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Interior a priori estimates for solutions of fully nonlinear equations. (English) Zbl 0692.35017
The author presents interior \(W_{2,p}\); \(C_{1,\alpha}\) resp. \(C_{2,\alpha}\)-regularity results for bounded viscosity solutions of fully nonlinear second order uniformly elliptic equations \(F(D^ 2u,x)=g(x),\) provided interior estimates of corresponding type for solutions of \(F(D^ 2w,0)=0\) are valid and some smallness condition with regard to the dependence on x holds. For example it is shown that u is of class \(C_{1,\alpha}\) at the origin, if all solutions to \(F(D^ 2w,0)=0\) satisfy \[ \| w\|_{C_{1,\alpha +\epsilon}(B_ r)}\leq Cr^{-(1+\alpha +\epsilon)}\sup_{B_{2r}}| w| \quad and\quad \oint_{B_ r}| g|^ n dx\leq Cr^{(\alpha -1)n}, \] \(\oint_{B_ r}\beta^ n dx\) small, with \(\beta (x)=\sup_{M\in S}| F(M,x)-F(M,0)| \| M\|^{-1}\), S the class of all symmetric matrices. For local and global \(C_{1,\alpha}\)-estimates for viscosity solutions of \(F(D^ 2u,Du,u,x)=0\), see also the earlier paper of N. S. Trudinger [Proc. R. Soc. Edinb., Sect. A 108, No. 1/2, 57–65 (1988; Zbl 0653.35026)].
Reviewer: M.Wiegner

MSC:
35B45 A priori estimates in context of PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)
35J60 Nonlinear elliptic equations
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