ABSTRACT

This chapter presents preliminary material from functional analysis that will be used subsequently. The results are stated without proofs, since they are standard and can be found in many references. We start with a review of definitions and properties of several function spaces, including spaces of continuous, continuously differentiable and p-integrable functions, Sobolev spaces, and spaces of vector-valued functions. All of the function spaces used in the book are real. We then recall the Banach fixedpoint theorem and some standard results on variational inequalities and evolution equations that will be applied repeatedly in proving existence and uniqueness results for the contact problems. Finally, we list several Gronwall-type inequalities that will be used repeatedly. We assume that the reader has some familiarity with the the notions of linear spaces, the Lebesgue measure, norms, Banach spaces, inner products, Hilbert spaces, and Sobolev spaces and their basic properties. This material can be found in many books on functional analysis, e.g., [1, 22, 35, 41], or in a concise form that is sufficient for this work in [7]. A comprehensive treatment of functional analysis and its applications is Zeidler [126-130], of which [127] and [128] are of particular relevance to this work. A list of books and surveys on variational inequalities and nonlinear partial differential equations include [12, 13, 21, 42, 70, 72, 108].