Paul Nicolae; Rehberg, JoachimAn one-dimensional Kohn-Sham system for spin particles is considered which effectively describes semiconductor nanostructures and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schrödinger operator with effective Kohn-Sham potential and obtain $W^1,2$-bounds of the associated particle density operator. Afterwards, compactness and continuity results allow to apply Schauder's fixed point theorem. In case of vanishing exchange-correlation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero.
- ItemCoercivity for elliptic operators and positivity of solutions on Lipschitz domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Haller-Dintelmann, Robert; Rehberg, JoachimWe show that usual second order operators in divergence form satisfy coercivity on Lipschitz domains if they are either complemented with homogeneous Dirichlet boundary conditions on a set of non-zero boundary measure or if a suitable Robin boundary condition is posed. Moreover, we prove the positivity of solutions in a general, abstract setting, provided that the right hand side is a positive functional. Finally, positive elements from $W^-1,2$ are identified as positive measures
- ItemDirect computation of elliptic singularities across anisotropic, multi-material edges(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Haller-Dintelmann, Robert; Kaiser, Hans-Christoph; Rehberg, JoachimWe characterise the singularities of elliptic div-grad operators at points or edges where several materials meet on a Dirichlet or Neumann part of the boundary of a two- or three-dimensional domain. Special emphasis is put on anisotropic coefficient matrices. The singularities can be computed as roots of a characteristic transcendental equation. We establish uniform bounds for the singular values for several classes of three- and fourmaterial edges. These bounds can be used to prove optimal regularity results for elliptic div-grad operators on three-dimensional, heterogeneous, polyhedral domains with mixed boundary conditions. We demonstrate this for the benchmark Lshape problem.
- ItemOptimal control for the thermistor problem(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Hömberg, Dietmar; Meyer, Christian; Rehberg, Joachim; Ring, WolfgangThis 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.
- ItemOptimal regularity for elliptic transmission problems including C1 interfaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Elschner, Johannes; Rehberg, Joachim; Schmidt, GuntherWe 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.
- ItemAnalyticity for some operator functions from statistical quantum mechanics : dedicated to Günter Albinus(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Hoke, Kurt; Kaiser, Hans-Christoph; Rehberg, Joachim; Albinus, GünterFor rather general thermodynamic equilibrium distribution functions the density of a statistical ensemble of quantum mechanical particles depends analytically on the potential in the Schrödinger operator describing the quantum system. A key to the proof is that the resolvent to a power less than one of an elliptic operator with non-smooth coefficients, and mixed Dirichlet/Neumann boundary conditions on a bounded up to three-dimensional Lipschitz domain factorizes over the space of essentially bounded functions.
- ItemQuasilinear parabolic systems with mixed boundary conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Hieber, Matthias; Rehberg, JoachimIn this paper we investigate quasilinear systems of reaction-diffusion equations with mixed Dirichlet-Neumann bondary conditions on non smooth domains. Using techniques from maximal regularity and heat-kernel estimates we prove existence of a unique solution to systems of this type.
- ItemModeling and simulation of strained quantum wells in semiconductorlasers(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2000) Bandelow, Uwe; Kaiser, Hans-Christoph; Koprucki, Thomas; Rehberg, JoachimA model allowing for efficiently obtaining band structure information on semiconductor Quantum Well structures will be demonstrated which is based on matrix-valued kp-Schrödinger operators. Effects such as confinement, band mixing, spin-orbit interaction and strain can be treated consistently. The impact of prominent Coulomb effects can be calculated by including the Hartree interaction via the Poisson equation and the bandgap renormalization via exchange-correlation potentials, resulting in generalized (matrix-valued) Schrödinger-Poisson systems. Band structure information enters via densities and the optical response function into comprehensive simulations of Multi Quantum Well lasers. These device simulations yield valuable information on device characteristics, including effects of carrier transport, waveguiding and heating and can be used for optimization.
- ItemMaximal parabolic regularity for divergence operators including mixed boundary conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Haller-Dintelmann, Robert; Rehberg, JoachimWe show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.