Vol. 19, No. 2, 1966

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An embedding theorem for function spaces

Colin W. Clark

Vol. 19 (1966), No. 2, 243–251

Let G be an open set in En, and let H0m(G) denote the Sobolev space obtained by completing C0(G) in the norm

∥u ∥ =  {   ∑   |Dαu (x)|2dx}1∕2.
m     G

We show that the embedding maps H0m+1(G) H0m(G) are completely continuous if G is “narrow at infinity” and satisfies an additional regularity condition. This generalizes the classical case of bounded sets G.

As an application, the resolvent operator Rλ, associated with a uniformly strongly elliptic differential operator A with zero boundary conditions is completely continuous in 2(G) provided G satisfies the same conditions. This generalizes a theorem of A. M. Molcanov.

Mathematical Subject Classification
Primary: 46.38
Received: 12 May 1965
Published: 1 November 1966
Colin W. Clark