A generalized form of anisotropic surface energy for the description of biaxial nematic liquid crystals is proposed, and the effects of weak anchoring on the elastic distortion of the biaxial nematic material on surface grooves is investigated as an extension of our previous work on the effects of strong anchoring [Zhang Z. D., & Ye W. J. (2009). Liq. Cryst., 36, 885888]. With some approximations for the elastic constants, we obtain a more general expression for the additional elastic distortion energy per unit area induced by the minor director . The results show that finite polar anchoring can provide an important contribution to the additional elastic distortion energy, and the additional term reduces to our previous result with strong polar anchoring.