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# 7.2.1: Conclusions from the Representation Formula

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Similar to the theory of functions of one complex variable, we obtain here results for harmonic functions from the representation formula, in particular from (7.2.5), (7.2.6). We recall that a function $$u$$ is called harmonic if $$u\in C^2(\Omega)$$ and
$$\triangle u=0$$ in $$\Omega$$.

Proposition 7.1. Assume $$u$$ is harmonic in $$\Omega$$. Then $$u\in C^\infty(\Omega)$$.}

Proof. Let $$\Omega_0\subset\subset\Omega$$ be a domain such that $$y\in\Omega_0$$. It follows from representation formulas (7.2.5), (7.2.6), where $$\Omega:=\Omega_0$$, that $$D^lu(y)$$ exist and are continuous for all $$l$$ since one can change differentiation with integration in right hand sides of the representation formulas.

$$\Box$$

Remark. In fact, a function which is harmonic in $$\Omega$$ is even real analytic in $$\Omega$$, see an exercise.

Proposition 7.2 (Mean value formula for harmonic functions). Assume $$u$$ is harmonic in $$\Omega$$. Then for each $$B_\rho(x)\subset\subset\Omega$$
$$u(x)= \frac{1}{\omega_n\rho^{n-1}}\int_{\partial B_\rho(x)}\ u(y)\ dS_y.$$

Proof. Consider the case $$n\ge3$$. The assertion follows from (7.2.6) where $$\Omega:=B_\rho(x)$$ since $$r=\rho$$ and
\begin{eqnarray*}
\int_{\partial B_\rho(x)}\frac{1}{r^{n-2}}\frac{\partial u}{\partial n_y}\ dS_y&=&\frac{1}{\rho^{n-2}}\int_{\partial B_\rho(x)}\frac{\partial u}{\partial n_y}\ dS_y\\
&=&\frac{1}{\rho^{n-2}}\int_{B_\rho(x)}\ \triangle u\ dy\\
&=&0.
\end{eqnarray*}

$$\Box$$

We recall that a domain $$\Omega\in\mathbb{R}^n$$ is called connected if $$\Omega$$ is not the union of two nonempty open subsets $$\Omega_1$$, $$\Omega_2$$ such that $$\Omega_1\cap\Omega_2=\emptyset$$. A domain in $$\mathbb{R}^n$$ is connected if and only if its path connected.

Proposition 7.3 (Maximum principle). Assume $$u$$ is harmonic in a connected domain and achieves its supremum or infimum in $$\Omega$$. Then $$u\equiv const.$$ in $$\Omega$$.

Proof. Consider the case of the supremum. Let $$x_0\in\Omega$$ such that
$$u(x_0)=\sup_\Omega u(x)=:M.$$
Set
$$\Omega_1:=\{x\in\Omega:\ u(x)=M\}$$ and $$\Omega_2:=\{x\in\Omega:\ u(x)<M\}$$. The set $$\Omega_1$$ is not empty since $$x_0\in\Omega_1$$. The set $$\Omega_2$$ is open since $$u\in C^2(\Omega)$$. Consequently, $$\Omega_2$$ is empty if we can show that $$\Omega_1$$ is open. Let $$\overline{x}\in\Omega_1$$, then there is a $$\rho_0>0$$ such that $$\overline{B_{\rho_0}(\overline{x})}\subset\Omega$$ and $$u(x)=M$$ for all $$x\in B_{\rho_0}(\overline{x})$$. If not, then there exists $$\rho>0$$ and $$\widehat{x}$$ such that
$$|\widehat{x}-\overline{x}|=\rho$$, $$0<\rho<\rho_0$$ and $$u(\widehat{x})<M$$. From the mean value formula, see Proposition 7.2, it follows
$$M=\frac{1}{\omega_n\rho^{n-1}}\int_{\partial B_\rho(\overline{x})}\ u(x)\ dS <\frac{M}{\omega_n\rho^{n-1}}\int_{\partial B_\rho(\overline{x})}\ \ dS=M,$$
which is a contradiction. Thus, the set $$\Omega_2$$ is empty since $$\Omega_1$$ is open.

$$\Box$$

Corollary. Assume $$\Omega$$ is connected and bounded, and $$u\in C^2(\Omega)\cap C(\overline{\Omega})$$ is harmonic in $$\Omega$$. Then $$u$$ achieves its minimum and its maximum on the boundary $$\partial\Omega$$.

Remark. The previous corollary fails if $$\Omega$$ is not bounded as simple counterexamples show.

## Contributors

• Integrated by Justin Marshall.