On the lowest eigenvalue of general operators for the intersection of two domains

E. Lieb (Invent. Math. 74 (1983), 441-448) has proved that if A and B are t wo bounded domains in R-N, then there exists a translation tau(y) such that lambda(1)(A boolean AND tau(y)B) < lambda(1)(A) + lambda(1)(B), where lamb da(1) is the first eigenvalue of the laplacian with Dirichlet boundary cond itions. Here we extend this result to elliptic operators in divergence from L := -div Q(x)del + c(x) with mixed Dirichlet-Neumann boundary conditions on A. If mu(L)(A) is the corresponding eigenvalue, we show that there exist s a translation tau(y) such that mu(L)(A boolean AND tau(y)B) < mu(L)(A) beta lambda(1)(B), if (t)xi Q(x) xi less than or equal to beta\xi\(2) for a ll x is an element of A, xi is an element of R-N. One can further improve t he estimate for non-isotropic operators (where beta can be large) by taking into account rotations of B. in that case, a similar inequality holds if ( t)xi (Q) over bar(x)xi less than or equal to beta\xi\(2), where (Q) over ba r is a "mean value of Q in different directions." (C) 1999 Academic Press.