Nodal Sets of Harmonic Functions

Abstract

If u is a homogenous harmonic polynomial, its frequency is exactly its degree. In general, the frequency controls the growth of the harmonic functions. In the present lecture notes, we shall discuss how the frequency controls the size of nodal sets. As we know, plane harmonic functions are simply the real parts of holomorphic functions in the complex plane. This simple identification is not enjoyed by harmonic functions in higher dimensional spaces. However, harmonic functions in Rn, as analytic functions with interior estimates on derivatives, can be extended as holomorphic functions in Cn. It turns out that such an extension is extremely important in the discussion of nodal sets of harmonic functions. This is because the nodal sets of (general analytic) functions in Rn are not stable in the sense that a simple perturbation may change the structure of nodal sets. In particular, the dimension of nodal sets may change by perturbations. However, this never happens for holomorphic functions in Cn. In order to discuss nodal sets of harmonic functions, we shall first discuss complex nodal sets of the holomorphic extensions. It is not surprising that complex analysis plays an important role in our study. For example, we shall use repeatedly Rouche Theorem in C and in C2. It asserts that if an equidimensional holomorphic map has isolated zeroes then its holomorphic perturbation enjoys the same property and the number of isolated zeroes is preserved. Another property we shall use is the behavior of polynomials away from their zeroes. For a suitably normalized polynomial, a positive lower bound can be established for the modulus of the polynomial outside some balls around its zeroes. The foundation of our discussion is a monotonicity formula for harmonic functions. Corollaries of such a monotonicity include the doubling condition of L2-integrals and finite vanishing order. In fact, an integral quantity of harmonic functions in the unit ball controls the vanishing order of harmonic functions inside the ball.