Some pathological examples of solutions to a Beltrami equation

Abstract

We construct an example of a bounded solution to a uniformly elliptic Beltrami equation that has no nontangential limit values almost everywhere on the boundary of the unit disk and also an example of a solution to such an equation that is not identically zero and has zero nontangential limit values almost everywhere on the boundary of the unit disk. These examples show that, in the general case of the Hardy spaces of solutions to a uniformly elliptic Beltrami equation (and to more general noncanonical first-order elliptic systems), the usual statement of boundary value problems used for holomorphic and generalized analytic functions is ill-posed.