# Il principio di minimo e sue applicazioni alle equazioni by L. Nirenberg (auth.), S. Faedo (eds.)

By L. Nirenberg (auth.), S. Faedo (eds.)

L. Nirenberg: On ellliptic partial differential equations.- S. Agmon: The Lp method of the Dirichlet problems.- C.B. Morrey, Jr.: a number of imperative difficulties within the calculus of adaptations and similar topics.- L. Bers: Uniformizzazione e moduli.

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**Example text**

G)and Hj,hp(Ox). Also, to compute the strong derivatives of u* one applies the usual chain rule. The same remark applies to the trace a t the boundary of derivatives of order sj- 1 when G is of cl&ss COJ Most of the following lemmas are the Lp modified versions of the calculus L, lemmas given in Nirenberg [23]. Unless otherwise stated we shall assume in these lemmas that G is of class COJ. 1. Let u E Lp (G),p 1. #uppose that k is a weak limit in Lp of a sequence of &unctions (N)which belong to H j V L p ( fand f ) possess unijormly bounded norms uk Ilj,Lp(~) Then, u E HjqL;(G) and its derivatives of order < j are the weak Lp limits of the corresponding derivatives of the jicnctions uk.

Ik+, diferertial eyuations We now summarize the resnlts without specifying the exact smoothness oor~ditionsqn the boundary; k will denote a non-negative integer. The integral estimates will be stated only for p = 2 . tes : If uE H2, and sntis$es, for simplicity, homoye~ceous Dirichlet dtrta, i. e. y j = 0 , aiid if L u E Hk, aicd tile coef3oients of L belong and to Ck, then u belongs to Similar reszilts hold jbr equations in integral, or variational, form. Sohauder Estimates :I j y o r some positive a 1 ,uE C 2m+"((i3),LU E 0 k+a(q), yj E C2~$+~+l-j+~, awd the coefjcie~tsof L belong to C ~ + Rthen , u E C2m+Na and < Si9)tilar result^ hold for vqrctctions in varicctional form.

2. Suppose that u belongs to HjILp(G) and that its j' th order derivatives belong to Hk,Lp(O) then u E H j + k , ~(G). ~ NOTATION : Let h = (h, h,,) be a real non-vanishing vector. 3. Let u E Hj,Lp(G) ( j2 0 p > 1). ~Yupposcthat there exists a constant C such that for every subdomain G,; G, c G : the Diriohlst problettb for all suficently small vectors h. Tile% $6 E Hj+l,Lp(G) and Proof: Consider first the case j = 0 . 3) and t,he weak compactness of the unit sphere in I;, it follows that there exists a sequence of vectors (h7m)z-,in the direction of the ~6 axis, hm 0 , such that the sequence dnm u (wt sufficiently large) tends weakly in Lp (GI) to a function ui; and this in every fixed subdomain G, G, c G.