## Dissertations from ProQuest

#### Title

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

1994

Article

#### Degree Name

Doctor of Philosophy (Ph.D.)

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