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COLLOQUIUM MATHEM 4TICUhI VOL LXTV FASC 2 1993 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 differentiability 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 i11 It aiso seems tnat the proof presen edin this paper is simpler ar d more natural than the origlnal m e it works ior anv class of eiliptic equations In divergence form for which cne s abie 70 prove the local 5oundedness o i weak solutions provided the dzuferenceqwotzent x z 4 hX i u z c hsatisfies an equation belonging 0 the class in qilestion whenever u x does Finaily x e want i stress he fact that in she case q n Reshe nyak s theorem n fact yieids some nontrivial geometric information aoout W e a K solutions of First of all the Ydicer ex onent r o v i d e aSv 5 As very ciose to zero On the other hand there e x s t continuous aowhere ciiikrenrlable f wictionst t 71 L lfi see the example of Serrin 6 For 7 n he result becomes trxial by a well kaow ltneorem of Caideron 2 all The work or both authors was partiaily supporteu 3 7 a K9N yrant 288 P H A J E A S Z AYD P S T R Z E L E C K I elements of I Q are differentiable a e The case q 1 is rather troublesome 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 f l 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 able functions each in some LS f g are positive measur 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 Sy A x U VU i B xU h j dx o 4 I2 wii c for each y E PV IZ where 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 f h ax The result of Reshesnya reads as foliows TXEORELZ 1 Each weak solutzon of 1 s dzflerentiable almost everywhere wzth respect to the Lebesgue measure zn G Our p r o is f 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 I s s m ethat u E VVL 2 B 2 c f2 solves the equatZon i l l Then where he constanz C l i u l c iu lQ l 312 K depends on Q q u E Ibli jc ijdjl and 11 I he n o r m s o f b g 5ezng take 11 q 1 f 1 13i q the appropmate L S spaces 289 O CrASIL NEAR E Q U A T I O N S 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 T H E O R E3 MLei R be a n open d o m a i n in Rn and u E w R T h e n for iz 0 and for almost all xo E R the following function of X E B 2 u x0 au hX u zoi C l Xi h i l dxi tends t o 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 arbitrary functzon defined on a n open set 9 c J f2 Rn Define R be a n T h e n E is Lebesgue measurable and u is dzfferentiable a e in E P r o o f of T h e o r e m 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 B a X v p h B x o hX zto h v p i for YE 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 a i l iirn suy r i 4 B x r dl 1 290 P H A J E A S Z AXD P S T R Z E L E C K I 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 Ahand Bh saiisfy the growth conditions 2 with the same constant b 3 replaced by b h gh bh X 2q 1lh q lb x0 hX eh X 2 uo lq lb xo 4 hX e xo thX a and ch Y IFLIc xO h Y j dh X 2 q 1 1 h q d l …