Lutz Recke

Linear Elliptic Boundary Value Problems with Non – Smooth Data: Normal Solvability on Sobolev – Campanato Spaces

In this paper linear elliptic boundary value problems of second order with non-smooth data (L∞-coefficients, Lipschitz domains, regular sets, non-homogeneous mixed boundary conditions) are considered. It is shown that such boundary value problems generate Fredholm operators between appropriate Sobolev–Campanato spaces, that the weak solutions are Holder continuous up to the boundary and that they depend smoothly (in the sense of a Holder norm) on the coefficients and on the right hand sides of the equations and boundary conditions.

Frequency regions for forced locking of self-pulsating multi-section DFB lasers

A method is developed which allows for the calculation of locking regions of self-pulsating multi-section lasers which are exposed to external optical data sequences. In particular, resonant locking is investigated where both wavelength detuning and detuning of the modulation frequency are important.

Fredholm alternative for periodic-Dirichlet problems for linear hyperbolic systems

This paper concerns hyperbolic systems of two linear first-order PDEs in one space dimension with periodicity conditions in time and reflection boundary conditions in space. The coefficients of the PDEs are supposed to be time independent, but allowed to be discontinuous with respect to the space variable. We construct two scales of Banach spaces (for the solutions and for the right-hand sides of the equations, respectively) such that the problem can be modeled by means of Fredholm operators of index zero between corresponding spaces of the two scales.

Dynamics of Multisection Semiconductor Lasers

We consider a mathematical model (the so-called traveling-wave system) which describes longitudinal dynamical effects in semiconductor lasers. This model consists of a linear hyperbolic system of PDEs, which is nonlinearly coupled with a slow subsystem of ODEs. We prove that a corresponding initial-boundary value problem is well posed and that it generates a smooth infinite-dimensional dynamical system. Exploiting the particular slow-fast structure, we derive conditions under which there exists a low-dimensional attracting invariant manifold. The flow on this invariant manifold is described by a system of ODEs. Mode approximations of that system are studied by means of bifurcation theory and numerical tools.

Smooth continuation of solutions and eigenvalues for variational inequalities based on the implicit function theorem

The implicit function theorem is applied in a nonstandard way to abstract variational inequalities depending on a (possibly infinite-dimensional) parameter. In this way, results on smooth continuation of solutions as well as of eigenvalues and eigenvectors are established under certain particular assumptions. The abstract results are applied to a linear second order elliptic eigenvalue problem with nonlocal unilateral boundary conditions (Schrodinger operator with the potential as the parameter).

APPLICATIONS OF THE IMPLICIT FUNCTION: THEOREM TO QUASILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS WITH NON-SMOOTH DATA

In this paper we consider boundary value problems for systems of quasilinear elliptic equations of the type {minus}{partial_derivative}{sub j}[a{sub ija{beta}}(x, u, {lambda}){partial_derivative}{sub i}u{sub a} + b{sub j{beta}}(x, u, {lambda})]++c{sub ia{beta}}(x, u, {lambda}){partial_derivative}{sub i}u{sub a} + d{sub {beta}}(x, u, {lambda}) = {integral}{beta}. In (1.1) (and in the sequel) the summation over the repeated subscripts i,j = 1, ..., N and a = 1,...,n is understood, and the {open_quotes}free{close_quotes} subscript {beta} varies from 1 to n. The independent variable x = (x{sub 1},...,x{sub N}) belongs to a bounded open domain {Omega} {contained_in} IR{sup N}, and {partial_derivative}{sub i} denotes the partial derivative with respect to x{sub i}. The unknown function u = (u{sub 1},...,u{sub n}) maps {Omega} into IR{sup n}. For the system (1.1) we consider mixed boundary conditions: [a{sub ija{beta}}(x,u,{lambda}){partial_derivative}{sub i}u{sub a} + b{sub j{beta}}(x,u,{lambda})]{nu}{sub j} = g{sub {beta}} for x {epsilon} {Gamma} u{sub {beta}} = h{sub {beta}} for x {epsilon} {partial_derivative}{Omega}/{Gamma}. 16 refs.

Smooth bifurcation for variational inequalities based on the implicit function theorem

Abstract We present a certain analog for variational inequalities of the classical result on bifurcation from simple eigenvalues of Crandall and Rabinowitz. In other words, we describe the existence and local uniqueness of smooth families of nontrivial solutions to variational inequalities, bifurcating from a trivial solution family at certain points which could be called simple eigenvalues of the homogenized variational inequality. If the bifurcation parameter is one-dimensional, the main difference between the case of equations and the case of variational inequalities (when the cone is not a linear subspace) is the following: For equations two smooth half-branches bifurcate, for inequalities only one. The proofs are based on scaling techniques and on the implicit function theorem. The abstract results are applied to a fourth order ODE with pointwise unilateral conditions (an obstacle problem for a beam with the compression force as the bifurcation parameter).

Centre manifold reduction approach for the lubrication equation

The goal of this study is the reduction of the lubrication equation, modelling thin film dynamics, onto an approximate invariant manifold. The reduction is derived for the physical situation of the late phase evolution of a dewetting thin liquid film, where arrays of droplets connected by an ultrathin film of thickness e undergo a slow-time coarsening dynamics. With this situation in mind, we construct an asymptotic approximation of the corresponding invariant manifold, that is parametrized by a family of droplet pressures and positions, in the limit when e → 0.The approach is inspired by the paper by Mielke and Zelik (2009 Mem. Am. Math. Soc. 198 1–97), where the centre manifold reduction was carried out for a class of semilinear systems. In this study this approach is considered for quasilinear degenerate parabolic PDEs such as lubrication equations.While it has previously been shown by Glasner and Witelski (2003 Phys. Rev. E 67 016302), that the system of ODEs governing the coarsening dynamics can be obtained via formal asymptotic methods, the centre manifold reduction approach presented here pursues the rigorous justification of this asymptotic limit.

Moving fronts in integro-parabolic reaction-advection-diffusion equations

We consider initial-boundary value problems for a class of singularly perturbed nonlinear integro-differential equations. In applications, they are referred to as nonlocal reactionadvection-diffusion equations, and their solutions have moving interior transition layers (fronts). We construct the asymptotics of such solutions with respect to a small parameter and estimate the accuracy of the asymptotics. To justify the asymptotics, we use the asymptotic differential inequality method.

Asymptotic stability via the Krein-Rutman theorem for singularly perturbed parabolic periodic dirichlet problems

We consider singularly perturbed semilinear parabolic periodic problems and assume the existence of a family of solutions. We present an approach to establish the exponential asymptotic stability of these solutions by means of a special class of lower and upper solutions. The proof is based on a corollary of the Krein-Rutman theorem.