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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



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