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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
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:
- 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
- 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).
(b) Lie Group Analysis of Singular PDEs
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
See [Obe1], [Hör1],[Hör2],
[Hör3] and the literature cited therein for an introduction
to this field.
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
- 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
(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],
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 http://ejde.math.swt.edu/Volumes/2004/14/abstr.html)
- [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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