$L^{\infty}$ norms of Robin eigenfunctions

Question

Let $\Omega\subseteq \mathbb{R}^2$ be a bounded Lipschitz planar domain. Suppose that $u$ is a Robin eigenfunction of the (negative) Laplacian on $\Omega$: $-\Delta u=\lambda u$ with $\partial_{\nu}u+\alpha u=0$ on $\partial \Omega$ where $\alpha$ is a piecewise constant non-negative function of the boundary. If we suppose that $u$ is $L^2$-normalized ($\int_{\Omega}u^2=1$), are there any estimates on the $L^{\infty}$ norm of $u$? There are many such results in the literature for Dirichlet eigenfunctions, but I cannot seem to find any for the Robin case. I am also particularly interested in estimates on the $L^{\infty}$ norm of $u$ restricted to the boundary.