Nonvariational layer potentials with respect to H¨older continuous vector fields

Nontangential a.e. vanishing of the oblique derivative of a harmonic function with respect to a Hölder continuous vector field on a Lipschitz boundary is shown to imply that the harmonic function is constant. In this article we prove a uniqueness result for an oblique derivative problem with respect to a transverse Hölder continuous vector field defined on the boundary of a Lipschitz domain Ω ⊂ R. The result is for harmonic functions when vanishing data is prescribed nontangentially almost everywhere, with respect to surface measure, in L(∂Ω) rather than everywhere as in the classical formulation. It is motivated in part by the following result due to A.P. Calderón. Theorem 0.1 ([1]). Let Ω ⊂ R (n ≥ 2) be a bounded Lipschitz domain with connected complement. Let α be a continuous transverse unit vector field on ∂Ω. Then there exist a finite number of linearly independent functions f1, . . . , fl ∈ L(∂Ω) so that if g ∈ L(∂Ω) satisfies