# behaviour of first eigenfunction near the boundary

Consider $\Omega$ a smooth bounded domain in $R^N$ and suppose $\phi_1(x)>0$ is the first eigenfunction of $-\Delta$ in $H_0^1(\Omega)$ normalized however one chooses.

My interest is in how $\nabla \phi_1(x)$ behaves for $x$ near $\partial \Omega$. So lets set $\delta(x)=dist(x, \partial \Omega)$ denote the Euclidean distance from $x$ to $\partial \Omega$ and note that near $\partial \Omega$ that $\delta(x)$ is smooth. So lets define $\delta_0(x)$ to be a smooth version of $\delta(x)$ which is equal to $\delta(x)$ near $\partial \Omega$. So we can define the vector field $b(x)$ as $$\nabla \phi_1(x)= | \nabla \phi_1(x)| \nabla \delta_0(x) + b(x).$$ So note that $b(x)$ is smooth away from critical points of $\phi_1(x)$ (in particular near $\partial \Omega$) and $b=0$ on $\partial \Omega$.

I am interested in the behaviour of $b$ near $\partial \Omega$. In particular I am interested in finding some upper bounds on how quickly $b(x) \cdot \nabla \delta_0(x) \rightarrow 0$ as $x \rightarrow x_0 \in \partial \Omega$.

thanks