DIANA Research Group
DIfferential Algebras and Nonlinear Analysis

  START Project Description: Nonlinear Distributional Geometry





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The aim of this project is to substantially advance nonsmooth differential geometry and nonsmooth geometric analysis of partial differential operators in the setting of algebras of generalized functions. Nonlinear distributional geometry is to be developed both as a mathematical theory and as a tool for applications, in particular in mathematical physics (general relativity and mathematical geophysics). Research on the foundations of the theory will therefore be carried out in close connection to relevant fields of applications. The project rests on three cornerstones:

  1. Nonlinear Distributional Geometry

    Applications in mathematical physics have demonstrated the need for a theory capable of treating nonlinear problems involving singularities in a geometrical setting. A suitable framework for studying such questions is provided by Colombeau's theory of algebras of generalized functions (see [Col1], [Col2], [Obe1]). A diffeomorphism-invariant version of the theory has recently been introduced (cf. [Gro1], [Gro2], [Gro3]). Building on these foundations, in this part of the project nonsmooth differential geometry is to be developed further by studying geometric constructions (e.g., singular connections, geodesics in nonsmooth Riemannian geometries, etc.) and embedding questions of linear (vector valued) distribution spaces into suitable spaces of Colombeau generalized functions.

  2. Nonsmooth Geometric Analysis of Partial Differential Operators

    (a) Microlocal analysis in nonsmooth geometries

    The concept of a regular (Colombeau) generalized function is based on regularization growth bounds uniformly over all derivatives and extends the distributional regularity property of smoothness. Combining such localized space-time properties with (Fourier) spectral estimates of generalized functions leads to the notion of microlocal regularity, i.e., generalized wave front sets. A first phase in the project will assess the feasibility of establishing the latter as a genuine geometric object in the cotangent bundle and investigate fundamental nonlinear properties (e.g. behavior under nonsmooth pullbacks).
    The new geometric generalized microlocal analysis is then to be applied to partial differential operators with nonsmooth symbols, such as wave equations on nonsmooth curved space-times. This will involve intensive research on a theory of generalized bicharacteristic flows and their relation to transport of wave front sets of generalized solutions, i.e., propagation of singularities in nonsmooth geometries. See [Obe1], [Hör1],[Hör2], [Hör3] and the literature cited therein for an introduction to this field.

    (b) Lie Group Analysis of Singular PDEs

    In recent years an extension of Lie group analysis of differential equations to nonsmooth differential operators and/or solutions has been initiated in the Colombeau framework (cf. [Gro1], Ch. 4 for an introduction). Here we intend to further pursue this line of research with the aim of developing a global theory of generalized symmetries of partial differential equations and of group invariant generalized functions extending the purely distributional aproach.

  3. Nonsmooth General Relativity

    The overall aim of this branch of the project is to establish conditions which allow for a unique solution of the Einstein equations in the presence of weak singularities thereby deepening the understanding of the ''real'' physical and mathematical issues in the context of the cosmic censorship hypothesis (CCH). More precisely we aim at tackling the following two strongly interconnected issues.

    (a) The Cauchy Problem in singular space-times

    Following a proposal of C.J.S. Clarke ([Cla1]) who defined ``genuine'' singularities as points which disrupt the local evolution of linear test fields the initial value problem of the wave equation in conical space-times was solved in nonlinear genarlized functions by Vickers and Wilson ([Vic1]) thus showing this singularity to be "non-genuine" resp. "weak". Building upon this work and the methods which currently are developed in the course of the relativity branch of Project-P16742 ''Geometric Theory of Generalized Functions'' (mainly higher order energy estimates) we are going to investigate the issues of local existence and uniqueness of solutions to Einstein's equations in the presence of such ''weak'' singularities (e.g. thin cosmic strings, impulsive gravitational waves, focusing gravitational waves). These results should then be applied to generally establish sufficient conditions on the geometry of the singularity to admit local existence and uniqueness of solutions to Einstein's equations. Subsequently singularities will be classified according to whether or not they disrupt local evolution of the field equations. The formalism will also be extended towards handling the global context.

    (b) Generalized Singularity Theorems

    The topic of this part of the project is a study of geodesics and their (non-)extendibility in the presence of singularities, resp. in space-times of low differentiability using, in particular, the techniques of [KS1], [KS2] to introduce a notion of ''strong'' singularities which are obstructions to unique solvability of the geodesic equation in the generalized sense. These are clearly stronger than the ''weak'' ones which form the topic of (a) and hence may not even be regarded as internal points of a generalized space time. This notion of a ''strong'' singularity makes it possible to precisely write out a ''genericity'' condition in the context of the CCH and will enable us to tackle the CCH also from this side. These ''strong'' or ''genuine'' singularities of part (b) should represent the breakdown of classical physics, while the ''weak'' ones of part (a) representing physically realistic scenarios would be no obstructions to generalized solvability of Einstein's equations.

  • [Cla1] C. J. S. Clarke, Generalized hyperbolicity in singular spacetimes, Class. Quant. Grav. 15, 974-984, (1998). (electronically available as gr-qc/9702033)
  • [Col1] J. F. Colombeau, New Generalized Functions and Multiplication of Distributions, North Holland, (Amsterdam, 1984).
  • [Col2] J. F. Colombeau, Elementray Introduction to New Generalized Functions, North Holland, (Amsterdam, 1985).
  • [Gro1] M. Grosser, M. Kunzinger, M. Oberguggenberger, R. Steinbauer, Geometric Theory of Generalized Functions, Mathematics and its Applications 537, Kluwer Academic Publishers (Dordrecht, 2001).
  • [Gro2] M. Grosser, E. Farkas, M. Kunzinger, R. Steinbauer, On the foundations of nonlinear generalized functions I, II, Mem. Amer. Math. Soc. 153 (729), 2001. (electronically available as math.FA/9912214)
  • [Gro3] M. Grosser, M. Kunzinger, R. Steinbauer, J. Vickers, A global theory of algebras of generalized functions, Adv. Math. 166(1), 50-72, (2002). (electronically available as math.FA/0102019)
  • [Hör1] G. Hörmann, M. DeHoop, Microlocal analysis and global solutions of some hyperbolic equations with discontinuous coefficients, Acta Appl. Math. 67, 173-224, (2001).
  • [Hör2] G. Hörmann, M. Oberguggenberger, Elliptic regularity and solvability for partial differential equations with Colombeau coefficients, Electr. Jour. Diff. Equ. Vol. 2004, no. 14, 1-30, (2004). (electronically available at
  • [Hör3] G. Hörmann, M. Oberguggenberger, S. Pilipovic , Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients, Trans. Amer. Math. Soc., to appear. (electronically available as math.AP/0303248)
  • [KS1] M. Kunzinger, R. Steinbauer, A rigorous solution concept for geodesic and geodesic deviation equations in impulsive gravitational waves, J. Math. Phys. 40, 1479--1489, (1999). (electronically available as gr-qc/9806009)
  • [KS2] M. Kunzinger, R. Steinbauer, Generalized pseudo-Riemannian geometry, Trans. Amer. Math. Soc. 354(10), 4179--4199, (2002). (electronically available as math.FA/0107057)
  • [Obe1] M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes in Mathematics 259, Longman (Halow, U.K, 1992).
  • [Vic1] J. Vickers, J. Wilson, Generalized hyperbolicity in conical spacetimes, Class. Quant. Grav. 17, 1333-1360, (2000). (electronically available as gr-qc/9907105)

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Last changed January 3, 2007 by M.K. and R.S.