Abstract
For one-dimensional 
-Laplacian with weights in 
balls, we are interested in the extremal values of the 
th positive half-eigenvalues associated with Dirichlet, Neumann, and generalized periodic
boundary conditions, respectively. It will be shown that the extremal value problems
for half-eigenvalues are equivalent to those for eigenvalues, and all these extremal
values are given by some best Sobolev constants.
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