Absence of $L^p$ spectrum for asymptotically flat diffusions in region with cavities

TL;DR

This paper proves that solutions to certain variable-coefficient elliptic equations in exterior domains with asymptotically flat metrics must be trivial if they belong to specific $L^p$ spaces, extending classical spectral results.

Contribution

It extends Rellich's classical $L^2$ spectral result to variable-coefficient elliptic equations with asymptotically flat metrics, establishing a sharper integrability threshold.

Findings

Solutions in $L^p$ with $0<p<2n/(n-1)$ are trivial.

New monotonicity formulas are developed based on weighted energies.

Results demonstrate a sharper integrability threshold in variable-coefficient settings.

Abstract

We study solutions to variable-coefficient elliptic equations of the form $-\D(A(x) \nabla u) = \kappa u$, $\kappa>0$, in an exterior domain $\Om\subset \Rn$, where $A(x)$ is uniformly elliptic and asymptotically flat. Extending Rellich's classical $L^2$ result for the Laplacian, we show that if $u\in L^p(\Om)$ for some $0<p<\frac{2n}{n-1}$, then $u\equiv 0$. The proof uses new monotonicity formulas based on weighted energies and vector fields adapted to the geometry of $A(x)$. Our results highlight a sharper integrability threshold in the variable-coefficient setting.