4 Prank Jone s that D i s a "reasonable " bounde d ope n set i n Rn wit h closur e D, an d suppose g is a smooth real-valued function define d onD x Wl+l. The n for an y C 1 functio n D R w e can for m th e integra l f(u)= / g{x,u,du/dx\,. .. ,du/dx n )dx. JD Usually it will be the case that u has to satisfy som e restrictions on the boundary dD o f D. Thes e restrictions are called boundary conditions. A common exampl e i s the so-calle d Dirichlet condition , i n whic h th e restriction o f u t o dD i s a given functio n define d o n dD. We ar e the n goin g to tr y t o investigat e possibl e critica l "points " for/. Since u mus t satisf y th e boundar y conditions , w e ar e somewha t limited i n permitted variation s u + tp of u. Thes e variations wil l cer - tainly b e allowed i f the functions p w e use are infinitely differentiat e on D an d ar e zero near dD. W e call such functions test functions an d we writ e v e C?{D). For each such test functio n w e can form th e directional derivativ e dt d_ dt / g(x,u + tip,u Xl +tpXl,...)dx\ JD = I ^ # 0 , u + tip, uXl + tpXl,... )dx | t = 0 chain rul e f (dg dg \ Next w e integrate b y parts t o ge t ri d o f all the term s dp/dx tl notin g that n o integratio n ove r dD i s required, thank s t o th e fac t tha t p i s zero near dD. Th e resul t i s n (3) D v f(u) = / {dg/du -J]d(dg/du Xz )/dx^dx. In cas e u i s a critica l poin t fo r / , the n b y definitio n w e conclud e that (4) [{-}vdx = 0 JD
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