Abstract

Co-authors: Michiel van den Berg (U Bristol), J”urgen Voigt (TU Dresden)

For $d in {f N}$ and $Omega e mptyset$ an open set in ${f R}^d$, 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 $Omega’ ubset Omega$ 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.

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This is joint work with Michiel van den Berg (Bristol) and J”urgen Voigt (Dresden), with improvements by Hendrik Vogt (Dresden).