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Christoph Walker

Publications and source records attributed to Christoph Walker.

At least 73 records · Page 4Linked to original sources
arXiv2013

A free boundary problem modeling electrostatic MEMS: II. nonlinear bending effects

Well-posedness of a free boundary problem for electrostatic microelectromechanical systems (MEMS) is investigated when nonlinear bending effects are taken into account. The model describes the evolution of the deflection of an electrically conductive elastic membrane suspended above a fixed ground plate together with the electrostatic potential in the free domain between the membrane and the fixed ground plate. The electrostatic potential is harmonic in that domain and its values are held fixed along the membrane and the ground plate. The equation for the membrane deflection is a parabolic quasilinear fourth-order equation, which is coupled to the gradient trace of the electrostatic potential on the membrane.

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arXiv2013

A free boundary problem modeling electrostatic MEMS: I. Linear bending effects

The dynamical and stationary behaviors of a fourth-order evolution equation with clamped boundary conditions and a singular nonlocal reaction term, which is coupled to an elliptic free boundary problem on a non-smooth domain, are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes two positive parameters $λ$ and $\varepsilon$ related to the applied voltage and the aspect ratio of the device, respectively. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as a criterion for global existence excluding the occurrence of finite time singularities which are not physically relevant. Existence of a stable steady state is shown for sufficiently small $λ$. Non-existence of steady states is also established when $\varepsilon$ is small enough and $λ$ is large enough (depending on $\varepsilon$).

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arXiv2013

Finite time singularity in a free boundary problem modeling MEMS

The occurrence of a finite time singularity is shown for a free boundary problem modeling microelectromechanical systems (MEMS) when the applied voltage exceeds some value. The model involves a singular nonlocal reaction term and a nonlinear curvature term accounting for large deformations.

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arXiv2013

A fourth-order model for MEMS with clamped boundary conditions

The dynamical and stationary behaviors of a fourth-order equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes a positive voltage parameter $λ$. It is shown that there is a threshold value $λ_*>0$ of the voltage parameter such that no radially symmetric stationary solution exists for $λ>λ_*$, while at least two such solutions exist for $λ\in (0,λ_*)$. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as the occurrence of finite time singularities when $λ>λ_*$.

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arXiv2013

Sign-Preserving Property for Some Fourth-Order Elliptic Operators in One Dimension and Radial Symmetry

For a class of one-dimensional linear elliptic fourth-order equations with homogeneous Dirichlet boundary conditions it is shown that a non-positive and non-vanishing right-hand side gives rise to a negative solution. A similar result is obtained for the same class of equations for radially symmetric solutions in a ball or in an annulus. Several applications are given, including applications to nonlinear equations and eigenvalue problems.

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arXiv2013

Dynamics of a free boundary problem with curvature modeling electrostatic MEMS

The dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems (MEMS) is investigated. The model couples the electric potential to the deformation of the membrane, the deflection of the membrane being caused by application of a voltage difference across the device. More precisely, the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane. The gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the membrane deformation. The main feature of the model considered herein is that, unlike most previous research, the small deformation assumption is dropped, and curvature for the deformation of the membrane is taken into account what leads to a quasilinear parabolic equation. The free boundary problem is shown to be well-posed, locally in time for arbitrary voltage values and globally in time for small voltages values. Furthermore, existence of asymptotically stable steady-state configurations is proved in case of small voltage values as well as non-existence of steady-states if the applied voltage difference is large. Finally, convergence of solutions of the free boundary problem to the solutions of the well-established small aspect ratio model is shown.

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arXiv2012

A parabolic free boundary problem modeling electrostatic MEMS

The evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system (MEMS) is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a harmonic function in an angular domain, the deformable membrane being a part of the boundary. The former solves a heat equation with a right hand side that depends on the square of the trace of the gradient of the electric potential on the membrane. The resulting free boundary problem is shown to be well-posed locally in time. Furthermore, solutions corresponding to small voltage values exist globally in time while global existence is shown not to hold for high voltage values. It is also proven that, for small voltage values, there is an asymptotically stable steady-state solution. Finally, the small aspect ratio limit is rigorously justified.

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arXiv2012

Some Remarks About the Semigroup Associated to Age-Structured Diffusive Populations

We consider linear age-structured population equations with diffusion. Supposing maximal regularity of the diffusion operator, we characterize the generator and its spectral properties of the associated strongly continuous semigroup. In particular, we provide conditions for stability of the zero solution and for asynchronous exponential growth.

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arXiv2011

A stationary free boundary problem modeling electrostatic MEMS

A free boundary problem describing small deformations in a membrane based model of electrostatically actuated MEMS is investigated. The existence of stationary solutions is established for small voltage values. A justification of the widely studied narrow-gap model is given by showing that steady state solutions of the free boundary problem converge toward stationary solutions of the narrow-gap model when the aspect ratio of the device tends to zero.

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arXiv2011

Two-Phase Flow in Rotating Hele-Shaw Cells with Coriolis Effects

The free boundary problem of a two phase flow in a rotating Hele-Shaw cell with Coriolis effects is studied. Existence and uniqueness of solutions near spheres is established, and the asymptotic stability and instability of the trivial solution is characterized in dependence on the fluid densities.

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arXiv2011

Weak solutions to a thin film model with capillary effects and insoluble surfactant

The paper focuses on a model describing the spreading of an insoluble surfactant on a thin viscous film with capillary effects taken into account. The governing equation for the film height is degenerate parabolic of fourth order and coupled to a second order parabolic equation for the surfactant concentration. It is shown that nonnegative weak solutions exist under natural assumptions on the surface tension coefficient.

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arXiv2011

A Note on a Nonlocal Nonlinear Reaction-Diffusion Model

We give an application of the Crandall-Rabinowitz theorem on local bifurcation to a system of nonlinear parabolic equations with nonlocal reaction and cross-diffusion terms as well as nonlocal initial conditions. The system arises as steady-state equations of two interacting age-structured populations.

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arXiv2010

Positive solutions of some parabolic system with cross-diffusion and nonlocal initial conditions

The paper is concerned with a system consisting of two coupled nonlinear parabolic equations with a cross-diffusion term, where the solutions at positive times define the initial states. The equations arise as steady state equations of an age-structured predator-prey system with spatial dispersion. Based on unilateral global bifurcation methods for Fredholm operators and on maximal regularity for parabolic equations, global bifurcation of positive solutions is derived.

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arXiv2010

On Nonlocal Parabolic Steady-State Equations of Cooperative or Competing Systems

Some systems of parabolic equations with nonlocal initial conditions are studied. The systems arise when considering steady-state solutions to diffusive age-structured cooperative or competing species. Local and global bifurcation techniques are employed to derive a detailed description of the structure of positive coexistence solutions.

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arXiv2010

On Positive Solutions of Some System of Reaction-Diffusion Equations with Nonlocal Initial Conditions

The paper focuses on positive solutions to a coupled system of parabolic equations with nonlocal initial conditions. Such equations arise as steady-state equations in an age-structured predator-prey model with diffusion. By using global bifurcation techniques, we describe the structure of the set of positive solutions with respect to two parameters measuring the intensities of the fertility of the species. In particular, we establish co-existence steady-states, i.e. solutions which are nonnegative and nontrivial in both components.

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