6 WEAK CONVERGENCE Â. Convergence of Averages Before proceeding further it may be useful to pause and develop some further insight into the meaning of weak convergence. So assume now 1 q oo and (1.2) fk-f i n L\U). Then if Å c U is a bounded measurable set, we deduce upon setting g = ÷ in (1.1) that (1.3) /"/***- f fax. JE JE This implies that the averages of the functions {fk}T_ over the set Å con- verge to the average of / over Å. (Conversely, it is straightforward to check oo Q that if {fk} is bounded in L (U) and verifies (1.3) for each bounded, measurable set Å c U, then (1.2) holds.) Á problem we will continually confront in PDE applications is that this convergence of averages does not imply norm or even a.e. convergence. It may very well be that the sequence {fk} °°_ effects its weak convergence to / by virtue of perhaps unbounded, very high frequency and quite irregulär oscillations. Such behavior utterly ex- cludes any simple analysis of nonlinear functionals of the sequence {fk } °°_ . In particular we note that (1.2) does not imply F(fk) » F(f) for any nonlinear, real-valued function F. To see this, select real numbers á b and 0 ë 1 so that F{Xa + (1 - X)b) ö ÄF{a) + (1 - X)F{b). Assuming U = (0, 1) c R, we set /*(*) - { Then fk -^ / = ëá + (1 - X)b in L , whereas F(fk) -1 F = XF(a) + (1 - X)F(b) ø F(f). C. Compactness in Sobolev Spaces 1. Embeddings. For later reference we record here the Gagliardo-Nirenberg- Sobolev inequality, which asserts that if 1 q ç and q* = qn/(n - q) is the Sobolev conjugate of q, then (1-4) ||/H L , V ) Cq\\Df\\Lq^X) for all functions f e Cc (R ), the optimal constant Cq depending only on q and ç . Invoking usual approximations, we see that this estimate is also valid provided f £ L , Df e L . á i f £ x ^ , V = 0 , . . . , / c - l b otherwise.
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