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$L_1$-Estimates for Eigenfunctions of the Dirichlet Laplacian

Wednesday 24th June 2015 - 15:00 to 16:00
INI Seminar Room 1
Co-authors: Michiel van den Berg (U Bristol), J\"urgen Voigt (TU Dresden)

For $d \in {\bf N}$ and $\Omega \ne \emptyset$ an open set in ${\bf R}^d$, we consider the eigenfunctions $\Phi$ of the Dirichlet Laplacian $-\Delta_\Omega$ of $\Omega$. We do {\it not} require $\Omega$ to be of finite volume. % 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 the $L_2$-norm of $\Phi$ and suitable spectral data of $-\Delta_\Omega$. The main idea in obtaining such estimates consists in finding a---sufficiently small---subset $\Omega' \subset \Omega$ where $\Phi$ is localized in the sense that $\Phi$ decays exponentially as one moves away from $\Omega'$. 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. \vskip.5em This is joint work with Michiel van den Berg (Bristol) and J\"urgen Voigt (Dresden), with improvements by Hendrik Vogt (Dresden).

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons