335 research outputs found
Characterization of distributions having a value at a point in the sense of Robinson
We characterize Schwartz distributions having a value at a single point in
the sense introduced by means of nonstandard analysis by A. Robinson. They
appear to be distributions continuous in a neighborhood of the point.Comment: 5 page
Group analysis of differential equations and generalized functions
We present an extension of the methods of classical Lie group analysis of
differential equations to equations involving generalized functions (in
particular: distributions). A suitable framework for such a generalization is
provided by Colombeau's theory of algebras of generalized functions. We show
that under some mild conditions on the differential equations, symmetries of
classical solutions remain symmetries for generalized solutions. Moreover, we
introduce a generalization of the infinitesimal methods of group analysis that
allows to compute symmetries of linear and nonlinear differential equations
containing generalized function terms. Thereby, the group generators and group
actions may be given by generalized functions themselves.Comment: 27 pages, LaTe
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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