What does $f\in L^p(\partial \Omega )$ mean

Question

Let $\Omega $ a bounded smooth surface with boundary of $\mathbb R^n$. What does $$f|_{\partial \Omega }\in L^p(\partial \Omega )\ \ ?$$ Is it $$\int_{\partial \Omega }|f|_{\partial \Omega }|^p<\infty \ \ ?$$ To be honnest, I don't really understand the meaning of $$\int_{\partial \Omega }|f|_{\partial \Omega }|^p,$$ because $f:\mathbb R^n\to \mathbb R$ and here we integrate on "$n-1$" value.

Answer 1

Here thing are different. Do you know trace operator ? $\mathcal C^0(\bar \Omega )$ is dense in $L^p(\Omega )$. Let $(f_n)$ a sequence of $\mathcal C^0(\bar \Omega )$ functions that converge to $f$ in $L^p$. Then, by definition $$f|_{\partial \Omega }=\lim_{n\to \infty }f_n|_{\partial \Omega },$$ in $L^p(\partial \Omega )$ sense, that mean $$\lim_{n\to \infty }\int_{\partial \Omega }|f-f_n|=0.$$ For your second question, notice that $\partial \Omega $ is a subset of $\mathbb R^n$, and thus $\int_{\partial \Omega }f|_{\partial \Omega }$ make sense.