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# section10 notes - LECTURE NOTES FOR 18.155 FALL 2004 63 10...

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± ± ± LECTURE NOTES FOR 18.155, FALL 2004 63 10. Sobolev embedding The properties of Sobolev spaces are brieﬂy discussed above. If m is a positive integer then u H m ( R n ) ‘means’ that u has up to m derivatives in L 2 ( R n ) . The question naturally arises as to the sense in which these ‘weak’ derivatives correspond to old-fashioned ‘strong’ derivatives. Of course when m is not an integer it is a little harder to imagine what these ‘fractional derivatives’ are. However the main result is: Theorem 10.1 (Sobolev embedding) . If u H m ( R n ) where m > n/ 2 then u → C 0 0 ( R n ) , i.e., H m (10.1) ( R n ) C 0 0 ( R n ) , m > n/ 2 . Proof. By deFnition, u H m ( R n ) means v → S ( R n ) and π m ˆ u ( π ) L 2 ( R n ). Suppose Frst that u → S ( R n ). The ±ourier inversion formula shows that ix · α ˆ ² (2 ) n | u ( x ) | = ± e u ( π ) 1 / 2 ³± π 2 m | ˆ( π ) | 2 · ´ µ π 2 m 1 / 2 . u R n R n Now, if m > n/ 2 then the second integral is Fnite. Since the Frst integral is the norm on H m ( R n ) we see that (10.2) sup | u ( x ) | = u (2 ) n u H m , m > n/ 2 . L R n This is all for u S ( R n ), but S ( R n ) γ ± H m ( R n ) is dense. The estimate (10.2) shows that if u j ± u in H m ( R n ) , with u j S ( R n ), 0 then u j ± u in C 0 ( R n ) . In fact u = u in S ( R n ) since u j ± u in 0 L 2 ( R n ) and u j ± u in C 0 ( R n ) both imply that R u j α converges, so u j α ± = u α ² α → S ( R n ) .

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section10 notes - LECTURE NOTES FOR 18.155 FALL 2004 63 10...

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