In geometric measure theory, the monotonicity formula for the area functional is a basic tool upon which many other basic fundamental facts depend. Some of them also follow from a weaker analytic tool, which is the Michael-Simon inequality. For anisotropic integrands (which generalize the area), monotonicity does not hold, while the latter inequality is conjectured to be true (under appropriate assumptions); actually, the latter is more essential to geometric measure theory, in that it turns out to be equivalent to the compactness of the classes of rectifiable and integral varifolds. In this talk we present some partial results, one of which is a slight improvement of a posthumous result of Almgren, namely the validity of this inequality for convex integrands close enough to the area, for surfaces in $R^3$. Our technique relies on a nonlinear inequality bounding the $L^1$-norm of the determinant of a function, from the plane to $2x2$ matrices, with the $L^1$-norms of the divergence of the rows, provided the matrix obeys some pointwise nonlinear constraints. This is joint work with Guido De Philippis (NYU).