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I don't understand here why: $2(\Delta(u_x)^2+\Delta(u_y)^2) \geq 0$.

Here $\Delta= \nabla^2, \quad u'_x=u_x $ etc

Snickett
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1 Answers1

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Hint: Just apply to $u'_x$ and $u'_y$ what you proved the line above for $u$.

Siminore
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  • Are we allowed to say $u_x$ and $u_y$ are harmonic? If I have that $u_{xx}+u_{yy}=0$ does this imply $u_{xxx}+u_{xyy}=0$? If I just take the partial derivative wrt $x$ of $u_{xx}+u_{yy}=0$ and change the order of differentiation it is clear, but how do I know these derivatives exist? – Snickett May 02 '15 at 09:50
  • It is standard, although not really trivial: http://math.stackexchange.com/questions/55117/smoothness-of-harmonic-functions – Siminore May 02 '15 at 11:23