# zbMATH — the first resource for mathematics

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
Full Text: