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COLLOQUIUM MATHEMATICUM

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COLLOQUIUM MATHEM.4TICUhI VOL. LXTV 1993 FASC. 2 ON THE DIFFERENTIL4BILITY OF SOL UTIONS OF & L-ASILINE,1R ELLIPTIC EQUATIONS 1. Introduction. In this note we consider weak solutioris (in the Sobolev .i,9( space Wi,, ,,(2), 1 < q 5 n) of quasilinear elliptic equations of the type where A and B satisfy certain grcwth conditions given in Section 2. In [3] E'u. G. Reshetn:,-ak proved that weak solutions of (1) are totally differentlable almost everywhere. For a lillear equation div(a(z)Gu) = O this was proved independently by B. Rojarski [I]. We will show how to simplify Reshetnyak's proof by adopting the method of Bojarski. ,2s was shown by Reshetnvak. the theorem on almost everywhere differ- entiability IS a -;imple consequence of a di%cu!t theorem of Serrin 15; which asserts the Holder continuity of weak solutions of (1'1. We shall use instead a weaker (and milch easier to prove) result on local boundedness of weak soiutions. Then the final argument, is provided by the classical Stepanov differentiability criterion (Theorem 4 below). This shows :hat the a.e. differentiability is. in some sense; independent cf Hoider continuity of sveak soiutions of i 11. It aiso seems tnat the proof presen~ed in this paper is simpler ar^d more natural than the origlnal me: it works ior anv class of eiliptic equations In divergence form for which cne :s abie 70 prove the local 5oundedness oi weak solutions. provided the dzuference qwotzent ( x( z,, 4- hX'i - u(zc!),'h satisfies an equation (belonging $0 the class in qilestion; whenever u(x' does. Finaily. xe want :i, stress :he fact that. in she case ! < q < n. Reshe+,- nyak's theorem :n fact yieids some nontrivial geometric information aoout WeaK solutions of ;;,. First of all. the Ydicer ex?onent ~rovidea Sv -5; As very ciose to zero. On the other hand. there exst continuous, aowhere ciiikr- enrlable f~wictions tt .; %71,'L?lfi' (see the example of Serrin 6!'. (For 7 > n he result becomes trxial: by a well-kaow~l tneorem of Caideron .2,, all The work or' both authors was partiaily supporteu '3:7 a K9N yrant.288 P. HAJEASZ AYD P. STRZE,LECKI elements of I+;~:(Q) are differentiable a.e. The case q = 1 is rather trou- blesome, mainly due to the fact that the spaces L1(Q) and $V1.l(fl) are not reflexive; equation (1) can then admit quite irregular solutions and the results of Serrin are, in general, not valid.) 2. Assumptions and the result. A and B are respectively Rn-and R-valued functions of (x, u,p) E fl x R x Rn. Moreover, we assume that A(x, u(x), p(x)) and B (x, U! x), P(L)) are measurable for any measurable functions u(x) and p(z], and where a is some positive constant, while b, c, d, e, f, g are positive measur- able functions each in some LS: for some E E (0, min{l, q - 1)). A function u E VC;?~(Q) is called a weak solution of (11 if and only if (4) ('Sy, A(x, U, VU) -i- ~B(x, U, hj) dx = o I2 for each y E PV~;;~(IZ). where wii,c(~) denotes the closure of G'r (a) in w~:;"(R). In the sequel B(x, r) will denote the Euclidean ball with center x and radius r; we write B(r) if x = 0. By fA f(x) dx we denote the ayeraged integral ,ill -' $, fh) ax. The result of Reshesnya~ reads as foliows. TXEORELZ 1. Each weak solutzon of (1) ?,s dzflerentiable almost every- where wzth respect to the Lebesgue measure zn G. Our pr~of is very close to the original one. We shall need three theorems. The 5rst one is taken from Serrin [5, Theorems 1 and 31. TIIEOREII 2. .Iss?~me that u E VVL:,"(2). B(2') c f2. solves the equatZon ill. Then, liu~l~,~(l, c(iu~lQ 312) + K) where :he constanz C depends on Q, q, u. E. Ibli, ,jc;/, ijdjl and = - ;/f 11)';'q-1) 13i/'/q, (11 I $he norms of b, . . , g 5ezng take?, 1% the appropmate LS spaces.O CrASIL;NEAR EQUATIONS 289 The next theorem is a slightly weaker version of the LP-differentiability theorem of Calderjn and Zygmund 17, Chapter VIII, Theorem 11 (see also i41). THEOREM 3. Lei R be an open domain in Rn and u E w;,"(R). Then. for iz --+ 0, and for almost all xo E R, the following function of X E B(2): u(x0 + hX) - u(zoi au - C -(l.,)Xi h i= l dxi tends to zero in L"B[2) j. The following theorem is due to Stepanov 181 (we recall the statement from 17, Chapter VIII, Theorem 31). THEOREM 4 (Stepanov differentiability criterion). Let ?J : f2 - R be an arbitrary functzon defined on an open set 9 c Rn. Define Then E is Lebesgue measurable and u is dzfferentiable a.e. in E Proof of Theorem 1. Let u be a weak solution of (1). Define the difference quotients whern, uo = u(xo). For iz < tdist(x9. 3R) this is a well defined function of X E B(2), readily of class W11q(B(3)). Using the change of variables z = + hX and the definition of weak soiutions of (1) one easily proves that vh (X) solves the equation where Ah (-Ti. V. p) = A/zo t ,+XI ~0 h~. . Ba(X,v,p) = hB(xo hX,zto -?- hv,p'i, for -Y E 3(2), u E a. p E En. Thesrem 3 irnp1:es that Yotice that by changing & (311 a set of measure zero we can actuaily use supemurn instead of essential supremum in (5). Namely. it is enough r;a i>ut ail) := iirn suy 4 ~(~1 dl/ r-i) B(x,r)290 P. HAJEASZ AXD P. STRZELECKI We shall show that for almost all xo E R the right hand side of (5) remains bounded when h tends to zero. This will allow us to apply the Stepsnov differentiability criterion and finish the proof. S t e? I. Using the properties of A and B one can easily check that Ah and Bh saiisfy the growth conditions (2) with the same constant a and b.. . . .3 replaced by bh, . . . , gh: bh(X) = 2q-1 lh/q-lb( x0 - hX) . eh (X) = 2"' :uo lq-lb(xo 4 hX) + e(xo t hX) . ch(-Y) = IFLIc(xO + h,Y j . dh(X) = 2q-11h/qdl~o + hX) . fiLiXj = ihl(2q-11~oiq-1d(zo + hX) + f (xo -r hX)) , gh(Xj -- 2q-1i~G'qd(~q i hX) + gjxg + hay). Xow, choose xa to be an s-Lebesgue point of all the functions b, c, . . . , g (for each of them take s according to (3)). Then the Lebesgue differentiatior, theorem implies that the norms of Sh, ch. . . . , gh in the respective LS (B(2)) are bounded f~r h tending to zero. Fgr instance, if s = n/(q - F), then and obviously the remaining cases ca~ be treated in the same way. Hence, Ci, and ATh on the right hand side of (5) are bounded (by a constant independent cf hjr when 1 h/ is sufF,ciently smsil. S Y ep 2. Theorem 3 readily implies thac for h - 0. tends to zero in LQ(B(2;). hence the L"-norrn of u,k is bounded for …


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