Counterexamples and uniqueness for L^p (∂ Ω) oblique derivative problems

Harmonic functions defined in Lipschitz domains of the plane that have gradient nontangentially in L² and have nonnegative oblique derivative almost everywhere on the boundary with respect to a continuous transverse vector field are shown to be constant. Explicit examples that have almost everywhere vanishing oblique derivative are constructed when L² is replaced by L^p, p < 2. Explicit examples with vanishing oblique derivative are constructed when p ≤ 2 and the continuous vector field is replaced by large perturbations of the normal vector field. Optimal bounds on the perturbation, depending on p ≤ 2 and the Lipschitz constant, are given which imply that only the constant solution has nonnegative oblique derivative almost everywhere. Examples are constructed in higher dimensions and the Fredholm properties of certain nonvariational layer potentials discussed.