Abstract
Contact and friction problems, as well as many other problems involving elliptic partial differential equations and some specific boundary conditions, are formulated and studied appropriately using fractional order Sobolev spaces. This functional setting allows to provide some meaningful reformulations of the original mathematical models (in strong form) and also to study their properties in terms of existence, uniqueness of solutions, as well as their approximability. We start this chapter with some definitions and basic results about functions, distributions, and distributional derivatives. Then, we introduce the fundamental notions about fractional order Sobolev spaces and Lipschitz domains. This allows further to provide some details about trace operators, which play a particularly important role in the next chapters, since they allow to define appropriately the restriction of regular distributions on the boundary of a Lipschitz domain. As well, lifting operators are studied, since they allow, the other way round, to extend to a whole domain a function defined only on a boundary. Moreover, these operators allow to derive Green formulas, which is the basic tool to go from the strong form to the weak form, and conversely. We end the chapter with a presentation of polynomial approximation theory in fractional order Sobolev spaces, from which we will derive later on interpolation estimates for finite element spaces.
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