Introduction and Basic Results
The first chapter is an introduction. After a brief historical overview, we introduce the fundamental notions to be used later and the first simple existence results.
In particular we present a review of differential calculus for functionals, with many examples, and we introduce the fundamental notion of weak solution that allows one to interpret solutions of elliptic problems as critical points of functionals.
Convex functionals and their main properties are then described, providing the first examples of existence theorems.
Finally we recall the main results from the linear theory, particularly some spectral properties of elliptic operators that will be used throughout the book.
KeywordsBanach Space Weak Solution Classical Solution Elliptic Operator Differential Calculus
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