Sufficient conditions of local solvability for partial differential operators in the Colombeau context

TL;DR

This paper establishes sufficient conditions for the local solvability of partial differential operators with variable Colombeau coefficients, focusing on operators with pseudodifferential parametrices and bounded perturbations.

Contribution

It introduces new solvability criteria for PDEs with Colombeau coefficients, expanding the understanding of solutions in Colombeau algebras and their duals.

Findings

Provided solvability conditions for operators with pseudodifferential parametrices

Analyzed operators as bounded perturbations of constant coefficient operators

Extended solvability results to solutions in Colombeau algebra and its dual

Abstract

We provide sufficient conditions of local solvability for partial differential operators with variable Colombeau coefficients. We mainly concentrate on operators which admit a right generalized pseudodifferential parametrix and on operators which are a bounded perturbation of a differential operator with constant Colombeau coefficients. The local solutions are intended in the Colombeau algebra $\G(\Om)$ as well as in the dual $\LL(\Gc(\Om),\wt{\C})$.