If $u$ satisfies the Laplace equation, show that both $xu$ and $yu$ satisfy the biharmonic equation.

Question

If $u$ satisfies the Laplace equation $nabla^2uequiv u_{xx}+u_{yy}=0$, show that both $xu$ and $yu$ satisfy the biharmonic equation$$nabla^4begin{pmatrix}xu\yuend{pmatrix}=0$$but $xu$ and $yu$ will not satisfy the Laplace equation.

So, I don't really understand what that parenthesis next to the biharmonic equation is. Is it a combination of some sorts? or a vector? Either way, how do I go about operating it in the biharmonic equation? Any help/idea to get started with this problem is greatly appreciated.

Sharik · Answer

For the first equation, by direct computations we obtain:
$$
nabla^4(xu)=partial_x^4(xu)+2partial_x^2partial_y^2(xu)+partial_y^4(xu)=xpartial_x^2big(partial_x^2u+partial_y^2ubig)+xpartial_y^2big(partial_x^2u+partial_y^2ubig)+4partial_xbig(partial_x^2u+partial_y^2ubig)=0,
$$
where in the last equality we have used the fact that $partial_x^2u+partial_y^2u=0$. The second equations follows exactly the same computations.

If $u$ satisfies the Laplace equation, show that both $xu$ and $yu$ satisfy the biharmonic equation.

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