On the univalent solution of PDE u = f between spherical annuli

It is proved that if u is the solution of PDE u = f , that maps two annuli on the space R3, then the annulus in co-domain cannot be with arbitrary small modulus, providing that the annulus of domain is fixed. Also it is improved the inequality obtained in [D. Kalaj, On the Nitsche conjecture for harmonic mappings in R2 and R3, Israel J. Math. 150 (2005) 241–253] for harmonic functions in R3. Finally it is given the new conjecture for harmonic mappings in the space similar to the conjecture of J.C.C. Nitsche for harmonic mapping in the plane related to the modulus of annuli. © 2006 Elsevier Inc. All rights reserved.