arXiv subjects

Guenther Hoermann

Publications and source records attributed to Guenther Hoermann.

29 records · Page 2Linked to original sources

arXiv2008

Comparison of some solution concepts for linear first-order hyperbolic differential equations with non-smooth coefficients

We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on refined techniques concerning energy estimates. We provide a series of examples both as simple illustrations of the notions and conditions involved but also to show logical independence among the concepts.

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arXiv2007

Distributional solution concepts for the Euler-Bernoulli beam equation with discontinuous coefficients

We study existence and uniqueness of distributional solutions to the differential equation of the Euler-Bernoulli rod with discontinuous coefficients and right-hand side. Upon checking the validity of a solution the occurring products of singular coefficients with the distributional solution have no obvious meaning. When interpreted on the most general level of the so-called hierarchy of distributional products, it turns out that existence of a solution forces a minimum regularity to hold. Curiously, the choice of the distributional product concept is thus incompatible with the possibility of having a discontinuous displacement function as a solution. We also give conditions for unique solvability.

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arXiv2006

Generalized Oscillatory Integrals and Fourier Integral Operators

In this article, a theory of generalized oscillatory integrals (OIs) is developed whose phase functions as well as amplitudes may be generalized functions of Colombeau type. Based on this, generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is motivated by the need of a general framework for partial differential operators with non-smooth coefficients and distribution data. The mapping properties of these FIOs are studied, as is microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wave front sets.

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arXiv2005

On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets

Summarizing basic facts from abstract topological modules over Colombeau generalized complex numbers we discuss duality of Colombeau algebras. In particular, we focus on generalized delta functionals and operator kernels as elements of dual spaces. A large class of examples is provided by pseudodifferential operators acting on Colombeau algebras. By a refinement of symbol calculus we review a new characterization of the wave front set for generalized functions with applications to microlocal analysis.

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arXiv2004

Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients

We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth hypoelliptic symbols. Methodological novelties and technical refinements appear embedded into classical strategies of proof in order to cope with most delicate interferences by non-smooth lower order terms. We include simplified conditions which are applicable in special cases of interest.

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arXiv2004

Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities

We characterize microlocal regularity of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow scale generalized symbols. Thus we obtain an alternative, yet equivalent, way to determine generalized wave front sets, which is analogous to the original definition of the wave front set of distributions via intersections over characteristic sets. The new methods are then applied to regularity theory of generalized solutions of (pseudo-)differential equations, where we extend the general noncharacteristic regularity result for distributional solutions and consider propagation of generalized singularities for homogeneous first-order hyperbolic equations.

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arXiv2003

Hölder-Zygmund regularity in algebras of generalized functions

We introduce an intrinsic notion of Hoelder-Zygmund regularity for Colombeau generalized functions. In case of embedded distributions belonging to some Zygmund-Hoelder space this is shown to be consistent. The definition is motivated by the well-known use of Littlewood-Paley decomposition in characterizing Hoelder-Zygmund regularity for distributions. It is based on a simple interplay of differentiated convolution-mollification with wavelet transforms, which directly translates wavelet estimates into properties of the regularizations. Thus we obtain a scale of new subspaces of the Colombeau algebra. We investigate their basic properties and indicate first applications to differential equations whose coefficients are non-smooth but belong to some Hoelder-Zygmund class (distributional or generalized). In applications problems of this kind occur, for example, in seismology when Earth's geological properties of fractal nature have to be taken into account while the initial data typically involve strong singularities.

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arXiv2003

Elliptic regularity and solvability for partial differential equations with Colombeau coefficients

The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new notions of ellipticity and hypoellipticity, study their interrelation, and give a number of new examples and counterexamples. Using the concept of $\G^\infty$-regularity of generalized functions, we derive a general global regularity result in the case of operators with constant generalized coefficients, a more specialized result for second order operators, and a microlocal regularity result for certain first order operators with variable generalized coefficients. We also prove a global solvability result for operators with constant generalized coefficients and compactly supported Colombeau generalized functions as right hand sides.

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arXiv2003

First-order hyperbolic pseudodifferential equations with generalized symbols

We consider the Cauchy problem for a hyperbolic pseudodifferential operator whose symbol is generalized, resembling a representative of a Colombeau generalized function. Such equations arise, for example, after a reduction-decoupling of second-order model systems of differential equations in seismology. We prove existence of a unique generalized solution under log-type growth conditions on the symbol, thereby extending known results for the case of differential operators with generalized functions as coefficients.

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arXiv2000

Microlocal properties of basic operations in Colombeau algebras

The Colombeau algebra of generalized functions allows to unrestrictedly carry out products of distributions. We analyze this operation from a microlocal point of view, deriving a general inclusion relation for wave front sets of products in the algebra. Furthermore, we give explicit examples showing that the given result is optimal, i.e. its assumptions cannot be weakened. Finally, we discuss the interrelation of these results with the concept of pullback under smooth maps.

math.FA↗

arXiv1999

Regularized derivatives in a 2-dimensional model of self-interacting fields with singular data

The coupled Maxwell-Lorentz system describes feed-back action of electromagnetic fields in classical electrodynamics. When applied to point-charge sources (viewed as limiting cases of charged fluids) the resulting nonlinear weakly hyperbolic system lies beyond the scope of classical distribution theory. Using regularized derivatives in the framework of Colombeau algebras of generalized functions we analyze a two-dimensional analogue of the Maxwell-Lorentz system. After establishing existence and uniqueness of solutions in this setting we derive some results on distributional limits of solutions with $δ$-like initial values.

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