The authors consider, in 2 and 3 dimensions, the problem of determining a harmonic function \(\phi\) satisfying a boundary condition \(\partial\phi/\partial n=p\phi\), where \(\partial\phi/\partial n\) is the normal derivative of \(\phi\) and \(p\) is a real constant. They deal especially with the case where the boundary is smooth but highly convoluted. The work is motivated largely by problems occurring in biology. A general method of attacking the problem via conformal techniques is illustrated by its application to certain special domains with corrugated boundaries. An extension of the technique to 3- dimensional problems is given and illustrated.