Let $u: \Omega \longrightarrow \mathbb{R}$ be a harmonic function and $f:\Omega_1 \longrightarrow \Omega$ a diffeomorphism. Is $uf$ (composition) harmonic in $\Omega_1$? I know I could compute the laplacian, but is there any other method to do it?.
Let $u: \Omega \longrightarrow \mathbb{R}$ be a harmonic function and $f:\Omega_1 \longrightarrow \Omega$ a diffeomorphism. Is $uf$ (composition) harmonic in $\Omega_1$? I know I could compute the laplacian, but is there any other method to do it?.
You need to apply the chain rule here. Define $w = u\circ f$. You need to show that $Lw(\bold{x}) =$ where $L$ is the Laplacian operator for each $\bold{x}\in \Omega_1$. To prove this let $f (u,v) = (g(u,v),h(u,v))$. Thus, write $w=u(x,y)$ where $x=g(u,v)$ and $y=h(u,v)$. Now by the chain rule $w_u = u_x \cdot x_u + u_y \cdot y_u$ and $w_u = u_x\cdot x_v + u_y\cdot y_v$. Now go on and continue to $w_{uu}$ and $w_{vv}$ and add them up.