Browsing by Author "Jussinmäki, Akseli"

This thesis starts with the definition of the Sobolev space W^{k,p}. This space is a function space consisting of functions whose weak derivatives of order at most k belong to the L^p space. These spaces are complete, which is important in applications and also in the theory of the Sobolev spaces themselves. We review the basic properties of the Sobolev spaces and prove the existence of the trace operator, which assigns boundary values to Sobolev functions. The main result of the thesis is the boundedness and surjectivity of the trace operator from the space W^{1,p}(Ω) to the boundary space W^{1− 1/p}(∂Ω) in the case p > 1. We give the definition and basic properties of the fractional Sobolev spaces that act as a codomain of the trace operator. We then show the surjectivity in two parts, proving first the boundedness of the trace operator in Chapter 3 and then the surjectivity in Chapter 4 with an original proof. We also show the surjectivity of the trace operator from W^{1,1}(Ω) to L^1(∂Ω). Chapters 3 and 4 also contain additional material. In Chapter 3, we give the definition of the fractional Sobolev spaces W^{s,p} for s > 1, and introduce an alternative way to define the Sobolev spaces W^{s,2} = H^s that are also Hilbert spaces. We show the equivalence of the norms of these spaces for 0 < s < 1. In Chapter 4, we emphasize the importance of the assumptions about the boundary in the theorems concerning traces by examining a case that would otherwise violate one of the theorems. In Chapter 5, we examine the Sobolev classes of homeomorphisms on the unit circle and sphere. We show that a homeomorphism ϕ: S^1 → S^1 belongs to the space W^{1− 1/p}(S^1), but a homeomorphism from S^2 to S^2 does not necessarily belong to space W^{1-1/p}(S^2). These results have implications for the properties of the harmonic extensions of the homeomorphisms.