On the Lowest Eigenvalue of General Operators for the Intersection of Two Domains
- Lachand-Robert, T.
DOIAbstract
AbstractE. Lieb (Invent. Math.74(1983), 441–448) has proved that ifAandBare two bounded domains in RN, then there exists a translation τysuch that λ1(A∩τyB)<λ1(A)+λ1(B), where λ1is the first eigenvalue of the laplacian with Dirichlet boundary conditions. Here we extend this result to elliptic operators in divergence fromL≔−divQ(x)∇+c(x) with mixed Dirichlet–Neumann boundary conditions onA. If μL(A) is the corresponding eigenvalue, we show that there exists a translation τysuch that μL(A∩τyB)<μL(A)+βλ1(B), iftξQ(x)ξ≤β|ξ|2for allx∈A, ξ∈RN. One can further improve the estimate for non-isotropic operators (where β can be large) by taking into account rotations ofB. In that case, a similar inequality holds iftξQ̄(x)ξ≤β|ξ|2, whereQ̄is a “mean val...
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