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