Layer Potentials for the Harmonic Mixed Problem in the Plane

The mixed problem is to find a harmonic 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 mixed problem has been solved on smooth domains in the plane by Wendland et al. (Math Methods Appl Sci 1(3):265–321, 1979). Significant advances were later made on Lipschitz domains by Ott and Brown (2011) and Brown (Commun Partial Differ Equ 19(7–8):1217–1233, 1994). 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 paper shows that these counterexamples are an anomaly, in that the mixed problem in the plane 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.