Math 6410 • ODE

Laplace’s Equation. Let \(\varphi (y_1,y_2)\in C_0(\mathbb {R}^2)\) be continuous with compact support. Let \[ u(x)=\frac {x_3}{2\pi }\int _{\mathbb {R}^2} \frac {\varphi (y_1,y_2)\,dy_1\,dy_2} {\bigl [(y_1-x_1)^2+(y_2-x_2)^2+x_3^2\bigr ]^{3/2}}, \] for \(x\in \mathbb {R}^3_+=\{(x_1,x_2,x_3)\in \mathbb {R}^3:x_3>0\}\), the open upper three space. Show that \(u\in C^2(\mathbb {R}^3_+)\cap C(\overline {\mathbb {R}^3_+})\) and solves the boundary value problem for Laplace’s equation \[ \begin {cases} \Delta u=0, & \text {if }x\in \mathbb {R}^3_+,\\ u(x_1,x_2,0)=\varphi (x_1,x_2), & \text {if }(x_1,x_2)\in \mathbb {R}^2. \end {cases} \] [You may assume \[ \int _{\mathbb {R}^2}\frac {x_3\,dy_1\,dy_2} {\bigl [(y_1-x_1)^2+(y_2-x_2)^2+x_3^2\bigr ]^{3/2}}=2\pi . \] ]