On a class of partial differential equations with singular coefficients

Abstract

We study the equation div(fΔu)=0 for functions f which may become infinite at some subset of the domain of definition. After defining adequate Sobolev-like spaces of functions, we make a careful study of the behaviour of admissible solutions in the neighbourhood of singular points. This enables us to show that the Dirichlet problem possesses a space of solutions whose dimension is determined by the number and characteristics of the connected components of the set f=∞, a similar conclusion holds for the Neumann problem.

Keywords:

• Partial differentiation equation
• singular function
• Hilbert space

AMS(MOS)::

• 35A05
• 76E25