# Eigenvalue dependence on problem parameters for stieltjes sturm-liouville problems

Abstract (Summary)

This work examines generalized Stieltjes Sturm-Liouville boundary value problems
with particular consideration of self-adjoint problems. Of central importance
is determining conditions under which the eigenvalues depend continuously and
differentiably on the problem data. These results can be applied to various physical
problems, such as constructing beams to maximize the fundamental frequency of
vibration, or constructing columns to maximize the height without buckling. These
problems involve maximizing the smallest eigenvalues of Sturm-Liouville equations,
and the continuous dependence of the eigenvalues on the problem parameters
can be used to accomplish this.
We first consider the generalized 2n-dimensional initial value problem dy =
Aydt + dP z, dz = (dQ ? ?dW )y + Dzdt on an interval [a, b]. In the proof of
existence and uniqueness of a quasi-continuous solution, we establish some bounds
and continuity properties of the solution that will be used throughout this work.
Next we define a sequence of initial value problems and prove that the sequence of
solutions converges to the solution of the limit problem.
We then consider the eigenvalue problem, adding general boundary conditions
to the system of equations. The eigenvalues are shown to be the roots of an entire
function. Taking a sequence of eigenvalue problems, we show that a sequence of
eigenvalues converges. This result establishes conditions under which each eigenvalue
depends continuously on the coefficients and on the boundary data. We find
separate conditions for the continuous dependence on the endpoints of the interval.
We next turn to ascertaining conditions under which each eigenvalue depends
differentiably on the problem data. For this topic, we consider the less general 2-
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dimensional Stieltjes Sturm-Liouville problem dy = dP z, dz = (dQ ? ?dW )y
with separated boundary conditions. Considering each eigenvalue as a function of
the coefficients and of the boundary data, we conclude that these functions are differentiable
under the same conditions we found for continuity. Separate conditions
are found to guarantee the differentiability of each eigenvalue with respect to the
endpoints. In all cases, we find expressions for the derivatives of the eigenvalues
with respect to the problem parameters.
We conclude with an application to the problem of finding extremal values of
an eigenvalue. For the fourth order problem (ry??)?? + (py?)? + qy = ?wy with
boundary conditions y(a) = y?(a) = y(b) = y?(b) = 0, we consider the smallest
eigenvalue ?0 as a function of the coefficients. The continuous dependence of the
eigenvalue on the coefficients is used to find a sequence of coefficients converging
to a function that attains the supremum or infimum of ?0 over a certain class of
coefficient functions.
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Bibliographical Information:

Advisor:

School:The University of Tennessee at Chattanooga

School Location:USA - Tennessee

Source Type:Master's Thesis

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