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It is by now well-known that for a harmonic function $u : B_1^n(0) \to \mathbb{R}$, the ratio $$ N(r) := \frac{r\int_{B_r(0)}|\nabla u|^2}{\int_{\partial B_r(0)} u^2} $$ is a non-decreasing function of $r$.

This and analogous expressions have been used in many different fundamental works on elliptic equations, but to me, it has always seemed like a mystery and as a result I have trouble even remembering this expression and usually have to look it up.

What's the intuition behind the definition of $N$? What does this ratio of energy in the ball and L^2 norm on the sphere even represent?

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Observe that if $u$ is homogeneous of degree $\alpha$, then $N(r) \equiv \alpha$. (After integrating by parts, the numerator becomes $r \int_{\partial B_r} u u_{\nu} = \alpha \int_{\partial B_r} u^2$.) In addition, harmonic functions that are homogeneous of degree $\alpha$ tend to oscillate over distance $\sim \alpha^{-1}$ on $\partial B_1$ (think $r^{\alpha}\cos(\alpha\theta)$ in $\mathbb{R}^2$). Roughly, the monotonicity of $N$ says that the dominant frequency represented in $u$ at scale $r$ is increasing.

Some intuition comes from the expansion of $u$ in harmonic polynomials. For $r$ small, $N(r)$ only sees the lowest frequency (the first term in the expansion dominates), and as $r$ grows, the higher frequencies become more and more relevant. One proof of monotonicity comes directly from the expansion. In the context of unique continuation, monotonicity says "if the dominant frequency at $r = 0$ is $\infty$, then the same is true at all larger scales, so $u$ has to vanish in $B_1$."

Another perspective (which I find more general and illuminating) is variational. Note that $N$ is invariant under transformations that preserve harmonicity (that is, under $w \rightarrow \tilde{w} := Aw(rx)$ we have $N_w(r) = N_{\tilde{w}}(1)$). In particular, $N_w$ is constant when $w$ is homogeneous of any degree. To show monotonicity of $N_u$ when $u$ is harmonic, a useful idea is to compare $u$ with the function $w$ that is homogeneous of degree $\gamma:= N_u(1)$ (the dominant frequency of $u$ in $B_1$) and has the same boundary values as $u$ on $\partial B_1$. Roughly, since $u$ is a minimizer of the Dirichlet energy, $N_u$ has to grow faster than $N_w$ (which is constant) at $r = 1$. More precisely, the condition $N_u'(1) \geq 0$ becomes $$\int_{\partial B_1} (|\nabla u|^2 - 2\gamma u u_{\nu}) - (n-2) \int_{B_1} |\nabla u|^2 \geq 0.$$ The second term is larger than the corresponding quantity for $w$ by energy minimality. The first term is $$\int_{\partial B_1} |\nabla_{\partial B_1} w|^2 + (u_{\nu} - \gamma u)^2 - \gamma^2w^2 = \int_{\partial B_1} |\nabla w|^2 - 2\gamma ww_{\nu} + (u_{\nu} - \gamma u)^2$$ which is larger than the corresponding quantity for $w$, completing the proof.

In my view, monotonicity formulae in variational problems (free boundary problems, minimal surfaces, etc.) come from playing with quantities that are invariant under the natural scaling of the problem, and involve the energy. (This is the way I remember the Almgren quantity). The monotonicity of these quantities on minimizers comes from comparison with the scaling-invariant objects with the same boundary data. The clearest example is minimal surfaces: if $0$ is in a surface of dimension $n-1$, the quantity ($A(r)$:= area in $B_r$)/$r^{n-1}$ is dilation-invariant and constant on cones. The monotonicity on a minimal surface comes from comparing with the cone over the intersection of the surface with $\partial B_1$. Indeed, the area of the cone in $B_r$ grows more slowly at $r = 1$ since it crosses $\partial B_1$ orthogonally, and its area in $B_1$ is larger than that of the minimal surface, so $A'(1) - (n-1)A(1)$ for the minimal surface is larger than the corresponding quantity for the cone, which vanishes.

I hope this helps!

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  • $\begingroup$ Nice answer! To expand slightly on the conclusion of this variational proof, note that $N_w(1)\ge N_u(1)=\gamma$ (again because $u$ minimizes the energy), so the condition that $N_w'(1)=0$ gives $0=\int_{\partial B_1}(|\nabla w|^2-2N_w(1)ww_\nu)-(n-2)\int_{B_1}|\nabla w|^2\le\int_{\partial B_1}(|\nabla w|^2-2\gamma ww_\nu)-(n-2)\int_{B_1}|\nabla w|^2$ (as $ww_\nu=\gamma w^2\ge 0$); the last quantity bounds from below $N_u'(1)$, so we get indeed $N_u'(1)\ge 0$. $\endgroup$ – Mizar Jul 12 '19 at 13:51

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