Multipliers in the Classical Layer Potential Theory for Lipschitz Domains
In this chapter we give applications of Sobolev multipliers to the question of higher regularity in fractional Sobolev spaces of solutions to boundary integral equations generated by the classical boundary value problems for the Laplace equation in and outside a Lipschitz domain. Since the sole Lipschitz graph property of ∂Ω does not guarantee higher regularity of solutions, we are forced to select an appropriate subclass of Lipschitz domains whose description involves a space of multipliers. For domains of this subclass we develop a solvability and regularity theory analogous to the classical one for smooth domains. We also show that the chosen subclass of Lipschitz domains proves to be the best possible in a certain sense. We end the chapter with a brief discussion of boundary integral equations of linear elastostatics.
KeywordsUnique Solution Dirichlet Problem Laplace Equation Neumann Problem Boundary Integral Equation
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