Any inequality that relates the integrals over the boundary and over the domain?

Question

Let $\Omega$ be a non-empty bounded open subset of $R^n$. Let $x_0\in\Omega$ and $r>0$. I am trying to see if there is any inequality that can be used to relate the following two integrals: $$\int_{\partial\Omega\cap B_r(x_0)}u(s)ds$$ and $$\int_{\Omega\cap B_r(x_0)}u(x)dx$$ where $x\in R^n$, $s\in R^{n-1}$ and $u(x)$ is defined up to the boundary of $\Omega$. Basically, I want to compare the integral over some part of the boundary and the integral over some part of the domain. If anyone knows anything that might be helpful, please let me know.

Answer 1

I don't think you can in general expect an inequality like $$\int_{\Omega\cap B_r(x_0)}u(s)\, ds\leq C \int_{\partial\Omega\cap B_r(x_0)}u(x)\, dx,$$ since you may choose $u$ to have compact support in $\Omega\cap B_r(x_0)$ and scale it up as big as you want. For a concrete example of such a function see Friedrich's Mollifier (https://en.wikipedia.org/wiki/Mollifier#Concrete_example).

What you can do is compare the integral over the boundary to the integral of a derivative of $u$. For this you may employ the divergence theorem (see https://en.wikipedia.org/wiki/Divergence_theorem#Generalizations).