A lower bound for the Laplacian.

Abstract

In my dissertation I study the Dirichlet Laplacian in an unbounded Euclidean domain of dimension n, Rⁿ, and in an unbounded domain in a hyperbolic space of dimension 2, H². The basic problem is to obtain estimates from below for the spectrum of the Laplacian, σ(Δ). In the case of an Euclidean domain the spectrum is non-negative, that is the lower bound of the spectrum, λ, satisfies λ₁ ≥ 0; for a hyperbolic domain λ₁ ≥ 1/4. I establish geometric conditions under which λ₁ > 0 for an Euclidean domain and λ₁ ≥ ¼ for a hyperbolic domain. Polya established that a relevant geometric characterization that gives a lower bound for σ(Δ) is a maximal radius for a ball inscribed in the domain. On the other hand, it is known that a non-trivial lower bound for σ(Δ) cannot be proved using only the assumption that radii of all balls contained in Ω are bounded from above. Rauch's estimates of λ₁ for a domain of crushed ice gives a counterexample. Hayman obtained a lower bound for σ(Δ) in a bounded domain in Rⁿ in terms of the maximal radius of an inscribed ball. To prove this estimate he made certain geometric assumptions about the domain. In my dissertation I introduce a different condition, namely the uniform exterior cone condition. This means that there exists a solid cone of fixed shape such that for any point on the boundary, if this boundary point is the cone's vertex and the outward normal of the domain's differentiable boundary is its axis then this cone remains outside the domain. In my dissertation I prove the following Theorem: If an unbounded domain Q in Rⁿ or Ω' in H² has a differentiable boundary, contains an inscribed ball of maximal radius, and satisfies the uniform exterior cone condition then the spectrum of the Dirichlet Laplacian operating on L² functions defined on these domains is bounded from below by a positive number a in the case of an Euclidean 1 domain, and it is bounded from below by a number a' > 1/4 in the case of a hyperbolic domain. The proof of this theorem uses variational techniques and more or less follows the ideas of Hayman. I use a cocompact discrete subgroup of SL(2,R) to construct a cover of H² in the hyperbolic case.