I am trying to derive a monotonicity formula for a certain Dirichlet type energy, say for simplicity, an energy of the from

$$\int_{B_r} (A\nabla e(u),\nabla e(u)),$$

for constant symmetric 4th order tensor $A$ and vector valued $u:\mathbf{R}^n \to \mathbf{R}^n$. Here, $e$ is just the matrix symmetrization $e(\xi) = (\xi + \xi^T)/2$.

A monotonicity formula is understood as proving a function of the form

$$f(r) = \frac{1}{r^\alpha}\int_r (A\nabla e(u),\nabla e(u))$$

to be non-decreasing on an interval $(r/2,r)$ whenever $f(r)$ is sufficiently small.

Several techniques may be employed on a scalar setting, but the vector valued case is rather more complicated. During the process I was suggested the use of Carleman estimates. Unfortunately, I am not familiar with the topic. Does anyone know, or may provide literature on how such estimates may be applied to obtain monotonicity formulas, even for scalar or simpler energies?

Thanks in advance