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- Homework Statement
- please see below

- Relevant Equations
- please see below

(1) From "Radial solutions to Laplace's equation", we know that

$$

\Delta u(x) = v(r)''+\frac{n-1}{r}v(r)'

$$

we re-write the PDE

$$

- \Delta u+m^2u=0

$$

in terms of ##v(r)##

\begin{equation}

- v(r)''-\frac{n-1}{r}v(r)'+m^2v(r)=0

\end{equation}

to give a linear second order ODE with non-constant coefficients.

(2) To check that the function

$$v(r)=\frac{e^{-mr}}{r}$$

is a solution of our ODE, we compute

$$ v'(r)=\frac{-me^{-mr}r-e^{-mr}}{r^2}$$

and

$$v''(r)=\frac{e^{-mr}(m^2r^2+2(mr+1))}{r^3}$$

and perform pluginology into equation (1)

$$ \frac{-e^{-mr}(m^2r^2+2(mr+1))}{r^3}-\Big(\frac{n-1}{r}\Big)\Big(\frac{-me^{-mr}r-e^{-mr}}{r^2}\Big)+m^2\frac{e^{-mr}}{r}$$

set ##n=3## and simplify

$$\Rightarrow -m^2r^2-2mr-2+2mr+2+m^2r^2=0$$

(3) To define the fundamental solution to the PDE

$$

- \Delta u+m^2u=f

$$

using the function ##\Phi:R^n\backslash\{0\} \rightarrow R## given by

$$

\Phi(x)=\frac{1}{4\pi}\frac{e^{-m||x||}}{||x||}

$$

we modify theorem 3.18

**Proof of Theorem ##3.18##:**

We recall

\begin{equation}

\tilde u(x)=\int_{R^n}\Phi(x-y)[f(y)-m^2u(y)]dy

\end{equation}

\begin{equation}=\int_{R^n}\Phi(y)[f(x-y)-m^2u(x-y)]dy\end{equation}

To see ##u(x)\in C^2(R)## we differentiate expression (3) twice

$$\partial_{x_i}\partial_{x_j}u(x)=\int_{R^n}\Phi(y)(\partial_{x_i}\partial_{x_j})[f(x-y)-m^2u(x-y)]$$

to give \begin{equation}\Delta_xu(x)=\sum^n_{i=1}\partial^2_{x_i}u(x)=\int_{R^n }\Phi(y)\Delta_x[f(x-y)-m^2u(x-y)]dy\end{equation}

To show expression (4) equals ##f-m^2u##, we deconstruct the integral into two parts

$$\Rightarrow \int_{B_\epsilon(0)} \Phi(y)\Delta_x[f(x-y)-m^2u(x-y)]dy+\int_{R^n\backslash B_\epsilon(0)} \Phi(y)\Delta_x[f(x-y)-m^2u(x-y)]dy$$

We have the following estimate for the first integral

\begin{equation}\Big|\int_{B_\epsilon(0)} \Phi(y)\Delta_x[f(x-y)-m^2u(x-y)]\Big|dy\end{equation} \begin{equation}\leq \int_{B_\epsilon(0)} \Phi(y)\Big|\Delta_x[f(x-y)-m^2u(x-y)]\Big| dy\end{equation} \begin{equation} \leq sup_{y\in B_\epsilon(0)}\Big|\Delta_x[f(x-y)-m^2u(x-y)] \Big|\times \int_{B_\epsilon(0)}\Phi(y)dy\end{equation}

where after a computation in polar coordinates, we observe expression (7) is the size of the largest second derivative times ##-\epsilon ^2##.

$$\Rightarrow C(n)||D^2[f(x-y)-m^2u(x-y)]||_{L^\infty(R)}\times -\epsilon^2\rightarrow 0$$

We let ##\epsilon\rightarrow 0## and see that the term goes to ##0##.

Next we evaluate the integral in expression (6)

$$\int_{R^n\backslash B_\epsilon(0)} \Phi(y)\Delta_x[f(x-y)-m^2u(x-y)]dy$$$$\Rightarrow \int_{R^n\backslash B_\epsilon(0)} \Phi(y)\Delta_y[f(x-y)-m^2u(x-y)]dy$$

Using integration by parts gives

$$\Rightarrow \int_{\partial\epsilon(0)} \Phi(y) <D_y [f(x-y)-m^2u(x-y)], v(y)> ds(y) $$

\begin{equation} -\int_{R^n\backslash B_\epsilon(0)} <D_y\Phi(y), D_y [f(x-y)-m^2u(x-y)]> dy\end{equation}

We have the following estimate for expression (5)

\begin{equation}\int_{\partial B_\epsilon(0)} \Phi(y) <D_y [f(x-y)-m^2u(x-y)], v(y)> ds(y)\end{equation} \begin{equation}\leq C(n)\Big|\Big|D[f(x-y)-m^2u(x-y)]\Big|\Big|_{L^\infty(R)}\times \epsilon \end{equation}

we send ##\epsilon\rightarrow 0## and see that expression (10) goes to ##0##.

Next, we compute expression (9) by integration by parts

\begin{equation}\int_{R^n\backslash B_\epsilon(0)} \Delta \Phi(y) [f(x-y)-m^2u(x-y)]> dy\end{equation} \begin{equation}-\int_{\partial B_\epsilon(0)} <D_y\Phi(y), v(y)> [f(x-y)-m^2u(x-y)] ds(y)\end{equation}

and see expression (11) goes to zero on ##R\backslash B_\epsilon(0)##. For expression (12), we compute

$$v(y)=-\frac{y}{\epsilon}$$

$$\Phi(y)=\frac{1}{4\pi}\frac{exp(-m||x||)}{||x||}$$

$$D\Phi(y)=-\frac{y}{4\pi \epsilon^n}$$ to give

$$=-\int_{\partial B_\epsilon(0)} \frac{1}{\epsilon^n}\frac{1}{\epsilon}<y,y> [f(x-y)-m^2u(x-y)] dy$$

$$=\frac{1}{nw_ne^{n-1}}\int_{\partial B_\epsilon(0)}[f(x-y)-m^2u(x-y)] dy$$

where we let ##\epsilon\rightarrow 0## and finish our proof by stating ## -\Delta u = f-m^2u##