All ETDs from UAB

Advisory Committee Chair

Marius N Nkashama

Advisory Committee Members

Shangbing Ai

Ryoichi Kawai

Ian Knowles

James Wang

Document Type

Dissertation

Date of Award

2015

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

This dissertation presents some results on the solvability of second order elliptic equations with nonlinear boundary conditions at resonance, in which the nonlinear boundary conditions perturbation is not necessarily required to satisfy Landesman- Lazer conditions or the monotonicity assumption. The nonlinearity may be unbounded. The nonlinearity interacts in some sense with the Steklov spectrum. The proofs are based on a priori estimates for possible solutions to a homotopy on suitable trace spaces and topological degree arguments. We will also study solvability of nonlinear second-order elliptic systems of partial differential equations with nonlinear boundary conditions. We study the generalized Steklov-Robin eigensystem (with possibly matrix weights) in which the spectral parameter is both in the system and on the boundary. We prove the existence of solutions for nonlinear problems when both nonlinearities in the differential systems and on the boundary interact, in some sense, with the generalized spectrum. The proofs are based on variational methods and a priori estimates.

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