Regularization

$\displaystyle R_0(u,v)$	$\displaystyle =$	$\displaystyle \Vert u\Vert^2+\Vert v\Vert^2,$	(4.4)
$\displaystyle R_1(u,v)$	$\displaystyle =$	$\displaystyle \Vert\nabla u\Vert^2+\Vert\nabla v\Vert^2=\Vert\partial_x u\Vert^2+\Vert\partial_y u\Vert^2 + \Vert\partial_x v\Vert^2+\Vert\partial_y v\Vert^2,$	(4.5)
$\displaystyle R_{div}(u,v)$	$\displaystyle =$	$\displaystyle \Vert div(u,v)\Vert^2=\Vert\partial_x u+\partial_y v\Vert^2,$	(4.6)
$\displaystyle R_{curl}(u,v)$	$\displaystyle =$	$\displaystyle \Vert curl(u,v)\Vert^2=\Vert\partial_y u-\partial_x v\Vert^2,$	(4.7)
$\displaystyle R_{div/curl}(u,v)$	$\displaystyle =$	$\displaystyle \Vert div(u,v)\Vert^2+\Vert curl(u,v)\Vert^2 = \Vert\partial_x u+\partial_y v\Vert^2+\Vert\partial_y u-\partial_x v\Vert^2,$	(4.8)
$\displaystyle R_{\nabla div}(u,v)$	$\displaystyle =$	$\displaystyle \Vert\nabla div(u,v)\Vert^2=\Vert\partial^2_{xx} u+\partial^2_{xy} v\Vert^2+\Vert\partial^2_{xy} u+\partial^2_{yy} v\Vert^2,$	(4.9)
$\displaystyle R_{\nabla div/\nabla curl}(u,v)$	$\displaystyle =$	$\displaystyle \Vert\nabla div(u,v)\Vert^2+\Vert\nabla curl(u,v)\Vert^2$	(4.10)
	$\displaystyle =$	$\displaystyle \Vert\partial^2_{xx} u+\partial^2_{xy} v\Vert^2+\Vert\partial^2_{... ...xy} u-\partial^2_{xx} v\Vert^2+\Vert\partial^2_{yy} u-\partial^2_{xy} v\Vert^2.$