Elliptic equations with low regularity boundary data via the unified method

Abstract

We use novel integral representations developed by the second author to prove certain rigorous results concerning elliptic boundary value problems in convex polygons. Central to this approach is the so-called global relation, which is a non-local equation in the Fourier space that relates the known boundary data to the unknown boundary values. Assuming that the global relation is satisfied in the weakest possible sense, i.e. in a distributional sense, we prove there exist solutions to Dirichlet, Neumann and Robin boundary value problems with distributional boundary data. We show that the analysis of the global relation characterises in a straightforward manner the possible existence of both integrable and non-integrable corner singularities.

Keywords:

• distributions
• unified method
• corner singularities

AMS Subject Classifications:

• 35J25
• 35D30

Acknowledgements

The first author is grateful for the support of Pembroke College, University of Cambridge.

Notes

1 Meaning that if a sequence tends to zero in (i.e. for all ), it also tends to zero in (i.e. for all ).

2 Recall Watson’s lemma as .