3
$\begingroup$

If $A(t)$ denotes for each fixed $t$ a (smooth) surface in $\mathbb{R}^n$, what is the norm on the space $L^2\left(\cup_{t \in [0,T]} A(t)\times \{t\}\right)?$

Is it $\lVert f \rVert^2 = \int_0^T{\lVert f \rVert^2_{L^2(A(t))}}$ (the usual Bochner space thing) or is it $\lVert f \rVert^2 = \int_0^T\int_{A(t)}{f^2(x,t)}\;\mathrm{d}x\;\mathrm{d}t$

If it's the latter can I change the order of integration?

  • 0
    @MatthewDaws yeah you're right! The $A(t)$ are $C^{2,1}$ hypersurfaces, so locally $A(t)$ is the zero level set of a $C^{2,1}$ function $u(x,t)$.2012-06-14

0 Answers 0