Comparison and oscillation theorems for elliptic equations
W. Allegretto · cIRcle (University of British Columbia) · 2011
Thesis Supervisor: C. A. Swanson. New comparison and Sturm-type theorems are established which enable us to extend known oscillation and non-oscillation criteria to: (1) non-self-adjoint operators, (2) quasi-linear operators, (3) fourth order operators of a type not previously-considered. Since the classical principle of Courant does not hold for some of the operators considered, the comparison theorems involve, in part, new estimates on the location of the smallest eigenvalue of the operators in question. A description of the behaviour of the eigenvalue as the domain is perturbed is also given for such operators by the use of Schauder's "a priori" estimates. The Sturm-type theorems are proved by topological arguments and extended to quasi-linear as well as to non-self-adjoint operators. The fourth order operators considered are of a type which does not yield forms identical to those arising in second order problems. Some examples illustrating the theory are given.