Suppose $\Omega \subset {\mathbb R}^n$ be open and bounded with smooth boundary. Let $t_0 > 0$ be small enough so that for every $x \in \partial \Omega$, there exists a unique $y \in \Omega$ with $|x-y|=t_0=dist(y,\partial \Omega)$. Let $h \in L^1(\Omega)$ with $h \geq 0$. For $x \in \Omega$, let $n(x)$ be the inward normal vector to $\partial \Omega$. Let $\Omega_t = \{x\in \Omega \mid dist(x,\partial \Omega) >t\}$. I would like to show that there exists $C = C(\Omega)$ such that if $t \in (0,t_0)$, then $ \int_{\partial \Omega} \int_0^t h(x+sn(x))\,ds \leq C\int_{\Omega \setminus \Omega_t} h\,dx.$ This is not a homework problem.
elementary differential geometry question
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1$\Omega$ is open and bounded, and therefore has compact closure. – 2011-12-31
2 Answers
Fix $t
I'll define $C$ at the end. Define $\displaystyle f(t)=\int_{\partial \Omega}\int_{0}^{t}h(x+sn(x))ds-C\int_{\Omega- \Omega_t}hdx$. Then, f'(t)=\int_{\partial \Omega}h(x+tn(x))dx'-C\int_{\partial \Omega_t}hdx'. Here, dx' is surface area measure for the boundary. Since g(t)=\frac{\int_{\partial \Omega}h(x+tn(x))dx'}{\int_{\partial \Omega_t}hdx'} is continues, as you can see $lim_{t\to 0}g(t)=1$, therefore $g$ achieves its maximum, $M$, on any compact interval, for example, $[0,t_0].$ Define $C:=M$. Therefore $f$ is non-increasing so we must have $f(t)\leq f(0)=0.$
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0Tim: I am not sure this works, since I am assuming only $h \in L^1(\Omega)$, so $h|_{\partial \Omega_t}$ may not be well-defined. – 2011-12-31