Optimal Regularity for the Convex Envelope and Semiconvex Functions Related to Supersolutions of Fully Nonlinear Elliptic Equations

Pointwise properties of <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms

“…We recall that it is not even known if the ABP maximum principle in the version with the upper contact set is true for L p -viscosity subsolutions of P − (D 2 u) − γ(x)|Du| = f (x) if γ is not bounded. We remark that properties of the concave envelopes for L p -viscosity subsolutions of such equations were recently studied in [3].…”

“…However it appears that these properties have not been widely known and used. Here we extend the pointwise properties and pointwise characterization of L p -viscosity solutions to equations satisfying the more general conditions (2) and (3). We hope that the current manuscript will bring more attention to these pointwise properties as they create a very powerful set of tools.…”

“…This shows that there exists a set of full measure Ω 3 ⊂ Ω such that for every x ∈ Ω 3 and every (r, p, X) ∈ R × R n × S(n),…”

“…where Ω ⊂ R n is a bounded domain, F : Ω × R × R n × S(n) → R, where S(n) is the set of all n × n symmetric matrices and f ∈ L p (Ω). The precise assumptions on F are given below in (2) and (3). We will always assume that p > p 0 , where p 0 = p 0 (n, Λ λ ) ∈ [n/2, n) is the constant giving the range where the maximum principle holds (see e.g.…”

“…The main goal of the paper is to present a complete picture of pointwise properties of L p -viscosity solutions of (1), where F satisfies (2) and (3). These properties include pointwise twice super/sub-differentiability of L p -viscosity sub/super-solutions and the so called "pointwise maximum principle" for L p -viscosity sub/supersolutions.…”