C 1, 1 Regularity in semilinear elliptic problems

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 optical results. The setting we'll consider is the following. Let u be a solution to$$\Delta u = f(x, u)\,\,\,\, \rm{in} \,\it{B},$$where 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. © 2002 Wiley Periodicals, Inc.