Variational views of stokeslets and stresslets

TL;DR

This paper develops a new variational framework for layer potentials in the Stokes problem on Lipschitz boundaries, providing a robust structure that connects classical integral formulas with modern weak formulations.

Contribution

It introduces a novel variational approach to Stokes layer potentials on Lipschitz domains, linking them to Lamé and Laplace operators and enhancing the theoretical foundation.

Findings

Provides a variational theory for Stokes layer potentials

Establishes formulas relating Stokes, Lamé, and Laplace potentials

Offers a solid theoretical structure for non-smooth domains

Abstract

In this paper we present a self-contained variational theory of the layer potentials for the Stokes problem on Lipschitz boundaries. We use these weak definitions to show how to prove the main theorems about the associated Calder\'on projector. Finally, we relate these variational definitions to the integral forms. Instead of working these relations from scratch, we show some formulas parametrizing the Stokes layer potentials in terms of those for the Lam\'e and Laplace operators. While all the results in this paper are well known for smooth domains, and most might be known for non-smooth domains, the approach is novel a gives a solid structure to the theory of Stokes layer potentials.