Monotone and pseudomonotone operators with applications to variational problems
Master Thesis
2015
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University of Cape Town
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Abstract
This work is primarily concerned with investigating how monotone and pseudomonotone operators between Banach spaces are used to prove the existence of solutions to nonlinear elliptic boundary value problems. A well-known approach to solving nonlinear elliptic boundary value problems is to reformulate them as equations of the form A (u) = f, where A is a monotone or pseudomonotone operator from a Sobolev space to its dual. We seek to study the abstract theory which underpins this approach and proves the existence of a solution to the equation A (u) = f, implying the existence of a weak solution to the elliptic boundary value problem. Further, we examine properties of monotone and pseudomonotone operators, with an emphasis on a characterization, which involves the latter, and establishes a connection between the operator and the principal part of a partial differential equation. In addition, results relating monotone and pseudomonotone operators with variational inequalities are explored.
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Includes bibliographical references
Reference:
Alexander, B. 2015. Monotone and pseudomonotone operators with applications to variational problems. University of Cape Town.