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Mixed Problems And Layer Potentials For Harmonic And Biharmonic Functions, Moises Venouziou

Mathematics - Dissertations

The mixed problem is to find a harmonic or biharmonic function having prescribed Dirichlet data on one part of the boundary and prescribed Neumann data on the remainder. One must make a choice as to the required boundary regularity of solutions. When only weak regularity conditions are imposed, the harmonic mixed problem has been solved on smooth domains in the plane by Wendland, Stephan, and Hsiao. Significant advances were later made on Lipschitz domains by Ott and Brown. The strain of requiring a square-integrable gradient on the boundary, however, forces a strong geometric restriction on the domain. Well-known counterexamples by …