2011, Fiaschi, Alice, Knees, Dorothee, Reichelt, Sina

We 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.

2021, Hintermüller, Michael, Kröner, Axel

In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from [Lasiecka, Szulc, and Zochoswki, Nonl. Anal.: Real World Appl., 44, 2018]. An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generaziling the results of [Wollner and Wick, J. Math. Fluid Mech., 21, 2019] to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners.

2014, Disser, Karoline, Kaiser, Hans-Christoph, Rehberg, Joachim

On 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.

2018, Disser, Karoline, Rehberg, Joachim

We 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.

2016, Disser, Karoline

In 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.

2006, Glitzky, Annegret, Hünlich, Rolf

We 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.

2015, Scala, Riccardo, Schimperna, Giulio

We 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.

2018, Ceretani, Andrea N., Rautenberg, Carlos N.

A @stationary Boussinesq system for an incompressible viscous fluid in a bounded domain with a nontrivial condition at an open boundary is studied. We consider a novel non-smooth boundary condition associated to the heat transfer on the open boundary that involves the temperature at the boundary, the velocity of the fluid, and the outside temperature. We show that this condition is compatible with two approaches at dealing with the do-nothing boundary condition for the fluid: 1) the directional do-nothing condition and 2) the do-nothing condition together with an integral bound for the backflow. Well-posedness of variational formulations is proved for each problem.

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.

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.