Ratios of harmonic functions with the same zero set (Q311147)

This paper is concerned with the study of the ratio of harmonic functions having the same zero set. The first result of the paper is as follows: Assume \(u\) and \(v\) are two harmonic functions in the unit disc of \(\mathbb R^2\) such that their zero sets \(Z(u)\) and \(Z(v)\) coincide. Assume further that the number of nodal domains of \(u\) and \(v\) is less than a fixed number \(N\). Then, for any compact subset \(K\) of the disc, there are two positive constants \(C_i=C_i(K,N)\) such that the ratio \(f=\frac{u}{v}\) satisfies \[ \sup_K |f|\leq C_1\inf_K|f| \] and \[ \sup|\nabla f|\leq C_2\inf_K |\nabla f|. \] The second result of the paper provides some estimates for the derivatives of the ratio \(f\). More precisely, let \(H_Z\) denote the class of all harmonic functions in the unit ball \(B\) of \(\mathbb R^n\) whose zero set is \(Z\subset B\). Then, there exist \(A=A(Z)>0\), \(B=B(Z)>0\) and \(R=R(Z)>0\) such that for any \(u,v\in H_Z\) and any multi index \(\alpha\), the ratio \(f=\frac{u}{v}\) satisfies \[ \sup_{B_{1/2}} |D^\alpha f|\leq\alpha ! AR^{|\alpha|}\inf_{B_{1/2}}|f|. \]