How do I show that $u_x$, $u_y$, $u_{xx}$, $u_{yy}$ and $u_{xy}$ are also solution for the pde $u_{xx}+u_{yy}=0$?

Question

Title says it all. Let $u$ satisfy the patial differntial equation $u_{xx}+u_{yy}=0$(elliptic in linear 2nd order pde). How do I show that $u_x$, $u_y$, $u_{xx}$, $u_{yy}$ and $u_{xy}$ are also solution? I don't know where to start.

martini · Accepted Answer · 2013-10-14T14:21:22.487

4

Hint: By Schwarz, for example $$ (u_x)_{yy} = u_{xyy} = u_{yyx} = (u_{yy})_x $$

edited Oct 14 '13 at 14:21

answered Oct 14 '13 at 14:08

martini

84,101

1

Typo ... corrected, I referred to the "symmetry of higher derivatives" theorem by Hermann Schwarz. – martini Oct 14 '13 at 14:21

How do I show that $u_x$, $u_y$, $u_{xx}$, $u_{yy}$ and $u_{xy}$ are also solution for the pde $u_{xx}+u_{yy}=0$?

1 Answers1