Elliptic regularity and solvability for partial differential equations with Colombeau coefficients
نویسندگان
چکیده
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∞-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. AMS Mathematics Subject Classification: 46F30, 35D05, 35D10
منابع مشابه
Elliptic regularity and solvability for PDEs 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∞-regularity of...
متن کاملSolvability of Quasilinear Elliptic Equations with Strong Dependence on the Gradient
We study the problem of existence of positive, spherically symmetric strong solutions of quasilinear elliptic equations involving p-Laplacian in the ball. We allow simultaneous strong dependence of the right-hand side on both the unknown function and its gradient. The elliptic problem is studied by relating it to the corresponding singular ordinary integro-differential equation. Solvability ran...
متن کاملTransonic Shocks in Multidimensional Divergent Nozzles
We establish existence, uniqueness and stability of transonic shocks for steady compressible non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section, for a prescribed exit pressure. The proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equa...
متن کاملZygmund regularity of Colombeau generalized functions and applications to differential equations with nonsmooth coefficients
We introduce an intrinsic notion of Zygmund regularity for Colombeau algebras of generalized functions. In case of embedded distributions belonging to some Zygmund-Hölder space this is shown to be consistent. The definition is motivated by the well-known use of the wavelet transform as a tool in studying Hölder regularity. It is based on a simple mollifier-wavelet interplay which translates wav...
متن کاملElliptic Regularity Theory Applied to Time Harmonic Anisotropic Maxwell's Equations with Less than Lipschitz Complex Coefficients
Elliptic regularity theory applied to time harmonic anisotropic Maxwell's equations with less than Lipschitz complex coefficients The focus of this paper is the study of the regularity properties of the time harmonic Maxwell's equations with anisotropic complex coefficients, in a bounded domain with C 2,1 boundary. We assume that at least one of the material parameters is W 1,3+δ for some δ > 0...
متن کامل