In this paper, we are concerned with the existence and differentiability properties of the solutions of "quasi-linear" elliptic partial differential equations in two variables, i.e., equations of the form A(x, y, z, p, q)r + 2B(x, y, z, p, q)s + C(x, y, z, p, q)t = D(x, y, z, p, q) These equations are special cases of the general elliptic equation t - 0, 0r > 0. * J. Schauder, Über lineare elliptische Differentialgleichungen zweiter Ordnung, Mathematische Zeitschrift, vol. 38 (1933-34), pp.

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... 282. t E. Hopf, Zum analytischen Charakter der Lösungen regulärer zweidimensionaler Variations problème, Mathematische Zeitschrift, vol. 30, pp. 404-^113. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Then a necessary and sufficient condition that fix, y) be A.C.T. on R is that fix, y) be continuous and <p(7?) and \piD) be absolutely continuous on each subregion A for which Ac/?. When this is true, <f,iD) = \ \ -dxdy, tiD) = f f -dxdy J J d dx J J D dy for each rectangle in R. Definition 5. We say that a function <p(x, y) is of class Lv on a region R if | <p | r is summable over R.