A harmonic function bounded from below is constant

Question

I am learning PDE on myself as a beginner. It takes me like several hours to finally think out this proof. However, I feel something not right about my proof, especially choosing "$R$" part, it looks not sufficient at all.

My attempt:

$w(x)=u(x)-M\ge0$. Because $w(x)$ is bounded below by $0$, so we can find a point $x_0$ such that $w(x_0)=\min(w(x))\ge 0$. Suppose $w(x)$ is not bounded above, then for $N>10w(x_0)+10$, exists some $x_1$ such that $x_1 \ge N$. Consider the ball $B_R(x_0)$, where $R=\operatorname{dist}(x_1,x_0)$. Since $w(x)$ is harmonic, it satisfies the Mean Value equality. But $x_0$ is the minimum of $w(x)$ and $w(x)\ge 0$ and $N>10w(x_0)+10$, so it cannot satisfies the Mean Value equality on the ball $B_R(x_0)$. So $w(x)$ is bounded both below and above and harmonic on $R^n$. By Liouville's theorem $w(x)$ is constant, so $u(x)$ is constant.

Can anyone help me look at this? It is really important for me. Thanks so much!

Answer 1

Thm: Let $u \ge 0$ be harmonic, let $R > 0$, and assume $u \in C^1(\overline{B(x_0,R)})$. Then $$|\nabla u(x_0)| \le \frac{N}{R}u(x_0).$$

Proof: By assumption $u$ is harmonic, then $\Delta u = 0$. Differentiating both sides we get $0 = \frac{\partial \Delta u}{\partial x_i} = \Delta(\frac{\partial u}{\partial x_i})$, then $\frac{\partial u}{\partial x_i}$ is harmonic for $i = 1, \dots , N$. In particular the partial derivatives of $u$ satisfy the mean value property. This, together with the divergence theorem, gives the following computation $$\frac{\partial u}{\partial x_i}(x_0) = \frac{1}{\lambda(B(x_0,R))}\int_{B(x_0,R)}\frac{\partial u}{\partial x_i}(x)\, dx = \frac{1}{\lambda(B(x_0,R))}\int_{\partial B(x_0,R)}u(x)v_i\, dS(x)$$ Here $v_i(x)$ is the $i$-th component of the normal unit vector at $x$ pointing outward and $S$ is the Hausdorff measure. Take the absolute value in the previous equality and estimate as follows (recall also that $u$ is nonnegative by assumption!): $$\Big|\frac{\partial u}{\partial x_i}(x_0)\Big| \le \frac{1}{\lambda(B(x_0,R))}\int_{\partial B(x_0,R)}|u(x)|\, dS(x) = \frac{1}{\lambda(B(x_0,R))}\int_{\partial B(x_0,R)}u(x)\, dS(x).$$ One last application of the Mean Value Property shows $$\Big|\frac{\partial u}{\partial x_i}(x_0)\Big| \le \frac{N}{R\lambda(\partial B(x_0,R))}\int_{\partial B(x_0,R)}u(x)\, dS(x) = \frac{N}{R}u(x_0).$$ This concludes the proof of the theorem.

Finally, to get the desired result notice that if $u$ is harmonic in $\mathbb{R}^N$ we are allowed to send $R \to \infty$. This shows that every partial derivative equals zero, from which we deduce that $u$ is constant.

:D