Hi everyone. I want to learn LaTeX, so I'm trying to solve (a) problem (using 7.2.7, in a different way that the guy above me). I haven't figured it out fully, and I'm not even sure if this is right. Can someone help?

We begin with equation 7.2.7.

\begin{equation}

u(y)=\int_{\Omega}G(x,y) \Delta u(x) dV + \int_{\Sigma}-u(x) \frac{\partial G}{\partial \nu_x}(x,y) dS + \int_{\Sigma}G(x,y) \frac{\partial u}{\partial \nu}(x)dS

\end{equation}

Let's look at the second term. Let $y$ be at the centre of the sphere, $\Omega$ is a ball, and $\Sigma$ is it's bounding sphere. We can pull $\frac{\partial G}{\partial v_x}$ out of the integral, since when $y$ is at the centre of the sphere, $\frac{\partial G}{\partial \nu_x}(x,y)$ is constant over $\Sigma$. The second term becomes:

\begin{equation}

\frac{1}{\sigma_n r^{n-1}} \int_{\Sigma}u(x)dS

\end{equation}

This is straight out of the textbook. This term represents the average of $u(x,y)$ over the sphere.

All that's left is to prove the remaining terms are negative (or zero) when evaluated at $y$. I'm not sure how to solve this problem for $n<3$, but it is easy for $n\geq3$, because when $n\geq3$, $G(x,y)$ is always negative (see 7.2).

Let's look at the first term. Since $G(x,y)<0$ and $\Delta u(x)\geq0$ on $\Omega$, the integrand of the first term is always non-positive, so the integral is non-positive.

\begin{equation}

\int_{\Omega}G(x,y) \Delta u(x) dV \leq 0

\end{equation}

The third term is easy too. $G(x,y)$ depends on $|x-y|$ only. Thus, if $y$ is at the centre of the sphere, $G(x,y)$ is constant over the sphere $\Sigma$. Pulling it out of the integral, we get

\begin{equation}

G(x,y) \int_{\Sigma} \frac{\partial u}{\partial \nu} (x) dS

\end{equation}

By the divergence theorem, this term is

\begin{equation}

G(x,y) \int_{\Omega} \Delta u(x) dS

\end{equation}

Since $\Delta u(x)$ is non-negative on the domain $\Omega$, so is the integral. Since $G(x,y)$ is always negative at $y$ (at least for $n\geq3$), this term is negative (or zero).

\begin{equation}

\int_{\Sigma}G(x,y) \frac{\partial u}{\partial \nu}(x)dS \leq 0

\end{equation}

Any idea how to approach this problem for lower $n$? If 7.2.7 is true for $n=2$, it seems possible to create a function $u$ such both the first and third term are positive.