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Math Help - Harmonic functions

  1. #1
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    Harmonic functions

    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?.
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  2. #2
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    Quote Originally Posted by Jose27 View Post
    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.
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