Calculus of Variations and Its Applications by Graves L. (ed.)

By Graves L. (ed.)

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K~e X l 'X 2 ' ... in when v=t=,u. llxv

By the Borel-Lebesgue lemma a finite number of such neighborhoods can be chosen which cover all of supp 4>. 1 we now choose I/IjEC(f(X) with O~I/Ij~ 1 and 1/1 j = 1 in K j. Then the functions 4>1 =4>1/1 1 ,4>2=4>1/12(1-1/11)' ···,4>k=4>l/Ik(l-l/Il)···(l-l/Ik-l) have the required properties since k "'i4>j-4>= -4> 1 n(1-1/1)=0 k 1 because either 4> or some 1 -1/1j is zero at any point. 5. Let Xl' ... ' X k be open sets in 1Rnand K a compact subset of k k UXj. ;'O and 1 L

2 and it is shown there that distributions may be defined locally provided that the local definitions are compatible. In addition it is proved that if u is a distribution then there is a unique way to define u(cp) for all CPEC oo with supp un supp cp compact. 3. 6). 6 on. 1. 1) where faE C(X) and the sum is finite. Here X is an open set in ]R". 1). 1. A distribution u in X is a linear form on Cg'(X) such that for every compact set K c X there exist constants C and k such that 34 II. 2) lu(

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