$L^{P} ( \Omega ; W^{1,P}(Y))$ can someone define this space. $L^{P}$ and $W^{1,P}$ are sobolve and Lp spaces. How norm can define?

Question

I have this space $L^{P} ( \Omega ; W^{1,P}_{per}(Y))$ can someone define it. As far I understands it takes a function f from $L^{P}(\Omega)$ and maps it to Sobolov space with periodicity $Y$. Or $f : W^{1,P}_{per}(Y) \rightarrow L^{P}(\Omega)$. i am confused here??? here $L^{P}$ and $W^{1,P}$ are sobolov and Lp spaces. Y is the periodicity of domain $\Omega$. How norm can define?

$||f||_{L^{P} ( \Omega ; W^{1,P}_{per}(Y))} = \int_{\Omega} ||f||_{W^{1,P}_{per}(Y)} dx$

Answer 1

Functions in $L^P(\Omega; W_{per}^{1, P}(Y))$ are functions $f: \Omega \rightarrow W_{per}^{1, P}(Y)$ s.t. the norm on the space is well-defined at $f$. A suitable norm is $$ \lVert f \rVert_{L^P(\Omega; W_{per}^{1, P}(Y))} := \left (\int_\Omega \lVert f(x) \rVert_{W_{per}^{1, P}(Y)}^p ~\mathrm{d}x \right)^\frac{1}{p}. $$

Answer 2

A function $f \in L^{p}(\Omega ; W^{1,p}_{per}(Y))$ means

f takes its value from$\Omega$ (which is of the following form $Dom(f):= \Omega \times Y$ in periodic settings) and its answer $f(x) \in W^{1,p}_{per}(Y)$.

So $f(x)$ has the characteristics of $W^{1,p}_{per}(Y)$ this space, which is sobolev and periodic in $Y$.

Thus $||f||_{L^{p}(\Omega ; W^{1,p}_{per} (Y) ) } (X)= ( \int_{\Omega} ||f(x)||^{p}_{W^{1,p}_{per}(Y)} dx )^{\frac{1}{p}} = (\int_{\Omega} \int_{Y} |f(x,y)|^{2} + |\nabla f|^{2} dx dy )^{\frac{1}{p}}$