LAUSR

In this paper we prove optimal regularity for the convex envelope of supersolutions to general fully nonlinear elliptic equations with unbounded coefficients. More precisely, we deal with coefficients and right hand sides (RHS) in Lq with $${q \geq n}$$q≥n. This extends the result of Caffarelli on the $${C_{loc}^{1,1}}$$Cloc1,1 regularity of the convex envelope of supersolutions of fully nonlinear elliptic equations with bounded RHS. Moreover, we also provide a regularity result with estimates for $${\omega}$$ω-semiconvex functions that are supersolutions to the same type of equations with unbounded RHS (i.e, RHS in $${L^{q}, q \geq n}$$Lq,q≥n). By a completely different method, our results here extend the recent regularity results obtained by Braga et al. (Adv Math 334:184–242, 2018) for $${q > n}$$q>n, as far as fully nonlinear PDEs are concerned. These results include, in particular, the apriori estimate obtained by Caffarelli et al. (Commun Pure Appl Math 38(2):209–252, 1985) on the modulus of continuity of the gradient of $${\omega}$$ω-semiconvex supersolutions (for linear equations and bounded RHS) that have a Hölder modulus of semiconvexity.