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In this note we show that the well-posedness range p ∈ (1, 2] for L p transmission boundary value problems for the Laplacian in the class of Lipschitz domains established by Escauriaza and Mitrea (2004) is sharp. Our approach relies on Mellin transform techniques for singular integrals naturally associated with the transmission problems and on a careful analysis of the L p spectra of such singular integrals.

Counterexamples to the well-posedness of 𝐿^{𝑝} transmission boundary value problems for the Laplacian