Sharp Estimates for the Eigenvalues of Some Differential Equations

Continuity in weak topology and extremal problems of eigenvalues of the $p$-Laplacian

“…Roughly speaking, the continuity of eigenvalues in weak topologies can yield the existence of minimizers and maximizers if only the domains are compact in weak topologies, while differentials of eigenvalues can be used to deduce the critical equations for minimal and maximal potentials or weights, as done in [31] for the smallest periodic eigenvalues of linear Hill's equations. Compared with the traditional approach to extremal problems of eigenvalues in [16,18,20,25], the approach here is quite different and is easy to handle. Some further applications to extremal problems of eigenvalues will be undertaken in future works.…”

“…Of particular interest is that we will use some methods which are completely different from those in [16,18,25]. Given γ ∈ [1, ∞] and r ∈ (0, ∞), one can define the positive semi-sphere…”

“…and, for t = 1, χ p (1) := lim In [16], Karaa has studied several extremal problems concerned with eigenvalues or weighted eigenvalues. One step is also the existence of minimizers or maximizers.…”

Upper bounds for the eigenvalues of differential equations

“…Many results concerning lower and upper bounds for eigenvalues have been obtained [2][3][4][5][9][10][11]. For instance in [5], lower bounds for the first eigenvalues of the equations (p(x)y ) + q(x)y + μy = 0 and (p(x)y ) + q(x)y + μy = 0, subject to Dirichlet boundary conditions, are found.…”

“…In [7], it is shown that the first eigenvalue of Euler problem decreases when one replaces the design (coefficient function) involved in the problem by its decreasing rearrangement. In [9], we derived upper bounds for the nth eigenvalue of the equation y + p(x)y + μq(x)y = 0 with Dirichlet boundary conditions, under some conditions on the coefficients p and q. The first goal of this paper is to complete the results obtained in [9], by deriving upper bounds for the eigenvalues of the problem…”