Title

On elliptic boundary value problems with indefinite weights: Variational formulations of the principal eigenvalue and applications

Date of Award

1994

Availability

Article

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

First Committee Member

George Christopher Cosner, Committee Chair

Abstract

A uniformly elliptic model $L : L\lbrack u\rbrack = \nabla \cdot \lbrack -a\ \nabla\ u + bu\rbrack = \lambda mu$, describing the stationary dynamics of a population with density u = u(x) subject to a diffusion matrix $a = (a\sb{ij}(x))$, a drift vector $b = (b\sb{i}(x))$ and a sign indefinite growth rate m = m(x) in a bounded region $\Omega \subset \IR\sp{n}$, is analyzed. Incorporation of the drift as part of the flux $J = - a\ \nabla\ u + bu$, is justified via analytic means and stochastic techniques. Dirichlet, Neumann (No-flux) and Robin conditions are treated throughout. Existence theory a la Hess-Kato, Brown-Lin and Senn-Hess is used to show existence of the principal eigenvalue ($\lambda\sp\*$) of model L. Coercivity conditions a la Cantrell-Cosner for model L are then obtained. Bounds a la Murray-Sperb and Protter-Weinberger, as well as variational formulations for the principal eigenvalue of L are developed, when the drift is conservative or potential-based. Questions related to persistence and extinction of populations subjected to a growth rate based drift ($b = 2\alpha a\ \nabla\ m$) are examined. Analysis is justified by an argument of Lazer. A conjecture of Cosner, concerning the persistence of a population living in the region $\Omega$ having a free boundary, is announced and discussed. Analytic and probabilistic theories treating elliptic-parabolic equations are then presented. Some of these methods are applied to the prototype H: H (G) = $\nabla\cdot$ ($A\ \nabla\ G + BG$) = 0, elliptic in $\Omega$ and degenerate on $\partial\Omega$, ($A > 0$ in $\Omega, A = 0$ on $\partial\Omega$), to prove the existence of a smooth non-trivial solution G. Using this fact, a minimax characterization of the Dirichlet principal eigenvalue ($\lambda\sbsp{D}{*}$) is given for model L when the coefficients a, b and m are in $C\sp{\infty}(\bar\Omega$). Similar variational formulations are obtained for the No-flux (Neumann) ($\lambda\sbsp{N}{*}$) and Robin ($\lambda\sbsp{R}{*}$) principal eigenvalues of model L, with only $C\sp{2+\theta}(\bar\Omega$) regularity on the coefficients. The eigenvalue formulations are shown to yield the Rayleigh-Ritz quotient when the drift is potential based. These formulations coincide with those of Manes-Micheletti when the operator is self-adjoint, those of Donsker-Varadhan when m = 1, and those of Holland when the weight m is positive. The minimax expressions are then shown to be maximins, thus allowing for two sided bounds, depending on the choice of given test functions.

Keywords

Applied Mechanics; Mathematics

Link to Full Text

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