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 $x \rightarrow x_0 \in \partial \Omega$.




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