# Weak Solutions to PDES

I am working through some practice problems for my PDE class and I came across the following:

Let $U\in \mathbb{R}^n$ be a smooth, bounded, connected open set. Let $\Gamma_1$, $\Gamma_2$, be two disjoint subsets of $\partial U$ of positive $(n-1)$-dimensional measure such that $\Gamma_1\cup\Gamma_2=\partial U$. Define the set

$$H=\{\phi\in C^{\infty}(\overline{U}) : \text{dist}(\text{spt}\phi,\Gamma_1)>0$$ and define the Hilbert space $\check{H}^1(U)$ as the closure of H in the standard $H^1(U)$ norm.

a) Prove the following Poincare inequality for functions in $\check{H}^1(U)$: there exists a constant $C>0$ such that

$$\int_U u^2dx\le C\int_U|Du|^2dx$$