Continuous minimizer of eigenvalues for eigenvalue problem with equimeasurable weights.

The problem in this paper is motivated by physical problems concerned with the case when a class of continuous and equimeasurable densities of a string is given then how to find minimal frequencies among these given densities, that is, what kind of densities minimize the frequencies. By taking Dirichlet eigenvalues into account, given a certain weight function <italic>ω</italic>, we will show the minimizer of the <italic>m</italic>th eigenvalue is the <italic>m</italic>-degree continuous symmetrical decreasing rearrangement of <italic>ω</italic>. The main result of this paper can be viewed as complementary to Schwarz’s work (Schwarz in J. Math. Mech. 10:401-422, <xref>1961</xref>). [ABSTRACT FROM AUTHOR]