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Some differentiability properties of solutions of elliptic equations with measurable coefficients. (English. Russian original) Zbl 0614.35023
Math. USSR, Izv. 27, 601-606 (1986); translation from Izv. Akad.Nauk SSSR, Ser. Mat. 49, No. 6, 1329-1335 (1985).
Regular elliptic equations of second order with measurable coefficients are discussed from the viewpoint of the existence of derivatives of their solutions in a given form. Non-divergent equations (a) and divergent equations (b) are considered separately. In case (a) it is proved that the solution u of the above type of elliptic equations whose operator is of the form \[ P=\sum^{n}_{i,j=1}a_{ij}(x)\partial^ 2/\partial x_ i\partial x_ j+\sum^{n}_{i=1}b_ i(x)\partial /\partial x_ i, \] defined in the unit sphere \(Q^ 1\) has a derivative of order 2. Under certain conditions the form of this derivative is predicted. A convex function v in \(Q^ 1\) also has the derivative of order 2. In case (b) the solution u has an ordinary derivative for \(x_ 0\in Q^ 1\).
Reviewer: V.Burjan

MSC:
35J15 Second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
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