Isotropic logarithmic domains for nonquadratic BBM rate limits

For 1 < p < 2, we identify the effective domain of the BBM rate functional for zero extensions from a bounded C¹ domain Ω. The domain is characterized by the isotropic condition ∫_{R^d} log(2+|ξ|) |F[V_p(∇u)](ξ)|² dξ < ∞, where F denotes the Fourier transform and V_p(z) = |z|^{p/2−1}z. The underlying comparison holds for every L^p vector field: the logarithmic energy of nonlinear projected increments taken in the projection direction is equivalent, after addition of the L^p term, to the isotropic energy of the transformed field. Angular H¹ regularity and an equatorial logarithmic weight estimate provide the comparison. For the rate (E_0−G_s)/(1−s), where G_s is the rescaled fractional energy and E_0 its local limit, we prove weak L^p lower bounds, constant strong recovery sequences, and strong W_0^{1,p} compactness under an L^p bound. We obtain convergence of constrained minima of the rate energies. Scaling and concentrating sequences describe the correction scale and the effect of reversing the rate's sign.