On the Lowest Eigenvalue of General Operators for the Intersection of Two Domains

E. Lieb In¨ent. Math. 74 1983., 441]448. has proved that if A and B are two bounded domains in RN, then there exists a translation ty such that l1 Alty B. -l1 A.ql1 B., where l1 is the first eigenvalue of the laplacian with Dirichlet boundary conditions. Here we extend this result to elliptic operators in divergence from L[ydivQ x.=qc x. with mixed Dirichlet]Neumann boundary condi- tions on A. If mL A. is the corresponding eigenvalue, we show that there exists a translation tysuch that mL Alty B.-mL A.qbl1 B., if jt Q x.j Fb