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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 ω, we will show the minimizer of the mth eigenvalue is the m-degree continuous symmetrical decreasing rearrangement of ω. The main result of this paper can be viewed as complementary to Schwarz’s work (Schwarz in J. Math. Mech. 10:401–422, 1961).

Continuous minimizer of eigenvalues for eigenvalue problem with equimeasurable weights