Elliptic regularity and solvability for partial differential equations with Colombeau coefficients
Date
2004-02-03
Authors
Hormann, Gunther
Oberguggenberger, Michael
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
This 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∞-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.
Description
Keywords
Algebras of generalized functions, Regularity, Solvability
Citation
Hörmann, G., & Oberguggenberger, M. (2004). Elliptic regularity and solvability for partial differential equations with Colombeau coefficients. <i>Electronic Journal of Differential Equations, 2004</i>(14), pp. 1-30.
Rights
Attribution 4.0 International