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Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities
We characterize microlocal regularity, in the script G sign ∞-sense, 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 of...
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| Format: | Default Article |
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2005
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| Online Access: | https://hdl.handle.net/2134/17302 |
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| _version_ | 1818172975343468544 |
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| author | Claudia Garetto Gunther Hormann |
| author_facet | Claudia Garetto Gunther Hormann |
| author_sort | Claudia Garetto (1253247) |
| collection | Figshare |
| description | We characterize microlocal regularity, in the script G sign ∞-sense, 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 of determining generalized wavefront sets that is analogous to the original definition of the wavefront 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 non-characteristic regularity result for distributional solutions and consider propagation of script G sign ∞-singularities for homogeneous first-order hyperbolic equations. |
| format | Default Article |
| id | rr-article-9385772 |
| institution | Loughborough University |
| publishDate | 2005 |
| record_format | Figshare |
| spelling | rr-article-93857722005-01-01T00:00:00Z Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities Claudia Garetto (1253247) Gunther Hormann (7159799) Other mathematical sciences not elsewhere classified Algebras of generalized functions Pseudodifferential operators Wavefront sets Mathematical Sciences not elsewhere classified We characterize microlocal regularity, in the script G sign ∞-sense, 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 of determining generalized wavefront sets that is analogous to the original definition of the wavefront 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 non-characteristic regularity result for distributional solutions and consider propagation of script G sign ∞-singularities for homogeneous first-order hyperbolic equations. 2005-01-01T00:00:00Z Text Journal contribution 2134/17302 https://figshare.com/articles/journal_contribution/Microlocal_analysis_of_generalized_functions_pseudodifferential_techniques_and_propagation_of_singularities/9385772 CC BY-NC-ND 4.0 |
| spellingShingle | Other mathematical sciences not elsewhere classified Algebras of generalized functions Pseudodifferential operators Wavefront sets Mathematical Sciences not elsewhere classified Claudia Garetto Gunther Hormann Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities |
| title | Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities |
| title_full | Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities |
| title_fullStr | Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities |
| title_full_unstemmed | Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities |
| title_short | Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities |
| title_sort | microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities |
| topic | Other mathematical sciences not elsewhere classified Algebras of generalized functions Pseudodifferential operators Wavefront sets Mathematical Sciences not elsewhere classified |
| url | https://hdl.handle.net/2134/17302 |