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 $n-1$.

  • $\begingroup$ Can you indicate some sources for the "several techniques" that one can apply in the scalar case? (I just want to see if an example will jog my memory a bit.) $\endgroup$ – Willie Wong Jun 4 '14 at 11:49
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    $\begingroup$ Also, do you mean $e(\nabla u)$ instead of $\nabla e(u)$? The latter doesn't make sense as $u$ is a vector and $e$ acts on square matrices. Are you also assuming that your $u$ solves some sort of an equation? Or perhaps $A$ has certain ellipticity? For arbitrary $u$ I cannot see any estimates being possible for $f(r)$, even in the scalar case. And in the elliptic case then the monotonicity is trivially true for $\alpha \leq 0$, so perhaps you have a certain rate in mind for your $\alpha$? $\endgroup$ – Willie Wong Jun 4 '14 at 11:52
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    $\begingroup$ You may want to look at this thread, which may tell you whether you will find the estimates useful. Two other side notes: most versions of Carleman estimates I am aware of requires the operator to have smooth coefficients, and off the top of my head I am not 100% sure whether you have a chance with discontinuous coefficients to even use those inequalities. Also, it is notoriously difficult to get Carleman estimates for coupled systems: most proofs I have seen only apply to (effectively) scalar equations. $\endgroup$ – Willie Wong Jun 6 '14 at 7:49
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    $\begingroup$ ...and especially at Nicolas Lerner's lecture notes mentioned in a comment there; Chapter 2 deals with elliptic operators with jumps in the principal coefficient. $\endgroup$ – Christian Clason Jun 6 '14 at 9:20
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    $\begingroup$ Regarding Carleman estimates for coupled systems: This seems to be the main current direction for people working on Carleman estimates, especially in the inverse problem and controllability communities; you might want to look there for ideas (see, e.g., Masahiro Yamamoto's survey paper (Section 7.8) or this paper by Jean-Pierre Raymond). $\endgroup$ – Christian Clason Jun 6 '14 at 9:31

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