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 Brown show this restriction to be a necessary condition.

This thesis proves that these harmonic counterexamples are an anomaly, in that the mixed problem can be solved for all data modulo a finite dimensional subspace. The geometric restriction now required is significantly less stringent than the one referred to above. This result is proved by representing solutions in terms of single and double layer potentials, establishing a mixed Rellich inequality, and applying functional analytic arguments to solve a two-by-two system of equations. These results are then extended to allow Robin data in place of Neumann data.

This thesis also establishes counterexamples for the biharmonic mixed problem with Poisson ratio in the interval [-1, -.5]. These counterexamples are biharmonic analogues to the harmonic ones referred to above. Their exact form is obtained by solving a four-by-four system of equations.