We study the ratio of harmonic functions u,v which have the same zero set Z in the unit ball B⊂ Rn. The ratio f= u/ v can be extended to a real analytic nowhere vanishing function in B. We prove the Harnack inequality and the gradient estimate for such ratios in any dimension: for a given compact set K⊂ B we show that sup K| f| ≤ C1inf K| f| and sup K|∇ f| ≤ C2inf K|f| , where C1 and C2 depend on K and Z only. In dimension two we specify the dependence of the constants on Z in these inequalities by showing that only the number of nodal domains of u, i.e. the number of connected components of B\ Z, plays a role.
All Science Journal Classification (ASJC) codes
- Geometry and Topology
- Divisors of harmonic functions
- Gradient estimates
- Harmonic functions
- Nodal set
- Łojasiewicz exponent