%0 Journal Article %T L_1-Estimates for Eigenfunctions of the Dirichlet Laplacian %A Michiel van den Berg %A Rainer Hempel %A Juergen Voigt %J Mathematics %D 2013 %I arXiv %X For $d \in \N$ and $\Omega \ne \emptyset$ an open set in $\R^d$, we consider the eigenfunctions $\Phi$ of the Dirichlet Laplacian $-\Delta_\Omega$ of $\Omega$. If $\Phi$ is associated with an eigenvalue below the essential spectrum of $-\Delta_\Omega$ we provide estimates for the $L_1$-norm of $\Phi$ in terms of its $L_2$-norm and spectral data. These $L_1$-estimates are then used in the comparison of the heat content of $\Omega$ at time $t>0$ and the heat trace at times $t' > 0$, where a two-sided estimate is established. We furthermore show that all eigenfunctions of $-\Delta_\Omega$ which are associated with a discrete eigenvalue of $H_\Omega$, belong to $L_1(\Omega)$. %U http://arxiv.org/abs/1308.4788v3