By Giovanni P. Galdi (auth.)
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Extra resources for An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Volume I: Linearised Steady Problems
2. 17). The following result due to Lax (1955, §2) ensures that for q E (1, oo) the two spaces W 0 m,q' (0) and (W;''9 (0))' can be identifi ed (see also Mirand a 1978, §57). 5. The spaces W 0 m,q' (0) and (W;'' 9 (0))', 1 < q < oo, are algebraically and isometr ically isomorphic. 7 Throug hout this book the value of a functional F E W m,q' (0) at uE 0 W;'• 9 (0) will be denoted by < F,u > (duality pairing). If, in particular, FE £9' (0), we have < F, u >= (F, u). 3 Boun dary Inequalities and the Trace of Functions of wm,q As a next problem, we wish to investigate if, analogously to what happen s for smooth functions, it is possible to ascribe a value at the bounda ry (the trace) to functions in wm,q(O).
4. Show that a domain having the segment property cannot lie simultaneously on both sides of its boundary. 1. 1t) by function s in cm(n) n < oo, can 30 II. Basic Function Spaces and Related Inequalities wm,q(O). Moreover, i£0 has the segment property, it can be approximated in the same norm by elements of (0). 18). 1). 4). To this end, we propose an elementary inequality due to Nirenberg (1959). 1. 2) Proof. 2) for n = 3, the general case being treated analogously. We have and similar estimates for x 2 and x 3 .
Supp08e, for instance, n:: JRn and consider the sequence Um Show that = exp[-lxl/(m + l)J, llumllq IIVumllq m E JN. 1) for domain or a half-space. 1) plays an important role in several applications. l. 1. l. 3) an (see Sobolev 1963a, Chapter II, §16). An estimate of >. 1 ~ 4/[6(0)) 2 • However, a better estimate can be obtained as a consequence of the following simple argument due to E. Picard (Picone 1946, §160). 1 In fact, assume as before 0 c Ld for some d > 0 and consider the function u(x) oo + d/2)/d], u E Co (0).
An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Volume I: Linearised Steady Problems by Giovanni P. Galdi (auth.)