On the criticality and the principal eigenvalue of almost periodic elliptic operators

We review the notion and the properties of the generalised \pe\ for elliptic operators in unbounded domains, and we relate it with the criticality theory. We focus on operators with almost periodic coefficients. We present a Liouville-type result in dimension $N\leq2$. Next, we show with a counter-example that criticality is not equivalent to the existence of an almost periodic principal eigenvalue, even for self-adjoint operators. Finally, we exhibit an almost periodic operator which is subcritical but which admits a critical limit operator. This is a manifestation of the instability character of the criticality property in the almost periodic setting.

💡 Research Summary

The paper investigates the relationship between the generalized principal eigenvalue and the notion of criticality for second‑order elliptic operators with almost‑periodic (a.p.) coefficients defined on the whole space ℝⁿ. In bounded domains the principal eigenvalue is classically obtained via the Krein‑Rutman theorem, guaranteeing existence, uniqueness and simplicity of a positive eigenfunction. On unbounded domains the resolvent is not compact, so this theory breaks down and the principal eigenvalue must be defined in a generalized sense.

The authors adopt the Berestycki–Nirenberg–Varadhan definition
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