11 results
Search Results
Now showing 1 - 10 of 11
- ItemStationary solutions to an energy model for semiconductor devices where the equations are defined on different domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Glitzky, Annegret; Hünlich, RolfWe discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain $Omega_0$ of the domain of definition $Omega$ of the energy balance equation and of the Poisson equation. Here $Omega_0$ corresponds to the region of semiconducting material, $OmegasetminusOmega_0$ represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the two-dimensional stationary energy model. For this purpose we derive a $W^1,p$-regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem.
- ItemSobolev-Morrey spaces associated with evolution equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Griepentrog, Jens A.In this text we introduce new classes of Sobolev-Morrey spaces being adequate for the regularity theory of second order parabolic boundary value problems on Lipschitz domains of space dimension n ≥ 3 with nonsmooth coefficients and mixed boundary conditions. We prove embedding and trace theorems as well as invariance properties of these spaces with respect to localization, Lipschitz transformation, and reflection. In the second part [11] of our presentation we show that the class of second order parabolic systems with diagonal principal part generates isomorphisms between the above mentioned Sobolev-Morrey spaces of solutions and right hand sides.
- ItemOptimal Sobolev regularity for linear second-order divergence elliptic operators occurring in real-world problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Disser, Karoline; Kaiser, Hans-Christoph; Rehberg, JoachimOn bounded three-dimensional domains, we consider divergence-type operators including mixed homogeneous Dirichlet and Neumann boundary conditions and discontinuous coefficient functions. We develop a geometric framework in which it is possible to prove that the operator provides an isomorphism of suitable function spaces. In particular, in these spaces, the gradient of solutions turns out to be integrable with exponent larger than the space dimension three. Relevant examples from real-world applications are provided in great detail.
- ItemA contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Scala, Riccardo; Schimperna, GiulioWe consider a three-dimensional viscoelastic body subjected to external forces. Inertial effects are considered; hence the equation for the displacement field is of hyperbolic type. The equation is complemented with Dirichlet and Neuman conditions on a part the boundary, while on another part the body is in adhesive contact with a solid support. The boundary forces acting on the latter part due to the action of elastic stresses are responsible for delamination, i.e., progressive failure of adhesive bonds. This phenomenon is mathematically represented by a nonlinear ODE which describes the evolution of the delamination order parameter z. Following the lines of a new approach introduced by the authors in a preceding paper and based on duality methods in Sobolev-Bochner spaces, we define a suitable concept of weak solutions to the resulting PDE system. Correspondingly, we prove an existence result on finite time intervals of arbitrary length.
- ItemMaximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Griepentrog, Jens A.This text is devoted to maximal regularity results for second order parabolic systems on Lipschitz domains of space dimension n ≥ 3 with diagonal principal part, nonsmooth coefficients, and nonhomogeneous mixed boundary conditions. We show that the corresponding class of initial boundary value problems generates isomorphisms between two scales of Sobolev–Morrey spaces for solutions and right hand sides introduced in the first part [12] of our presentation. The solutions depend smoothly on the data of the problem. Moreover, they are Hölder continuous in time and space up to the boundary for a certain range of Morrey exponents. Due to the complete continuity of embedding and trace maps these results remain true for a broad class of unbounded lower order coefficients.
- ItemThe 3D transient semiconductor equations with gradient-dependent and interfacial recombination(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Disser, Karoline; Rehberg, JoachimWe establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on chargecarrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators.
- 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.
- ItemA unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Disser, Karoline; Meyries, Martin; Rehberg, JoachimIn this paper we consider scalar parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the corresponding abstract Cauchy problem.
- ItemGlobal higher integrability of minimizers of variational problems with mixed boundary conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Fiaschi, Alice; Knees, Dorothee; Reichelt, SinaWe consider integral functionals with densities of p-growth, with respect to gradients, on a Lipschitz domain with mixed boundary conditions. The aim of this paper is to prove that, under uniform estimates within certain classes of p-growth and coercivity assumptions on the density, the minimizers are of higher integrability order, meaning that they belong to the space of first order Sobolev functions with an integrability of order p+e for a uniform e >0. The results are applied to a model describing damage evolution in a nonlinear elastic body and to a model for shape memory alloys.
- ItemWell-posedness for coupled bulk-interface diffusion with mixed boundary conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Disser, KarolineIn this paper, we consider a quasilinear parabolic system of equations describing coupled bulk and interface diffusion, including mixed boundary conditions. The setting naturally includes non-smooth domains. We show local well-posedness using maximal Ls-regularity in dual Sobolev spaces of type W 1,q (Omega) for the associated abstract Cauchy problem.