I am trying to derive a monotonicity formula for a certain Dirichlet critical point (or even maybe a minimizer) of an energy of the type, say for simplicity, an energy of the from

$$\int_{B_r} (Ae(u),e(u)),$$

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

A monotonicity formula is understood as proving for a certain $\alpha \ge 0$, the function

$$f(r) = \frac{1}{r^\alpha}\int_r (Ae(u),e(u)), \tag{*}$$

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

As requested, I illustrate an easy example for the scalar case, say the laplacian. Let $u:\mathbf R^{n} \to \mathbf R$ solving, $$-\Delta u = 0.$$

Observe, for $\alpha = n-1$, the derivative $f$ is given by $$\frac{-(n-1)}{r}\cdot\frac{1}{r^{n-1}}\int_{B_r} |\nabla u |^2 + \frac{1}{r^{n-1}} \int_{\partial B_r} |\nabla u |^2,$$ the desired monotonicity then will follow from the differential inequality (under a proper re-scaling)

$$\int_{B_1} |\nabla u |^2 \le \frac{1}{n-1} \int_{\partial B_1} |\nabla u |^2. \tag{**}$$

We consider $\{\phi_i\}$ an orthonormal basis of $L^2(\partial B_1)$ solutions to $(\lambda_1 \le \lambda_2 \le ...$ eigenvaules of the Laplacian in $\partial B_1$), $$-\Delta \phi_i = \lambda_i\phi \quad \text{on} \quad \partial B_1.$$ If $$u|_{\partial B_1}(r,\omega) = \sum a_i\phi_i(\omega),$$ we find after some easy calculations $$u(r,\omega) = \sum a_ir^{\alpha_i}\phi_i(\omega); \qquad \alpha_i(\alpha_i + (n-2)) = \lambda_i.$$ Hence, $$\int_{\partial B_1} |\nabla u |^2 = \int_{\partial B_1} |\partial_r u|^2 \left(= \sum \alpha_i^2a_i^2\right) + \sum\lambda_ia_i^2 \ge \sum (\alpha_i^2 + \lambda_i)a_i^2.$$ One uses that $\lambda_i \ge n-1$ to justify $\alpha_i \ge 1$ and from the latter conclude $$\int_{\partial B_1} |\nabla u |^2 \ge n\sum \alpha_ia_i^2 = \int_{B_1} |\nabla u |^2 .$$ As observation, $\alpha \le n-2$ is trivial, as re-scaling gives the answer. As well, one may prove it for $\alpha = n$, as it sees from the last inequality but for elliptic operators with non continuous coefficients the technique will only allow up to $n-1$.