2
$\begingroup$

Given a $C^2$ convex function $f$ and $u$ a harmonic function in an open subset of $\mathbb{R^2}$, how can I show that $f(u)$ is sub-harmonic?

  • 0
    How $u$ is important in your question? Are you just asking that $C^2$ convex function is subharmonic?2012-05-25
  • 0
    @Ilya, thank you. I meant $f(u)$. I want to show $-\Delta f(u) \leq 0$.2012-05-25
  • 0
    I hope, you meant $-\Delta f\leq 0$2012-05-25
  • 3
    use Jensen's inequality to show averages over balls smaller than value at center, (or bigger, whichever is right).2012-05-25

1 Answers 1

1

You can also simply compute derivatives since everything is differentiable enough: $$ (f(u))_x = f'\cdot u_x\Rightarrow (f(u))_{xx} = f'' u_x^2+f'u_{xx} $$ and clearly, $(f(u))_{yy} = f'' u_y^2+f' u_{yy}$. As a result, $$ \Delta f(u) = f'' (u_x^2+u_y^2)\geq 0 $$ since $f$ is a convex $C^2$ function.

  • 0
    just that simple! Thanks very much.2012-05-25