Abstract
Let D be a non-empty open subset of mathbb {R}^{m}, mge 2, with boundary ∂D, with finite Lebesgue measure |D|, and which satisfies a parabolic Harnack principle. Let K be a compact, non-polar subset of D. We obtain the leading asymptotic behaviour as ε↓ 0 of the L^{infty } norm of the torsion function with a Neumann boundary condition on ∂D, and a Dirichlet boundary condition on ∂(εK), in terms of the first eigenvalue of the Laplacian with corresponding boundary conditions. These estimates quantify those of Burdzy, Chen and Marshall who showed that D ∖ K is a non-trap domain.
Highlights
Introduction and Main ResultsLet D be an open, non-empty set in Rm, m ≥ 2, with finite Lebesgue measure |D|, and let K ⊂ D be a compact set with boundary ∂K, and with positive logarithmic capacity if m = 2 or with positive Newtonian capacity cap (K) if m ≥ 3
In Theorem 1 below we quantify this statement in terms of the first eigenvalue λ(K, D) of the Laplacian with boundary conditions Eqs. 1 and 2 in the case where K is scaled down by a factor ε with respect to a fixed point in D. Estimates of this type are well known for the torsion function u for an open set satisfying a 0 Dirichlet boundary condition on ∂
The first inequality in Eq 6 is due to Weinberger [11]
Summary
In Theorem 1 below we quantify this statement in terms of the first eigenvalue λ(K, D) of the Laplacian with boundary conditions Eqs. 1 and 2 in the case where K is scaled down by a factor ε with respect to a fixed point (the origin) in D. Estimates of this type are well known for the torsion function u for an open set satisfying a 0 Dirichlet boundary condition on ∂. For any non-polar compact set K ⊂ D, uK,D It was shown in Theorem 2.5(i) in [4] that if Eq 3 holds, the Neumann Laplacian on D has discrete spectrum. Where μ(D) is the first non-zero eigenvalue of the Neumann Laplacian acting in L2(D), and B is a ball of radius 1 in Rm
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