C^{1, 1} Regularity in semilinear elliptic problems

Abstract

In this paper we give an astonishingly simple proof of C 1,1 regularity in elliptic theory. Our technique yields both new simple proofs of old results as well as new optimal results.The setting we'll consider is the following. Let u be a solution towhere B is the unit ball in R n , f (x, t) is a bounded Lipschitz function in x, and f t is bounded from below. Then we prove that u ∈ C 1,1 (B 1/2 ). Our method is a simple corollary to a recent monotonicity argument due to Caffarelli, Jerison, and Kenig.

Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients

“…This theorem generalizes that of Shahgholian [Sha03] for equations of the type ∆u = f (x, u)χ Ω , |∇u| = 0 on Ω c . See also the work of Uraltseva [Ura01] for a similar result in a two-phase membrane problem.…”

Geometric and energetic criteria for the free boundary regularity in an obstacle-type problem

“…This is a particular case of the main result in Shahgholian [Sha03]. A different proof for f = 1 can be found in [CKS00], which can be also generalized to f ∈ Lip, see [CS04].…”

An overview of unconstrained free boundary problems

“…It was shown in [33] that solutions to this problem are C 1,1 . It is tantalizing to analyse the case of fully nonlinear equations F(D 2 u, u) = 0 with F u ≤ 0.…”