On critical points of harmonic functions in the plane.

We show that if u is a p harmonic function, 1 < p < ∞, in the unit disk and equal to a polynomial P of positive degree on the boundary of this disk, then ∇u has at most degP − 1 zeros in the unit disk. In this note we prove the following theorem. Theorem 1 Given p, 1 < p <∞, let u be a real valued weak solution to ∇ · (|∇u|∇u) = 0 (*) in D = {(x1, x2) : x1 + x 2 2 < 1} ⊂ R 2 with u = P on ∂ D where P is a real polynomial in x1, x2 of degree m ≥ 1. Then ∇u has at most m− 1 zeros in D counted according to multiplicity. In (*), ∇· denotes the divergence operator while ∇u denotes the gradient of u. The above theorem answers a question in the affirmative first posed by D. Khavinson in connection with determining the extremal functions for certain linear functionals in the Bergman space of p th power integrable analytic functions on D, 1 < p < ∞. We note that the differential operator in (*) is often called the p Laplacian and it is well known (see [GT]) that solutions to this equation are infinitely differentiable (in fact real analytic) at each point where ∇u 6= 0 while (*) is degenerate elliptic at each point where ∇u = 0. The above theorem appears to be the first of its kind to establish independent of p and the structure constants for the p Laplacian, a bound (m 1) for the number of points in D where (*) degenerates. Because of this independence we conjecture that our theorem also remains true for p = ∞ and the so called ∞ Laplacian (see [BBM] or [J] for definitions). Finally we remark that in [Al] a result, in the same spirit as ours, is obtained for smooth linear equations whose matrix of coefficients has determinant one. ∗1991 Mathematics Subject Classifications: 35J70, 35B05.