On Domain Monotonicity for the Principal Eigenvalue of the Laplacian with a Mixed Dirichlet-neumann Boundary Condition

Let Ω ⊂ Rd be a bounded domain with smooth boundary and let A ⊂⊂ Ω be a smooth, compactly embedded subdomain. Consider the operator − 1 2 ∆ in Ω − Ā with the Dirichlet boundary condition at ∂A and the Neumann boundary condition at ∂Ω, and let λ0(Ω, A) > 0 denote its principal eigenvalue. We discuss the question of monotonicity of λ0(Ω, A) in its dependence on the domain Ω. The main point of this note is to suggest an open problem that is in the spirit of Chavel’s question concerning domain monotonicity for the Neumann heat kernal. Let Ω ⊂ R be a bounded domain with smooth boundary and let A ⊂⊂ Ω be a smooth, compactly embedded subdomain. Consider the operator − 12∆ in Ω − Ā with the Dirichlet boundary condition at ∂A and the Neumann boundary condition at ∂Ω, and let λ0(Ω, A) > 0 denote its principal eigenvalue. If instead of the Neumann boundary condition, one imposes the Dirichlet boundary condition at ∂Ω, then it’s easy to see that λ0(Ω, A) is monotone decreasing in Ω and increasing in A. Similarly, in the case at hand, it is clear that λ0(Ω, A) is monotone increasing in A; however, the question of monotonicity in Ω is not easily resolved. The impetus for studying this question arose in part from a recent paper [5] in which one can find the asymptotic behavior of λ0(Ω, A) when A is a ball that shrinks to a point, 1991 Mathematics Subject Classification. 60C05, 60F05.