Well-posed problems in algebras of generalized functions

Abstract

We show that a well-defined generalized problem can be associated, in the framework of (𝒞, ℰ, 𝒫)-algebras, with an ill-posed one (with singular data, singular coefficients or in the characteristic case) by means of generalized operators associated with stability properties of the algebra. The question of the well-posedness, in the Hadamard sense, of the generalized regularization is also discussed.

Keywords:

• regularization of data
• regularization of problems
• distributions
• nonlinear generalized functions
• nonlinear problems
• characteristic problems

AMS Subject Classifications::

• 35A01
• 35A02
• 35A25
• 35D99
• 46F30
• 46T30