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By Giovanni P. Galdi (auth.)

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Extra info for An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Volume I: Linearised Steady Problems

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We shall give, for simplicity, the proof in the caBe n = 3. 5} for u E C 1 (0). r• :~(e,x2,x3)£1e ~ hl 3 :u(Yli'7•X3)dq '7 Integrating over they-variables and raising to the qth power, we deduce ju(x1,x2,x3) -ulq ~ ICI-q [a foa IVu(e,x2,x3)lde 3 +a2 foa loa jVu(Yli'7•X3)Idyld'7 +a Ia IVuldC] q. 4. v 53 which completes the proof. 3. 2). 1. Suppose 0 is a cube of side a and subdivide it into N equal cubes Ci, each having sides of length a/N 1fn. -q)/q Xi(x), with Xi characteristic function of the cube Ci, from the previous inequality one has the following result due to Friedrichs {1933).

Proof. Let Xo E an. By assumption there is Br(xo) and a function ( ((x'), x' = (x11 ... a in the integral will be omitted. 3. Boundary Inequalities and the Trace of FUnctions of wm,q for some K, > 0 and, moreover, points X= (x', Xn) Xn while points x E = ((x'), E 41 ann Br(xo) satisfy x' E D, n n Br(xo) satisfy Xn < ((x'), x' E D. We may (and will) take x 0 to be the origin of coordinates. Denote next, by Yo := (0, ... , 0, Yn) the point of n intersection of the Xn-axis with Br(xo) and consider the cone r(yo, a) with vertex at y0 , axis Xn, and semiaperture a < 1r /2.

N: -d/2 < Xn < d/2}. 1. Assume f2 C Ld, for some d > 0. 1) Proof. It is enough to show the theorem for u E Clf(O). 1) for q = oo. If q E [1, oo), employing the Holder inequality yields 50 II. 1). 1. 1} fails, in general, if n is not contained in some layer Ld. 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).

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