Rainer Picard

An elementary proof for a compact imbedding result in generalized electromagnetic theory

Propriete de plongement compact et certaines de ses consequences. On etudie la validite du plongement suivant: R 9 (G)∩e q −1 D q (G)→→L 2 q,loc (G)

On the boundary value problems of electro- and magnetostatics

The classically well-known relation between the number of linearly independent solutions of the electro- and magnetostatic boundary value problems (harmonic Dirichlet and Neumann vector fields) and topological characteristics (genus and number of boundaries) of the underlying domain in 3-dimensional euclidean space is investigated in the framework of Hilbert space theory. It can be shown that this connection is still valid for a large class of domains with not necessarily smooth boundaries (segment property). As an application the inhomogeneous boundary value problems of electro- and magnetostatics are discussed.

Time-Harmonic Maxwell Equations in the Exterior of Perfectly Conducting, Irregular Obstacles

The boundary value problem of total reflection of time-harmonic electromagnetic waves is considered in an exterior domain Ω. Α Fredholm type alternative is shown to be valid under rather general assumptions on boundary regularity and regularity of the coefficients. The solution theory is developed in suitably weighted spaces. A cornerstone of the reasoning used to obtain the solution theory is a local compact imbedding property assumed to hold for Ω. A large class of domains is characterized featuring this property. MOS classifications: 35Q60, 78A25, 78A40, 78A50

Partial differential equations : a unified Hilbert space approach

This book presents a systematic approach to a solution theory for linear partial differential equations developed in a Hilbert space setting based on a Sobolev Lattice structure, a simple extension of the well established notion of a chain (or scale) of Hilbert spaces. The focus on a Hilbert space (rather than an apparently more general Banach space) setting is not a severe constraint, but rather a highly adaptable and suitable approach providing a more transparent framework for presenting the main issues in the development of a solution theory for partial differential equations. In contrast to other texts on partial differential equations which consider either specific types of partial differential equations or apply a collection of tools for solving a variety of partial differential equations, this book takes a more global point of view by focussing on the issues involved in determining the appropriate functional analytic setting in which a solution theory can naturally be developed. Applications to many areas of mathematical physics are presented. The book aims to be largely self-contained. Full proofs to all but the most straightforward results are provided, keeping to a minimum references to other literature for essential material. It is therefore highly suitable as a resource for graduate courses and for researchers, who will find new results for particular evolutionary system from mathematical physics.

On evolutionary equations with material laws containing fractional integrals

A well-posedness result for a time-shift invariant class of evolutionary operator equations involving material laws with fractional time-integrals of order α ϵ ]0, 1[ is considered. The fractional derivatives are defined via a function calculus for the (time-)derivative established as a normal operator in a suitable L2 type space. Employing causality, we show that the fractional derivatives thus obtained coincide with the Riemann-Liouville fractional derivative. We exemplify our results by applications to a fractional Fokker-Planck equation, equations describing super-diffusion and sub-diffusion processes, and a Kelvin-Voigt type model in fractional visco-elasticity. Moreover, we elaborate a suitable perspective to deal with initial boundary value problems. Copyright

On a structural observation in generalized electromagnetic theory

are satisfied. Equations (0.2) describe the electromagnetic properties of the medium where (0.1) is considered. If this medium is confined to a certain region G of R3 then the electromagnetic properties of its complent CC are usually represented via boundary conditions on dG. We shall deal here with the boundary condition appearing if CC is assumed to be a perfect conductor, i.e., nxE(t)=O on C7G, 13 0, (0.3)

Error estimates for the finite element approximation to a maxwell-type boundary value problem

Maxwells boundary value problem for the time-harmonic case in a smooth, bounded domain G of R 2 is considered. The optimal asymptotic L2(G) and H1(G)-error estimates 0(h2) and 0(h) resp, are derived for a piecewise linear finite element solution.

On a comprehensive class of linear control problems

We discuss a class of linear control problems in a Hilbert space setting. This class encompasses such diverse systems as port-Hamiltonian systems, Maxwells equations with boundary control or the acoustic equations with boundary control and boundary observation. The boundary control and observation acts on abstract boundary data spaces such that the only geometric constraint on the underlying domain stems from requiring a closed range constraint for the spatial operator part, a requirement which for the wave equation amounts to the validity of a Poincare–Wirtinger-type inequality. We also address the issue of conservativity of the control problems under consideration.

On a selfadjoint realization of curl in exterior domains

The first order vectorial operator ‘curl’ occurs in many areas of applications, e.g. in electrodynamics, fluid mechanics (hydrodynamics), magnetohydrodynamics and elasticity theory. It is the purpose of this note to discuss a particular instance of ‘curl’ where a domain can be given to ‘curl’ so that it becomes a selfadjoint operator. The particular domain of selfadjointness under consideration is singled out by its relevance in applications. To establish this operator we initially consider ‘curl’ as an operator curl| ◦ C∞(Ω) (|... read ‘restricted to . . . ’) in L2(Ω) defined by curl|◦ C∞(Ω) : ◦ C∞(Ω) ⊂ L2(Ω) → L2(Ω) (0.1)

Zur Theorie der harmonischen Differentialformen

Generalizing harmonic differential forms (rot ω=0, div ω=0 in M, M being a smooth riemannian manifold with boundary) of first and second kind (ω=0 and *ω=0 on ∂M resp.) within the framework of Hilbert space notation, it is possible to extend the meaning of the boundary conditions to non-smooth boundaries. It turns out that in this case the classical result is still valid for certain open subregions G of M: The dimension of the space of harmonic differential forms of second kind is given by the q-th Betti number of G; *-duality leads to the respective result for harmonic differential forms of first kind.